Abstract
We construct the compatible system of l-adic representations associated to a regular algebraic cuspidal automorphic representation of \(GL_n\) over a CM (or totally real) field and check local–global compatibility for the l-adic representation away from l and a finite number of rational primes above which the CM field or the automorphic representation ramifies. The main innovation is that we impose no self-duality hypothesis on the automorphic representation.
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1 Introduction
Our main theorem is as follows (see Corollary 7.14).
Theorem A
Let p denote a rational prime and let \(\imath : \overline{{{\mathbb {Q}}}}_p \mathop {\rightarrow }\limits ^{\sim }{\mathbb {C}}\). Suppose that E is a CM (or totally real) field and that \(\pi \) is a cuspidal automorphic representation of \(GL_n({\mathbb {A}}_E)\) such that \(\pi _{\infty }\) has the same infinitesimal character as an irreducible algebraic representation \(\rho _\pi \) of \({{\text {RS}}}^E_{\mathbb {Q}}GL_n\). Then there is a unique continuous semi-simple representation
such that, if \(q \ne p\) is a rational prime above which \(\pi \) is unramified and if v|q is a prime of E, then \(r_{p,\imath }(\pi )\) is unramified at v and
Here \({\text {rec}}_{E_v}\) denotes the local Langlands correspondence for \(E_v\). It may be possible to extend the local–global compatibility to other primes v. Ila Varma is considering this question.
The key point is that we make no self-duality assumption on \(\pi \). In the presence of such a self-duality assumption (‘polarizability’, see [9]) the existence of \(r_{p,\imath }(\pi )\) has been known for some years (see [18, 51]). In almost all polarizable cases \(r_{p,\imath }(\pi )\) is realized in the cohomology of a Shimura variety, and in all polarizable cases \(r_{p,\imath }(\pi )^{\otimes 2}\) is realized in the cohomology of a Shimura variety (see [16]). In contrast, according to unpublished computations of one of us (M.H.) and of Laurent Clozel, in the non-polarizable case the representation \(r_{p,\imath }(\pi )\) will never occur in the Betti or etale cohomology of a Shimura variety. Rather we construct it as a p-adic limit of representations which do occur in such cohomology groups.
We sketch our argument. We may easily reduce to the case of an imaginary CM field F which contains an imaginary quadratic field in which p splits. For all sufficiently large integers N, we construct a 2n-dimensional representation \(R_p(\imath ^{-1}(\pi ||\det ||^N)^\infty )\) such that for good primes v we have
as a p-adic limit of (presumably irreducible) p-adic representations associated to polarizable, regular algebraic cuspidal automorphic representations of \(GL_{2n}({\mathbb {A}}_F)\). It is then elementary algebra to reconstruct \(r_{p,\imath }(\pi )\).
We work on the quasi-split unitary similitude group \(G_n\) associated to \(F^{2n}\). Note that \(G_n\) has a maximal parabolic subgroup \(P_{n,(n)}^+\) with Levi component
(We will give all these groups integral structures.) We set
Then our strategy is to realize \(\varPi (N)\), for sufficiently large N, in a space of overconvergent p-adic cusp forms for \(G_n\) of finite slope. Once we have done this, we can use an argument of Katz (see [35]) to find congruences modulo arbitrarily high powers of p to classical (holomorphic) cusp forms on \(G_n\) (of other weights). (Alternatively it is presumably possible to construct an eigenvariety in this setting, but we have not carried this out.) One can attach Galois representations to these classical cusp forms by using the trace formula to lift them to polarizable, regular algebraic, discrete automorphic representations of \(GL_{2n}({\mathbb {A}}_F)\) (see, e.g., [52]) and then applying the results of Chenevier and Harris [18], Shin [51].
We learnt the idea that one might try to realize \(\varPi (N)\) in a space of overconvergent p-adic cusp forms for \(G_n\) (of finite slope) from Chris Skinner. The key problem was how to achieve such a realization. To sketch our approach we must first establish some more notation.
To a neat open compact subgroup U of \(G_n\) we can associate a Shimura variety \(X_{n,U}/{{\text {Spec}}\,}{\mathbb {Q}}\). It is a moduli space for abelian \(n[F: {\mathbb {Q}}]\)-folds with an isogeny action of F and certain additional structures. It is not proper. It has a canonical normal compactification \(X_{n,U}^{{\text {min}}}\) and, to certain auxiliary data \(\varDelta \), one can attach a smooth compactification \(X_{n,U,\varDelta }\) which naturally lies over \(X_{n,U}^{{\text {min}}}\) and whose boundary is a simple normal crossings divisor. To a representation \(\rho \) of \(L_{n,(n)}\) (over \({\mathbb {Q}}\)) one can attach a locally free sheaf \({\mathcal {E}}_{U,\rho }/X_{n,U}\) together with a canonical (locally free) extension \({\mathcal {E}}_{U,\varDelta ,\rho }\) to \(X_{n,U,\varDelta }\), whose global sections are holomorphic automorphic forms on \(G_n\) ‘of weight \(\rho \) and level U’. (The space of global sections does not depend on \(\varDelta \).) The product of \({\mathcal {E}}_{U,\varDelta ,\rho }\) with the ideal sheaf of the boundary of \(X_{n,U,\varDelta }\), which we denote \({\mathcal {E}}_{U,\varDelta ,\rho }^{{\text {sub}}}\), is again locally free, and its global sections are holomorphic cusp forms on \(G_n\) ‘of weight \(\rho \) and level U’ (and again the space of global sections does not depend on \(\varDelta \)).
To the schemes \(X_{n,U}\), \(X_{n,U}^{{\text {min}}}\) and \(X_{n,U,\varDelta }\) one can naturally attach dagger spaces \(X_{n,U}^\dag \), \(X_{n,U}^{{{\text {min}}},\dag }\) and \(X_{n,U,\varDelta }^\dag \) in the sense of Grosse-Klönne [27]. These are like rigid analytic spaces except that one consistently works with overconvergent sections. If U is the product of a neat open compact subgroup of \(G_n({\mathbb {A}}^{\infty ,p})\) and a suitable open compact subgroup of \(G_n({\mathbb {Q}}_p)\), then one can define admissible open subdagger spaces (‘the ordinary loci’)
and
and
By an overconvergent cusp form of weight \(\rho \) and level U one means a section of \({\mathcal {E}}_{U,\rho }^{{\text {sub}}}\) over \(X_{n,U,\varDelta }^{{{\text {ord}}},\dag }\). (Again this definition does not depend on the choice of \(\varDelta \).)
We write \(G_n^{(m)}\) for the semi-direct product of \(G_n\) with the additive group with \({\mathbb {Q}}\)-points \({{\text {Hom}}}_F(F^m,F^{2n})\), and \(P_{n,(n)}^{(m),+}\) for the pre-image of \(P_{n,(n)}^+\) in \(G_n^{(m)}\). We also write \(L_{n,(n)}^{(m)}\) for the semi-direct product of \(L_{n,(n)}\) with the additive group with \({\mathbb {Q}}\)-points \({{\text {Hom}}}_F(F^m,F^n)\), which is naturally a quotient of \(P_{n,(n)}^{(m),+}\). (Again we will give these groups integral structures.) To a neat open compact subgroup \(U \subset G_n^{(m)}({\mathbb {A}}^\infty )\) with projection \(U'\) in \(G_n({\mathbb {A}}^\infty )\) one can attach a (relatively smooth, projective) Kuga–Sato variety \(A_{n,U}^{(m)}/X_{n,U'}\). For a cofinal set of U it is an abelian scheme isogenous to the m-fold self-product of the universal abelian scheme over \(X_{n,U'}\). To certain auxiliary data \(\varSigma \) one can attach a smooth compactification \(A_{n,U,\varSigma }^{(m)}\) of \(A_{n,U}^{(m)}\) whose boundary is a simple normal crossings divisor; which lies over \(X_{n,U}^{{\text {min}}}\); and which, for suitable \(\varSigma \) depending on \(\varDelta \), lies over \(X_{n,U',\varDelta }\). Thus
We define \(A_{n,U}^{(m),{{\text {ord}}},\dag }\) and \(A_{n,U,\varSigma }^{(m),{{\text {ord}}},\dag }\) to be the pre-image of \(X_{n,U'}^{{{\text {ord}}},{{\text {min}}},\dag }\) in the dagger spaces associated to \(A_{n,U}^{(m)}\) and \(A_{n,U,\varSigma }^{(m)}\).
We will define
to be the hypercohomology of the complex on \(A^{(m),{{\text {ord}}},\dag }_{n,U,\varSigma }\) which is the tensor product of the de Rham complex with log poles towards the boundary, \(A^{(m),{{\text {ord}}},\dag }_{n,U,\varSigma }-A^{(m),{{\text {ord}}},\dag }_{n,U}\), and the ideal sheaf defining the boundary. We believe it is a sort of rigid cohomology of the ordinary locus \(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U}\) in the special fibre of an integral model of \(A_{n,U}^{(m)}\); more specifically, cohomology with compact support towards the toroidal boundary, but not towards the non-ordinary locus, hence our notation. However we have not bothered to verify that this group only depends on ordinary locus in the special fibre. The theory of Shimura varieties provides us with sufficiently canonical lifts that this will not matter to us. Our proof that for N sufficiently large \(\varPi (N)\) occurs in the space of overconvergent p-adic cusp forms for \(G_n\) proceeds by evaluating \(H^i_{c-\partial }(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U}, \overline{{{\mathbb {Q}}}}_p)\) in two ways.
Firstly we use the usual Hodge spectral sequence. The higher direct images from \(A_{n,U,\varSigma }^{(m)}\) to \(X_{n,U',\varDelta }\) of the tensor product of the ideal sheaf of the boundary and the sheaf of differentials of any degree with log poles along the boundary, are canonically filtered with graded pieces sheaves of the form \({\mathcal {E}}_{U',\varDelta ,\rho }^{{\text {sub}}}\). Thus \(H^i_{c-\partial }(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U}, \overline{{{\mathbb {Q}}}}_p)\) can be computed in terms of the groups
A crucial observation for us is that for \(j>0\) this group vanishes (see Theorem 5.4 and Proposition 6.12). This observation seems to have been made independently, at about the same time, by Andreatta, Iovita and Pilloni (see [1, 2]). It seems quite surprising to us. It is false if one replaces \({\mathcal {E}}_{U,\varDelta ,\rho }^{{\text {sub}}}\) with \({\mathcal {E}}_{U,\varDelta ,\rho }^{{\text {can}}}\). Its proof depends on a number of apparently unrelated facts, including:
-
\(X_{n,U}^{{{\text {ord}}},{{\text {min}}},\dag }\) is affinoid.
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The stabilizer in \(GL_n({\mathcal {O}}_F)\) of a positive definite hermitian \(n \times n\) matrix over F is finite.
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Certain line bundles on self-products A of the universal abelian scheme over \(X_{n',U'}\) (for \(n'<n\)) are relatively ample for \(A/X_{n',U'}\).
This observation implies that \(H_{c-\partial }^i(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U}, \overline{{{\mathbb {Q}}}}_p)\) can be computed by a complex whose terms are spaces of overconvergent cusp forms. Hence it suffices for us to show that, for N sufficiently large, \(\varPi (N)\) occurs in
for some m and i (depending on N).
To achieve this we use a second spectral sequence which computes the cohomology group \(H^i_{c-\partial }(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U}, \overline{{{\mathbb {Q}}}}_p)\) in terms of the rigid cohomology of \(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U,\varSigma }\) and its various boundary strata. See Sect. 6.5. This is an analogue of the spectral sequence
where Y is a proper smooth variety over \({\mathbb {C}}\), where \(\partial Y\) is a simple normal crossings divisor on Y, and where \(Y^{(i)}\) is the disjoint union of the i-fold intersections of irreducible components of \(\partial Y\). (So \(Y^{(0)}=Y\).) Some of the terms in this spectral sequence seem a priori to be hard to control, e.g. \(H^i_{{\text {rig}}}(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U,\varSigma })\). However employing theorems about rigid cohomology due to Berthelot and Chiarellotto, we see that the eigenvalues of Frobenius on \(H^i_{c-\partial }(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U,\varSigma }, \overline{{{\mathbb {Q}}}}_p)\) are all Weil \(p^j\)-numbers for \(j\ge 0\). Moreover the weight 0 part, \(W_0 H^i_{c-\partial }(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U}, \overline{{{\mathbb {Q}}}}_p)\), equals the cohomology of a complex only involving the rigid cohomology in degree 0 of \(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U}\) and its various boundary strata. (See Proposition 6.24.) This should have a purely combinatorial description. More precisely we define a simplicial complex \({\mathcal {S}}(\partial \overline{{A}}^{(m),{{\text {ord}}}}_{n,U,\varSigma })\) whose vertices correspond to boundary components of \(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U,\varSigma }\) and whose j-faces correspond to j-boundary components with non-trivial intersection. For \(i>0\) we obtain an isomorphism
Thus it suffices to show that for N sufficiently large \(\varPi (N)\) occurs in
for some m and some \(i>0\) (possibly depending on N).
The boundary of \(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U,\varSigma }\) comes in pieces indexed by the conjugacy classes of maximal parabolic subgroups of \(G_n\). We shall be interested in the union of the irreducible components which are associated to \(P_{n,(n)}^+\). These correspond to an open subset \(|{\mathcal {S}}(\partial \overline{{A}}^{(m),{{\text {ord}}}}_{n,U,\varSigma })|_{=n}\) of \(|{\mathcal {S}}(\partial \overline{{A}}^{(m),{{\text {ord}}}}_{n,U,\varSigma })|\). As \(|{\mathcal {S}}(\partial \overline{{A}}^{(m),{{\text {ord}}}}_{n,U,\varSigma })|\) is compact, the interior cohomology
is naturally a subquotient of \(H^i(|{\mathcal {S}}(\partial \overline{{A}}^{(m),{{\text {ord}}}}_{n})|,\overline{{{\mathbb {Q}}}}_p)\). (By interior cohomology we mean the image of the cohomology with compact support in the cohomology. The interior cohomology of an open subset of an ambient compact Hausdorff space is naturally a subquotient of the cohomology of that ambient space.) Thus it even suffices to show that for N sufficiently large \(\varPi (N)\) occurs in
for some m and some \(i>0\) (possibly depending on N).
However the data \(\varSigma \) are a \(G_n^{(m)}({\mathbb {Q}})\)-invariant (glued) collection of polyhedral cone decompositions and \({\mathcal {S}}(\partial \overline{{A}}^{(m),{{\text {ord}}}}_{n})\) is obtained from \(\varSigma \) by replacing 1-cones by vertices, 2-cones by edges, etc. The cones corresponding to \(|{\mathcal {S}}(\partial \overline{{A}}^{(m),{{\text {ord}}}}_{n})|_{=n}\) are a disjoint union of polyhedral cones in the space of positive definite hermitian forms on \(F^n\). From this one obtains an equality
where
with U(n) denoting the usual \(n \times n\) compact unitary group. We deduce that
where
as \(U'\) runs over neat open compact subgroups of \(L_{n,(n)}^{(m)}({\mathbb {A}}^\infty )\). (The \({\mathbb {Z}}_p^\times \)-invariants result from a restriction on the open compact subgroups of \(G_n({\mathbb {A}}^\infty )\) that we are considering.) Thus it suffices to show that for all sufficiently large N, the representation \(1 \times (\pi ||\det ||^N)^{\infty ,p}\) occurs in \(H^i_{{\text {Int}}}({\mathfrak {T}}_{(n)}^{(m)}, {\mathbb {C}})\) for some \(i>0\) and some m (possibly depending on N).
We will write simply \({\mathfrak {T}}_{(n),U'}\) for \({\mathfrak {T}}^{(0)}_{(n),U'}\), a locally symmetric space associated to \(L_{n,(n)}\cong GL_1 \times {{\text {RS}}}_{\mathbb {Q}}^F GL_n\). If \(\rho \) is a representation of \(L_{n,(n)}\) over \({\mathbb {C}}\), then it gives rise to a locally constant sheaf \({\mathcal {L}}_{\rho ,U'}\) over \({\mathfrak {T}}_{(n),U'}\). We set
a smooth \(L_{n,(n)}({\mathbb {A}}^\infty )\)-module. The space \({\mathfrak {T}}^{(m)}_{(n), U'}\) is an \((S^1)^{nm[F: {\mathbb {Q}}]}\)-bundle over the locally symmetric space \({\mathfrak {T}}^{(0)}_{(n),U'}\) and if \(\pi ^{(m)}\) denotes the fibre map then
where \(L_{n,(n)}\) acts on \({{\text {Hom}}}_F(F^m,F^n)\) via projection to \({{\text {RS}}}^F_{\mathbb {Q}}GL_n\). Moreover the Leray spectral sequence
degenerates at the second page. (This can be seen by considering the action of the centre of \(L_{n,(n)}({\mathbb {A}}^\infty )\).) Thus it suffices to show that for all sufficiently large N, we can find non-negative integers j and m and an irreducible constituent \(\rho \) of \(\wedge ^j{{\text {Hom}}}_F(F^m,F^n)^\vee \otimes _{\mathbb {Q}}{\mathbb {C}}\) such that the representation \(1 \times (\pi ||\det ||^N)^{\infty ,p}\) occurs in \(H^i_{{\text {Int}}}({\mathfrak {T}}_{(n)}, {\mathcal {L}}_\rho )\) for some \(i \in {\mathbb {Z}}_{>0}\). Clozel [20] checked that (for \(n>1\)) this will be the case as long as \(1 \times (\pi ||\det ||^N)_\infty \) has the same infinitesimal character as some irreducible constituent of \(\wedge ^j{{\text {Hom}}}_F(F^m,F^n) \otimes _{\mathbb {Q}}{\mathbb {C}}\), i.e. if \(\rho _\pi \otimes ({\text{ N }}_{F/{\mathbb {Q}}} \circ \det )^N\) occurs in \(\wedge ^j{{\text {Hom}}}_F(F^m,F^n) \otimes _{\mathbb {Q}}{\mathbb {C}}\). From Weyl’s construction of the irreducible representations of \(GL_n\), for large enough N this will indeed be the case for some m and j.
We remark it is essential to work with N sufficiently large. It is not an artefact of the fact that we are working with Kuga–Sato varieties rather than local systems on the Shimura variety. We can twist a local system on the Shimura variety by a power of the multiplier character of \(G_n\). However the restriction of the multiplier factor of \(G_n\) to \(L_{n,(n)} \cong GL_1 \times {{\text {RS}}}^F_{\mathbb {Q}}GL_n\) factors through the \(GL_1\)-factor and does not involve the \({{\text {RS}}}^F_{\mathbb {Q}}GL_n\) factor.
We learnt from the series of papers [30–32] the key observation that \(|{\mathcal {S}}(\partial A_{n,U,\varSigma }^{(m)})|\) has a nice geometric interpretation involving the locally symmetric space for \(L_{n,(n)}\) and that this could be used to calculate cohomology.
Although the central argument we have sketched above is not long, this paper has unfortunately become very long. If we had only wanted to construct \(r_{p,\imath }(\pi )\) for all but finitely many primes p, then the argument would have been significantly shorter as we could have worked only with Shimura varieties \(X_{n,U}\) which have good integral models at p. The fact that we want to construct \(r_{p,\imath }(\pi )\) for all p adds considerable technical complications and also requires appeal to the recent work [44]. (Otherwise we would only need to appeal to [41, 42].)
Another reason this paper has grown in length is the desire to use a language to describe toroidal compactifications of mixed Shimura varieties that is different from the language used in [41, 42, 44]. We do this because at least one of us (R.T.) finds this language clearer. In any case it would be necessary to establish a substantial amount of notation regarding toroidal compactifications of Shimura varieties, which would require significant space. We hope that the length of the paper, and the technicalities with which we have to deal, won’t obscure the main line of the argument. On a first reading the reader might like to start with “Appendix A”, which summarizes the extensive notation we use, and then turn to Sects. (5 and) 6 and 7. These sections will provide reference back to the key results from earlier sections. We have added “Appendix B” to help comparison between the notation of this paper and the notation of [41, 42, 44], which we hope will make life easier for those readers that want to follow-up on our many references to these papers.
After we announced these results, but while we were writing up this paper, Scholze found another proof of Theorem A, relying on his theory of perfectoid spaces. His arguments seem to be in many ways more robust. For instance, he can handle torsion in the cohomology of the locally symmetric varieties associated to \(GL_n\) over a CM field. Scholze’s methods have some similarities with ours. Both methods first realize the Hecke eigenvalues of interest in the cohomology with compact support of the open Shimura variety by an analysis of the boundary and then show that they also occur in some space of p-adic cusp forms. We work with the ordinary locus of the Shimura variety, which for the minimal compactification is affinoid. Scholze works with the whole Shimura variety, but at infinite level. He (very surprisingly) shows that at infinite level, as a perfectoid space, (some compactification of) the Shimura variety has a Hecke invariant affinoid cover.
1.1 Notation
If \(G \twoheadrightarrow H\) is a surjective group homomorphism and if U is a subgroup of G we will sometimes use U to also denote the image of U in H.
If \(f: X \rightarrow Y\) and \(f': Y \rightarrow Z\) then we will denote by \(f' \circ f: X \rightarrow Z\) the composite map f followed by \(f'\). In this paper we will use both left and right actions. Suppose that G is a group acting on a set X and that \(g,h \in G\). If G acts on X on the left we will write gh for \(g \circ h\). If G acts on X on the right we will write hg for \(g \circ h\).
If f is an automorphism of \({{\text {Hom}}}(X,Y)\) we will sometimes use \((\circ f)\) to denote the map
We will sometimes use / to denote a quotient, and sometimes we will use it to denote the fact that the object to the left lives ‘over’ the object to the right. Both these usages are standard, and we hope no confusion will arise.
If G is a group (or group scheme) then Z(G) will denote its centre.
We will write \(S_n\) for the symmetric group on n letters. We will write U(n) for the group of \(n \times n\) complex matrices h with \({}^th({}^ch)=1_n\).
If G is an abelian group we will write \(G[\infty ]\) for the torsion subgroup of G, \(G[\infty ^p]\) for the subgroup of elements of order prime to p, and \(G^{{\text {TF}}}=G/G[\infty ]\). We will write \(TG= \lim _{\leftarrow N} G[N]\) and \(T^pG=\lim _{\leftarrow p {\not | }N}G[N]\). We will also write \(VG=TG \otimes _{\mathbb {Z}}{\mathbb {Q}}\) and \(V^pG=T^pG \otimes _{\mathbb {Z}}{\mathbb {Q}}\).
If A is a ring, if B is a locally free, finite A-algebra, and if \(X/{{\text {Spec}}\,}B\) is a quasi-projective scheme; then we will let \({{\text {RS}}}^B_A X\) denote the restriction of scalars (or Weil restriction) of X from B to A. (See, for instance, section 7.6 of [12].)
By a p-adic formal scheme we mean a formal scheme such that p generates an ideal of definition.
If X is an A-module and B is an A-algebra, we will sometimes write \(X_B\) for \(X \otimes _A B\). If X is reflexive over A, then we will also use X to denote the additive group scheme over A defined by
for all A-algebras B.
If X is a locally free \({\mathcal {O}}_F\)-module we will write \(GL(X/{\mathcal {O}}_F)\) for the group scheme over \({\mathbb {Z}}\) defined by
If Y is a scheme and if \(G_1,G_2/Y\) are group schemes then we will let
denote the Zariski sheaf on Y whose sections over an open W are
If in addition R is a ring then we will let \({\mathcal {H}}{{o}}{{m}}(G_1,G_2)_R\) denote the tensor product of sheaves \({\mathcal {H}}{{o}}{{m}}(G_1,G_2)\otimes _{\mathbb {Z}}R\) and we will let \({{\text {Hom}}}(G_1,G_2)_R\) denote the R-module of global sections of \({\mathcal {H}}{{o}}{{m}}(G_1,G_2)_R\). If Y is noetherian this is the same as \({{\text {Hom}}}(G_1,G_2) \otimes _{\mathbb {Z}}R\), but for a general base Y it may differ.
If \({\mathcal {S}}\) is a simplicial complex we will write \(|{\mathcal {S}}|\) for the corresponding topological space.
If F is a field then \(G_F\) will denote its absolute Galois group. If F is a number field and \(F_0 \subset F\) is a subfield and S is a finite set of primes of \(F_0\), then we will denote by \(G_F^S\) the maximal continuous quotient of \(G_F\) in which all primes of F not lying above an element of S are unramified.
Suppose that F is a number field and that v is a place of F. If v is finite we will write \(\varpi _v\) for a uniformizer in \(F_v\) and k(v) for the residue field of v. We will write \(| \,\,\,|_v\) for the absolute value on F associated to v and normalized as follows:
-
if v is finite then \(|\varpi _v|_v=(\# k(v))^{-1}\);
-
if v is real then \(|x|_v=\pm x\);
-
if v is complex then \(|x|_v={}^cxx\).
We write
We will write \({\mathcal {D}}_F^{-1}\) for the inverse different of \({\mathcal {O}}_F\).
If \(w \in {\mathbb {Z}}\) and p is a prime number then by a Weil \(p^w\)-number we mean an element \(\alpha \in \overline{{{\mathbb {Q}}}}\) which is an integer away from p and such that for each infinite place v of \(\overline{{{\mathbb {Q}}}}\) we have \(|\alpha |_v=p^w\).
Suppose that v is finite and that
is a continuous representation, which in the case v|l we assume to be de Rham. Then we will write \({{\text {WD}}}(r)\) for the corresponding Weil–Deligne representation of the Weil group \(W_{F_v}\) of \(F_v\) (see, for instance, section 1 of [56]). If \(\pi \) is an irreducible smooth representation of \(GL_n(F_v)\) over \({\mathbb {C}}\) we will write \({\text {rec}}_{F_v}(\pi )\) for the Weil–Deligne representation of \(W_{F_v}\) corresponding to \(\pi \) by the local Langlands conjecture (see, for instance, the introduction to [29]). If \(\pi _i\) is an irreducible smooth representation of \(GL_{n_i}(F_v)\) over \({\mathbb {C}}\) for \(i=1,2\) then there is an irreducible smooth representation \(\pi _1 \boxplus \pi _2\) of \(GL_{n_1+n_2}(F_v)\) over \({\mathbb {C}}\) satisfying
Suppose that G is a reductive group over \(F_v\) and that P is a parabolic subgroup of G with unipotent radical N and Levi component L. Suppose also that \(\pi \) is a smooth representation of \(L(F_v)\) on a vector space \(W_\pi \) over a field \(\varOmega \) of characteristic 0. We will define
to be the representation of \(G(F_v)\) by right translation on the set of locally constant functions
such that
for all \(h \in P(F_v)\) and \(g \in G(F_v)\). In the case \(\varOmega ={\mathbb {C}}\) we also define
where
If G is a linear algebraic group over F then the concept of a neat open compact subgroup of \(G({\mathbb {A}}_F^\infty )\) is defined, for instance, in section 0.6 of [49].
2 Some algebraic groups and automorphic forms
For the rest of this paper fix the following notation. Let \(F^+\) be a totally real field and \(F_0\) an imaginary quadratic field, and set \(F=F_0F^+\). Write c for the non-trivial element of \({{\text {Gal}}}(F/F^+)\). Also choose a rational prime p which splits in \(F_0\) and choose an element \(\delta _F \in {\mathcal {O}}_{F,(p)}\) with \({{\text {tr}}}_{F/F^+} \delta _F=1\) (which is possible as p is unramified in \(F/F^+\)).
Fix an isomorphism \(\imath : \overline{{{\mathbb {Q}}}}_p \mathop {\rightarrow }\limits ^{\sim }{\mathbb {C}}\). Fix a choice of \(\sqrt{p} \in \overline{{{\mathbb {Q}}}}_p\) by \(\imath \sqrt{p} >0\). If v is a prime of F and \(\pi \) an irreducible admissible representation of \(GL_m(F_v)\) over \(\overline{{{\mathbb {Q}}}}_p\) define
a Weil–Deligne representation of \(W_{F_v}\) over \(\overline{{{\mathbb {Q}}}}_p\).
Let n be a non-negative integer. We will often attach n as a subscript to other notation, when we need to record the particular choice of n we are working with, but, at other times when the choice of n is clear, we may drop it from the notation.
2.1 Three algebraic groups
Write \(\varPsi _n\) for the \(n \times n\)-matrix with 1’s on the anti-diagonal and 0’s elsewhere, and set
Let
and define a perfect pairing
by
We will write \(V_n\) for \(\varLambda _n \otimes {\mathbb {Q}}\). Let \(G_n\) denote the group scheme over \({\mathbb {Z}}\) defined by
for any ring R, and let \(\nu : G_n \rightarrow GL_1\) denote the multiplier character which sends \((g,\mu )\) to \(\mu \). Then \(G_n\) is a quasi-split connected reductive group scheme over \({\mathbb {Z}}[1/D_{F/{\mathbb {Q}}}]\) (where \(D_{F/{\mathbb {Q}}}\) denotes the discriminant of \(F/{\mathbb {Q}}\)) and splits over \({\mathcal {O}}_{F^{{\text {nc}}}}[1/D_{F/{\mathbb {Q}}}]\) (where \(F^{{{\text {nc}}}}\) denotes the normal closure of \(F/{\mathbb {Q}}\)). In particular \(G_0\) will denote \(GL_1\) and \(\nu : G_0 \rightarrow GL_1\) is the identity map.
If \(n>0\) set
Then there is a natural map
If \(n=0\) we set \(C_0={\mathbb {G}}_m\) and let \(G_0 \longrightarrow C_0\) denote the map \(\nu \). In either case this map identifies \(C_n\) with \(G_n/[G_n,G_n]\).
We will write \(\varLambda _{n,(i)}\) for the submodule of \(\varLambda _n\) consisting of elements whose last \(2n-i\) entries are 0, and \(V_{n,(i)}\) for \(\varLambda _{n,(i)} \otimes {\mathbb {Q}}\). If W is a submodule of \(\varLambda _n\) we will write \(W^\perp \) for its orthogonal complement with respect to \(\langle \,\,\,,\,\,\,\rangle _n\). Thus \(\varLambda _{n,(i)}^\perp \) is the submodule of \(\varLambda _n\) consisting of vectors whose last i entries are 0. Also write
and set \(V_n^{(m)}=\varLambda _n^{(m)} \otimes _{\mathbb {Z}}{\mathbb {Q}}\). Throughout this paper there will be various objects indexed by a superscript \({}^{(m)}\). In the case \(m=0\) we will sometimes simply drop it from the notation. For example, \(\varLambda _n=\varLambda _n^{(0)}\).
Define an additive group scheme \({{\text {Hom}}}_n^{(m)}\) over \({\mathbb {Z}}\) by
Then \({{\text {Hom}}}_n^{(m)}\) has an action of \(G_n \times {{\text {RS}}}^{{\mathcal {O}}_F}_{\mathbb {Z}}GL_m\) given by
Also define a perfect pairing
by
where \(e_1,\ldots ,e_m\) denotes the standard basis of \({\mathcal {O}}_F^m\). We have
Moreover \(G_n(R)\) is identified with the set of pairs
such that g commutes with the action of \(GL_m({\mathcal {O}}_F \otimes _{{\mathbb {Z}}} R)\) and such that
for all \(f,f' \in {{\text {Hom}}}_{{\mathcal {O}}_F}({\mathcal {O}}_F^m, \varLambda _n)(R)\). We set
Then \(G_n^{(m)}\) has an action of \({{\text {RS}}}^{{\mathcal {O}}_F}_{\mathbb {Z}}GL_m\) by
Moreover \(G_n^{(m)}\) acts on \(\varLambda _n^{(m)}\), by letting \(f \in {{\text {Hom}}}_n^{(m)}\) act by
and \(g \in G_n\) act by
Moreover \({{\text {RS}}}_{\mathbb {Z}}^{{\mathcal {O}}_F} GL_m\) acts on \(\varLambda _n^{(m)}\) by
We have \(\gamma \circ g=\gamma (g) \circ \gamma \).
If \(m_1 \ge m_2\) we embed \({\mathcal {O}}_F^{m_2} \hookrightarrow {\mathcal {O}}_F^{m_1}\) via
This gives rise to maps
and
It also gives rise to
Suppose that R is a ring and that X is an \({\mathcal {O}}_F \otimes _{\mathbb {Z}}R\)-module. We will write \({{\text {Herm}}}_X\) for the R-module of R-bilinear pairings
which satisfy
-
(1)
\((ax,y)=(x,{}^cay)\) for all \(a \in {\mathcal {O}}_F\) and \(x,y \in X\);
-
(2)
\((x,y)=(y,x)\) for all \(x,y \in X\).
If \(z \in {{\text {Herm}}}_X\) we will sometimes denote the corresponding pairing \((\,\,\,,\,\,\,)_z\). If S is an R-algebra we have a natural map
If \(X={\mathcal {O}}_F^m \otimes _{\mathbb {Z}}R\) then we will write
If \(X \rightarrow Y\) then there is a natural map \({{\text {Herm}}}_Y \rightarrow {{\text {Herm}}}_X\). In particular if \(m_1 \ge m_2\), then there is a natural map
induced by the map \({\mathcal {O}}_F^{(m_2)} \hookrightarrow {\mathcal {O}}_F^{(m_1)}\) described in the last paragraph. The group \(GL(X/{\mathcal {O}}_F)\) acts on the left on \({{\text {Herm}}}_X\) by
There is a natural isomorphism
under which an element (z, f, w) of the right hand side corresponds to
If X is an \({\mathcal {O}}_F \otimes _{\mathbb {Z}}R\)-module, there is a natural pairing
We further define
and
There is a natural map in the other direction
such that the composite \(S(X) \rightarrow X \otimes _{{\mathcal {O}}_F \otimes R,c \otimes 1} X \rightarrow S(X)\) is multiplication by 2. Note that if \(F/F^+\) is ramified above 2 then \(S({\mathcal {O}}_F^m)\) can have 2-torsion, but that \(S({\mathcal {O}}_{F,(p)}^m)\) is torsion free. (Either \(p>2\) or by assumption \(F/F^+\) is not ramified above 2.) There is a perfect duality
We will write
where \(e_1,\ldots ,e_m\) denotes the standard basis of \({\mathcal {O}}_F^m\).
Set \(N_n^{(m)}({\mathbb {Z}})\) to be the set of pairs
such that
for all \(x,y \in {\mathcal {O}}_F^m\). We define a group scheme \(N_{n}^{(m)}/{{\text {Spec}}\,}{\mathbb {Z}}\) by setting \(N_{n}^{(m)}(R)\) to be the set of pairs
with group law given by
where by \(\langle f\,\,\,,f'\,\,\,\rangle _n - \langle f'\,\,\,,f\,\,\,\rangle _n\) we mean the hermitian form
Note that \((f,z)^{-1}=(-f,-z)\). Thus there is an exact sequence
In fact, \(Z(N_{n}^{(m)})={{\text {Herm}}}^{(m)}\). The commutator in \(N_n^{(m)}\) induces an alternating map
under which \((f,f')\) maps to the pairing
If \(m_1 \ge m_2\) there is a natural map
compatible with the previously described maps
and
Note that \(G_n \times {{\text {RS}}}^{{\mathcal {O}}_F}_{\mathbb {Z}}GL_m\) acts on \(N_{n}^{(m)}\) from the left by
If 2 is invertible in R we see that
and
Set
which has an \({{\text {RS}}}^{{\mathcal {O}}_F}_{\mathbb {Z}}GL_m\)-action via
If \(m_1 \ge m_2\) then we get a natural map \(\widetilde{{G}}^{(m_1)}_n \rightarrow \widetilde{{G}}_n^{(m_2)}\). Note that
Let \(B_n\) denote the subgroup of \(G_n\) consisting of elements which preserve the chain \(\varLambda _{n,(n)} \supset \varLambda _{n,(n-1)} \supset \cdots \supset \varLambda _{n,(1)} \supset \varLambda _{n,(0)}\) and let \(N_n\) denote the normal subgroup of \(B_n\) consisting of elements with \(\nu =1\), which also act trivially on \(\varLambda _{n,(i)}/\varLambda _{n,(i-1)}\) for all \(i=1,\ldots ,n\). Let \(T_n\) denote the group consisting of the diagonal elements of \(G_n\) and let \(A_n\) denote the image of \({\mathbb {G}}_m\) in \(G_n\) via the embedding that sends t onto \(t1_{2n}\). Over \({\mathbb {Q}}\) we see that \(T_n\) is a maximal torus in a Borel subgroup \(B_n\) of \(G_n\) and that \(N_n\) is the unipotent radical of \(B_n\). Moreover \(A_n\) is a maximal split torus in the centre of \(G_n\).
If \(\varOmega \) is an algebraically closed field of characteristic 0 then set
Also let \(\varPhi _n \subset X^*(T_{n,/\varOmega })\) denote the set of roots of \(T_n\) on \({{\text {Lie}}\,}G_n\); let \(\varPhi ^+_n \subset \varPhi _n\) denote the set of positive roots with respect to \(B_n\) and let \(\varDelta _n \subset \varPhi _n^+\) denote the set of simple positive roots. We will write \(\varrho _n\) for half the sum of the elements of \(\varPhi _n^+\). If \(R \subset {\mathbb {R}}\) is a subring then \(X^*(T_{n,/\varOmega })^+_R\) will denote the subset of \(X^*(T_{n,/\varOmega })_R\) consisting of elements which pair non-negatively with the coroot \(\check{\alpha } \in X_*(T_{n,/\varOmega })\) corresponding to each \(\alpha \in \varDelta _n\). We will write simply \(X^*(T_{n,/\varOmega })^+\) for \(X^*(T_{n,/\varOmega })_{\mathbb {Z}}^+\). If \(\lambda \in X^*(T_{n,/\varOmega })^+\) we will let \(\rho _{n,\lambda }\) (or simply \(\rho _\lambda \)) denote the irreducible representation of \(G_n\) with highest weight \(\lambda \). When \(\rho _\lambda \) is used as a subscript we will sometimes replace it by just \(\lambda \).
There is a natural identification
This gives rise to the further identification
We will use this to identify \(X^*(T_{n,/\varOmega })\) with a quotient of
Under this identification \(X^*(T_{n,/\varOmega })^+\) is identified to the image of the set of
with
for all \(\tau \).
If R is a subring of \({\mathbb {R}}\) and H an algebraic subgroup of \(\widetilde{{G}}_n^{(m)}\) we will write \(H(R)^+\) for the subgroup of H(R) consisting of elements with positive multiplier. Thus \(G_n({\mathbb {R}})^+\) (resp. \(G_n^{(m)}({\mathbb {R}})^+\), resp. \(\widetilde{{G}}_n^{(m)}({\mathbb {R}})^+\)) is the connected component of the identity in \(G_n({\mathbb {R}})\) (resp. \(G_n^{(m)}({\mathbb {R}})\), resp. \(\widetilde{{G}}_n^{(m)}({\mathbb {R}})\)).
Let
with \(j^2=1\) and \(j(A_\tau ,B_\tau )j=(B_\tau ,A_\tau )\). Embed \(U_{n,\infty }\) in \(G_n({\mathbb {R}})\) by sending \((A_\tau ,B_\tau ) \in (U(n)^2)^{{{\text {Hom}}}(F^+,{\mathbb {R}})}\) to
and sending j to
(This map depends on identifications \(F \otimes _{F^+,\tau } {\mathbb {R}}\cong {\mathbb {C}}\), but the image of the map does not, and this image is all that will concern us.) Then \(U_{n,\infty }\) is a maximal compact subgroup of \(G_n({\mathbb {R}})\) (and even of \(\widetilde{{G}}_n^{(m)}({\mathbb {R}})\)). If \(L \supset T_n \times {{\text {Spec}}\,}{\mathbb {R}}\) is a Levi component of a parabolic subgroup \(P \supset B_n \times {{\text {Spec}}\,}{\mathbb {R}}\) then \(U_{n,\infty } \cap L({\mathbb {R}})\) is a maximal compact subgroup of \(L({\mathbb {R}})\). The connected component of the identity of \(U_{n,\infty }\) is \(U_{n,\infty }^0=U_{n,\infty } \cap G_n({\mathbb {R}})^+\).
We will write \({\mathfrak {p}}_n\) for the set of elements of \({{\text {Lie}}\,}G_n({\mathbb {R}})\) of the form
where \({}^{c,t}A_\tau =A_\tau \) and \({}^{c,t}B_\tau =B_\tau \) for all \(\tau \). Then
We give the real vector space \({\mathfrak {p}}_n\) a complex structure by letting i act by
We decompose
by setting
We also set
It is a parabolic subalgebra of \(({{\text {Lie}}\,}G_n({\mathbb {R}})) \otimes _{\mathbb {R}}{\mathbb {C}}\) with unipotent radical \({\mathfrak {p}}_n^-\) and Levi component \({{\text {Lie}}\,}(U_{n,\infty } A_n({\mathbb {R}})) \otimes _{\mathbb {R}}{\mathbb {C}}\). We will write \({\mathfrak {Q}}_n\) for the parabolic subgroup of \(G_n \times _{\mathbb {Q}}{\mathbb {C}}\) with Lie algebra \({\mathfrak {q}}_n\). Note that
Let \({\mathfrak {H}}_n^+\) (resp. \({\mathfrak {H}}_n^{\pm }\)) denote the set of I in \(G_n({\mathbb {R}})\) with multiplier 1 such that \(I^2=-1_{2n}\) and such that the symmetric bilinear form \(\langle I \,\,\, ,\,\,\, \rangle _n\) on \(\varLambda _n \otimes _{{\mathbb {Z}}} {\mathbb {R}}\) is positive definite (respectively positive or negative definite). Then \(G_n({\mathbb {R}})\) (resp. \(G_n({\mathbb {R}})^+\)) acts transitively on \({\mathfrak {H}}_n^\pm \) (resp. \({\mathfrak {H}}_n^+\)) by conjugation. Moreover \(J_n \in {\mathfrak {H}}_n^+\) has stabilizer \(U_{n,\infty }^0A_n({\mathbb {R}})^0\) and so we get an identification of \({\mathfrak {H}}_n^{\pm }\) (resp. \({\mathfrak {H}}_n^+\)) with \(G_n({\mathbb {R}})/U_{n,\infty }^0A_n({\mathbb {R}})^0\) (resp. \(G_n({\mathbb {R}})^+/U_{n,\infty }^0A_n({\mathbb {R}})^0\)). The natural map
is an open embedding and gives \({\mathfrak {H}}_n^\pm \) the structure of a complex manifold. The action of \(G_n({\mathbb {R}})\) is holomorphic, and the complex structure induced on the tangent space \(T_{J_n} {\mathfrak {H}}_n^\pm \cong {\mathfrak {p}}_n\) is the complex structure described in the previous paragraph.
If \(\rho \) is a finite dimensional algebraic representation of \({\mathfrak {Q}}_n\) on a \({\mathbb {C}}\)-vector space \(W_\rho \), then there is a holomorphic vector bundle \({\mathfrak {E}}_\rho /{\mathfrak {H}}_n^\pm \) together with a holomorphic action of \(G_n({\mathbb {R}})\), defined as the pull-back to \({\mathfrak {H}}^\pm \) of \((G_n({\mathbb {C}}) \times W_\rho )/{\mathfrak {Q}}_n({\mathbb {C}})\), where
-
\(h \in {\mathfrak {Q}}_n({\mathbb {C}})\) sends (g, w) to \((gh,h^{-1}w)\),
-
and where \(h \in G_n({\mathbb {R}})\) sends [(g, w)] to [(hg, w)].
If \(N_2 \ge N_1 \ge 0\) are integers we will write \(U_p(N_1,N_2)_n\) for the subgroup of \(G_n({\mathbb {Z}}_p)\) consisting of elements whose reduction modulo \(p^{N_2}\) preserves
and acts trivially on \(\varLambda _n/(\varLambda _{n,(n)} + p^{N_1} \varLambda _n)\). If \(N_2 \ge N_1 \ge N_1' \ge 0\) then \(U_p(N_1,N_2)_n\) is a normal subgroup of \(U_p(N_1',N_2)_n\) and
We will also set
and set \(\widetilde{{U}}_p(N_1,N_2)^{(m)}_n\) to be the pre-image of \(U_p(N_1,N_2)^{(m)}_n\) in \(\widetilde{{G}}_n^{(m)}({\mathbb {Z}}_p)\). Pictorially we can think of \(U_p(N_1,N_2)_n\) as
and of \(U_p(N_1,N_2)_n^{(m)}\) as
If \(U^p\) is an open compact subgroup of \(G_n({\mathbb {A}}^{\infty ,p})\) (resp. of \(G_n^{(m)}({\mathbb {A}}^{\infty ,p})\), resp. of \(\widetilde{{G}}_n^{(m)}({\mathbb {A}}^{\infty ,p})\)) we will set \(U^p(N_1,N_2)\) to be \(U^p \times U_p(N_1,N_2)_n\) (resp. \(U^p \times U_p(N_1,N_2)_n^{(m)}\), resp. \(U^p \times \widetilde{{U}}_p(N_1,N_2)_n^{(m)}\)), a compact open subgroup of \(G_n({\mathbb {A}}^{\infty })\) (resp. \(G_n^{(m)}({\mathbb {A}}^{\infty })\), resp. \(\widetilde{{G}}_n^{(m)}({\mathbb {A}}^{\infty })\)).
2.2 Maximal parabolic subgroups
We will write \(P_{n,(i)}^+\) (resp. \(P^{(m),+}_{n,(i)}\), resp. \(\widetilde{{P}}^{(m),+}_{n,(i)}\)) for the subgroup of \(G_n\) (resp. \(G_n^{(m)}\), resp. \(\widetilde{{G}}_n^{(m)}\)) consisting of elements which (after projection to \(G_n\)) take \(\varLambda _{n,(i)}\) to itself. We will also write \(N_{n,(i)}^+\) (resp. \(N^{(m),+}_{n,(i)}\), resp. \(\widetilde{{N}}^{(m),+}_{n,(i)}\)) for the subgroups of \(P_{n,(i)}^+\) (resp. \(P^{(m),+}_{n,(i)}\), resp. \(\widetilde{{P}}^{(m),+}_{n,(i)}\)) consisting of elements which act trivially on \(\varLambda _{n,(i)}\) and \(\varLambda _{n,(i)}^\perp /\varLambda _{n,(i)}\) and \(\varLambda _n/\varLambda _{n,(i)}^\perp \). Over \({\mathbb {Q}}\) the groups \(P_{n,(i)}^+\) (resp. \(P^{(m),+}_{n,(i)}\), resp. \(\widetilde{{P}}^{(m),+}_{n,(i)}\)) are maximal parabolic subgroups of \(G_n\) (resp. \(G_n^{(m)}\), resp. \(\widetilde{{G}}_n^{(m)}\)) containing the pre-image of \(B_n\). The groups \(N_{n,(i)}^+\) (resp. \(N^{(m),+}_{n,(i)}\), resp. \(\widetilde{{N}}^{(m),+}_{n,(i)}\)) are their unipotent radicals.
In some instances it will be useful to replace these groups by their ‘hermitian part’. We will write \(P_{n,(i)}\) for the normal subgroup of \(P_{n,(i)}^+\) consisting of elements which act trivially on \(\varLambda _n/\varLambda _{n,(i)}^\perp \). We will also write \(P^{(m)}_{n,(i)}\) for the normal subgroup
of \(P^{(m),+}_{n,(i)}\), and \(\widetilde{{P}}^{(m)}_{n,(i)}\) for the pre-image of \(P^{(m)}_{n,(i)}\) in \(\widetilde{{P}}^{(m),+}_{n,(i)}\). We will let
and
and
Over \({\mathbb {Q}}\) these are the unipotent radicals of \(P_{n,(i)}\) (resp. \(P^{(m)}_{n,(i)}\), resp. \(\widetilde{{P}}^{(m)}_{n,(i)}\)).
Pictorially one can think of \(P_{n,(i)}^+\) and \(P_{n,(i)}\) as matrices of the following shapes
and
respectively. If we picture an element of \(G^{(m)}_n\) as a pair of matrices
(the first \(2n \times 2n\) and the second \(2n \times m\)) then we can picture \(P_{n,(i)}^{(m),+}\) and \(P_{n,(i)}^{(m)}\) as consisting of matrices of the shape
and
respectively.
We have an isomorphism
To describe it let \(\varLambda _{n,(i)}'\) denote the subspace of \(\varLambda _n\) consisting of vectors with their first \(2n-i\) entries 0, so that
and
We define
by letting \(g \in G_{n-i}\) act as \(\nu (g)\) on \(\varLambda _{n,(i)}\), as g on \(\varLambda _{n-i} \cong \varLambda _{n,(i)}^\perp \cap (\varLambda _{n,(i)}')^\perp \) and as 1 on \(\varLambda _{n,(i)}'\), i.e.
We define
by sending h to the map
We also define
by sending z to the pairing
on \(\varLambda _{n,(i)}'\). In the other direction \((f,z) \in N_{n-i}^{(i)}\) is mapped to
where we think of \(f \in M_{2(n-i) \times i}(F)\) with first \(n-i\) rows in \(({\mathcal {D}}_F^{-1})^i\) and second \((n-i)\) rows in \({\mathcal {O}}_F^i\), and we think of \(z \in M_{i \times i}(F)^{t=c}\).
We also have isomorphisms
and
We will describe the second of these isomorphisms. Suppose \(f \in {{\text {Hom}}}_{n-i}^{(i)}\) and \(g \in {{\text {Hom}}}_{n-i}^{(m)}\). Also suppose that \(z \in \frac{1}{2}{{\text {Herm}}}^{(i)}\) and \(w \in \frac{1}{2}{{\text {Herm}}}^{(m)}\) and
so that
Let h(f, z) denote the element of \(P_{n,(i)}\) corresponding to \((f,z) \in N_{n,(i)}\). Think of g as a map
Define \(j(f,g,u) \in {{\text {Hom}}}({\mathcal {O}}_F^m, \varLambda _{n,(i)})\) by
for all \(x \in {\mathcal {O}}_F^m\) and \(y \in \varLambda _{n,(i)}' \cong {\mathcal {O}}_F^i\). Then
Note that
and that
Write \(L_{n,(i),{{\text {lin}}}}\) for the subgroup of \(P_{n,(i)}^+\) consisting of elements with \(\nu =1\) which preserve \(\varLambda _{n,(i)}'\subset \varLambda _n\) and act trivially on \(\varLambda _{n,(i)}^\perp /\varLambda _{n,(i)}\). We set \(N(L_{n,(i),{{\text {lin}}}}^{(m)})\) to be the additive group scheme over \({\mathbb {Z}}\) associated to
and write \(L_{n,(i),{{\text {lin}}}}^{(m)}\) for
and \(\widetilde{{L}}_{n,(i),{{\text {lin}}}}^{(m)}\) for
Note that
and
and
Also note that
via its action on \(\varLambda _{n,(i)}'\cong {\mathcal {O}}_F^i\), and that
Pictorially we can think of \(L_{n,(i),{{\text {lin}}}}\) as consisting of matrices of the form
and \(L_{n,(i),{{\text {lin}}}}^{(m)}\) as consisting of matrices of the form
We let \(L_{n,(i),{{\text {herm}}}}\) denote the subgroup of \(P_{n,(i)}\) consisting of elements which preserve \(\varLambda _{n,(i)}'\). Thus
In particular
Pictorially we can think of \(L_{n,(i),{{\text {herm}}}}\) as consisting of matrices of the form
Over \({\mathbb {Q}}\) it is a Levi component for \(P_{n,(i)}\) and \(P^{(m)}_{n,(i)}\) and \(\widetilde{{P}}^{(m)}_{n,(i)}\), so in particular
and
and
We also set
and
and
Over \({\mathbb {Q}}\) we see that \(L_{n,(i)}\) is a Levi component for each of \(P_{n,(i)}^+\) and \(P^{(m),+}_{n,(i)}\) and \(\widetilde{{P}}^{(m),+}_{n,(i)}\). Moreover
and
and
We will occasionally write \(P_{n,(i)}^{(m),-}\) (resp. \(L_{n,(i),{{\text {herm}}}}^-\)) for the kernel of the map \(P_{n,(i)}^{(m)}\rightarrow C_{n-i}\) (resp. \(L_{n,(i),{{\text {herm}}}} \rightarrow C_{n-i}\)).
We will write \(R_{n,(n),(i)}\) for the subgroup of \(L_{n,(n)}\) mapping \(\varLambda _{n,(i)}\) to itself. We will write \(N(R_{n,(n),(i)})\) for the subgroup of \(R_{n,(n),(i)}\) which acts trivially on \(\varLambda _{n,(i)}\) and \(\varLambda ^\perp _{n,(i)}/\varLambda _{n,(i)}\) and \(\varLambda _n/\varLambda ^\perp _{n,(i)}\).
We will also write \(R_{n,(n)}^{(m)}\) for the semi-direct product
If \(m'\le m\) we will fix \({\mathbb {Z}}^m \twoheadrightarrow {\mathbb {Z}}^{m-m'}\) to be projection onto the last \(m-m'\) coordinates and define \(Q_{m,(m')}\) for the subgroup of \(GL_m\) consisting of elements preserving the kernel of this map. We also define \(Q_{m,(m')}'\) to be the subgroup of \(Q_{m,(m')}\) consisting of elements which induce \(1_{{\mathbb {Z}}^{m-m'}}\) on \({\mathbb {Z}}^{m-m'}\). Thus there is an exact sequence
Moreover
We will also write \(A_{n,(i),{{\text {lin}}}}\) (resp. \(A_{n,(i),{{\text {herm}}}}\)) for the image of the map from \({\mathbb {G}}_m\) to \(L_{n,(i),{{\text {lin}}}}\) (resp. \(L_{n,(i),{{\text {herm}}}}\)) sending t to \(t 1_i\) (resp. \((t^2,t 1_{2(n-i)})\)). Moreover write \(A_{n,(i)}\) for \(A_{n,(i),{{\text {lin}}}} \times A_{n,(i),{{\text {herm}}}}\). Over \({\mathbb {Q}}\) the group \(A_{n,(i)}\) (resp. \(A_{n,(i),{{\text {lin}}}}\), resp. \(A_{n,(i),{{\text {herm}}}}\)) is the maximal split torus in the centre of \(L_{n,(i)}\) (resp. \(L_{n,(i),{{\text {lin}}}}\), resp. \(L_{n,(i),{{\text {herm}}}}\)).
Again suppose that \(\varOmega \) is an algebraically closed field of characteristic 0. Let \(\varPhi _{(n)} \subset \varPhi _n\) denote the set of roots of \(T_n\) on \({{\text {Lie}}\,}L_{n,(n)}\), and set \(\varPhi _{(n)}^+=\varPhi _n^+ \cap \varPhi _{(n)}\) and \(\varDelta _{(n)}=\varDelta _n \cap \varPhi _{(n)}\). We will write \(\varrho _{n,(n)}\) for half the sum of the elements of \(\varPhi _{(n)}^+\). If \(R \subset {\mathbb {R}}\) then \(X^*(T_{n,/\varOmega })^+_{(n),R}\) will denote the subset of \(X^*(T_{n,/\varOmega })_R\) consisting of elements which pair non-negatively with the coroot \(\check{\alpha } \in X_*(T_{n,/\varOmega })\) corresponding to each \(\alpha \in \varDelta _{(n)}\). We write \(X^*(T_{n,/\varOmega })_{(n)}^+\) for \(X^*(T_{n,/\varOmega })_{(n),{\mathbb {Z}}}^+\). If \(\lambda \in X^*(T_{n,/\varOmega })_{(n)}^+\) we will let \(\rho _{(n),\lambda }\) denote the irreducible representation of \(L_{n,(n)}\) with highest weight \(\lambda \). When \(\rho _{(n),\lambda }\) is used as a subscript we will sometimes replace it by just \((n),\lambda \).
Note that \({{\text {Lie}}\,}P_{n,(n)}({\mathbb {C}})\) and \({\mathfrak {q}}_n\) are conjugate under \(G_n({\mathbb {C}})\) and hence we obtain an identification (‘Cayley transform’) of \(({{\text {Lie}}\,}U_{n,\infty } A_n({\mathbb {R}})) \otimes _{\mathbb {R}}{\mathbb {C}}\) and \({{\text {Lie}}\,}L_{n,(n)}({\mathbb {C}})\), which is well defined up to conjugation by \(L_{n,(n)}({\mathbb {C}})\). Similarly \({\mathfrak {Q}}_n\) and \(P_{n,(n)}({\mathbb {C}})\) are conjugate in \(G_n \times _{\mathbb {Q}}{\mathbb {C}}\). Thus \(L_{n,(n)}({\mathbb {C}})\) can be identified with \({\mathfrak {Q}}_n\) modulo its unipotent radical, canonically up to \(L_{n,(n)}({\mathbb {C}})\)-conjugation. Thus if \(\rho \) is a finite dimensional algebraic representation of \(L_{n,(n)}\) over \({\mathbb {C}}\), we can associate to it a representation of \({\mathfrak {Q}}_n\) and of \({\mathfrak {q}}_n\), and hence a holomorphic vector bundle \({\mathfrak {E}}_\rho /{\mathfrak {H}}_n^\pm \) with \(G_n({\mathbb {R}})\)-action.
The isomorphism \(L_{n,(n)} \cong GL_1 \times {{\text {RS}}}^{{\mathcal {O}}_F}_{\mathbb {Z}}GL_n\) gives rise to a natural identification
and hence to identifications
and
Under this identification \(X^*(T_{n,/\varOmega })_{(n)}^+\) is identified to the set of
with
for all \(\tau \).
To compare this parametrization of \(X^*(T_{n,/\varOmega })\) with the one introduced in Sect. 1.1 note that the map
coming from \(L_{n,(n)} \hookrightarrow G_n\) sends
Thus the map
sends
A section is provided by the map
In particular we see that \(X^*(T_{n,/\varOmega })^+\subset X^*(T_{n,/\varOmega })_{(n)}^+\) is identified with the set of
with
and
for all \(\tau \).
Note that
We write \({{\text {Std}}}\) for the representation of \(L_{n,(n)}\) on \(\varLambda _n/\varLambda _{n,(n)}\) over \({\mathbb {Z}}\), and if \(\tau : F \hookrightarrow \overline{{{\mathbb {Q}}}}\) we write \({{\text {Std}}}_\tau \) for the representation of \(L_{n,(n)}\) on \((\varLambda _n/\varLambda _{n,(n)}) \otimes _{{\mathcal {O}}_F,\tau }{\mathcal {O}}_{\overline{{{\mathbb {Q}}}}}\). If \(\varOmega \) is an algebraically closed field of characteristic 0 and if \(\tau : F \hookrightarrow \varOmega \) we will sometimes write \({{\text {Std}}}_\tau \) for the representation of \(L_{n,(n)}\) on \((\varLambda _n/\varLambda _{n,(n)}) \otimes _{{\mathcal {O}}_F,\tau }\varOmega \). We hope that context will make clear the distinction between these two slightly different meaning of \({{\text {Std}}}_\tau \). We also let \({{\text {KS}}}\) denote the representation
of \(L_{n,(n)}\) over \({\mathbb {Z}}\). (See Sect. 1.1.) Note that over \(\overline{{{\mathbb {Q}}}}\) the representation \({{\text {Std}}}_\tau ^\vee \) is irreducible and in our normalizations has highest weight \((0,b_{\tau '})\) where
but \(b_{\tau '}=0\) for \(\tau ' \ne \tau \). Similarly the representation \(\wedge ^{n[F: {\mathbb {Q}}]} {{\text {Std}}}^\vee \) is irreducible with highest weight
Finally \({{\text {KS}}}\) is the direct sum of the \([F^+: {\mathbb {Q}}]\) irreducible representations indexed by \(\tau \in {{\text {Hom}}}(F^+, \overline{{{\mathbb {Q}}}})\) with highest weights \((1,b_{\tau '})\), where
if \(\tau '\) extends \(\tau \), and \(b_{\tau '}=0\) otherwise.
We will let \(\varsigma _p \in L_{n,(n),{{\text {herm}}}}({\mathbb {Q}}_p)\cong {\mathbb {Q}}_p^\times \) denote the unique element with multiplier \(p^{-1}\).
Set
and
Also set
and
and
Let \(U^p\) be an open compact subgroup of \(L_{n,(i)}({\mathbb {A}}^{\infty ,p})\) (resp. \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^{\infty ,p})\), resp. \(L_{n,(i)}^{(m)}({\mathbb {A}}^{\infty ,p})\), resp. \((P_{n,(i)}^{(m),+}/Z(N_{n,(i)}^{(m)}))({\mathbb {A}}^{\infty ,p})\), resp. \(P_{n,(i)}^{(m),+}({\mathbb {A}}^{\infty ,p})\), resp. \(\widetilde{{P}}_{n,(i)}^{(m),+}({\mathbb {A}}^{\infty ,p})\)). Then set
(resp.
resp.
resp.
resp.
resp.
In the case \(i=n\) these groups do not depend on \(N_2\), so we will write simply \(U^p(N_1)\).
For the study of the ordinary locus we will need a variant of \(G_n({\mathbb {A}}^\infty )\) and \(G_n^{(m)}({\mathbb {A}}^\infty )\) and \(\widetilde{{G}}_n^{(m)}({\mathbb {A}}^\infty )\). More specifically define a semigroup
Its maximal sub-semigroup that is also a group is
If H is an algebraic subgroup of \(\widetilde{{G}}_n^{(m)}\) (over \({{\text {Spec}}\,}{\mathbb {Q}}\)) we set
Its maximal sub-semigroup that is also a group is
Thus
and
If \(U^p\) is an open compact subgroup of \(H({\mathbb {A}}^{\infty ,p})\), we set
for any \(N' \ge N\). The group does not depend on the choice of \(N'\).
2.3 Base change
We will write \(B_{GL_m}\) for the subgroup of upper triangular elements of \(GL_m\) and \(T_{GL_m}\) for the subgroup of diagonal elements of \(B_{GL_m}\).
We will also let \(G_n^1\) denote the group scheme over \({\mathcal {O}}_{F^+}\) defined by
Thus
We will write \(B_n^1\) for the subgroup of \(G_n^1\) consisting of upper triangular matrices and \(T_n^1\) for the subgroup of \(B_n^1\) consisting of diagonal matrices. There is a natural projection \(B_n^1 \twoheadrightarrow T_n^1\) obtained by setting the off diagonal entries of an element of \(B_n^1\) to 0.
Suppose that q is a rational prime. Let \(u_1,\ldots ,u_r\) denote the primes of \(F^+\) above q which split \(u_i=w_i{}^cw_i\) in F and let \(v_1,\ldots ,v_s\) denote the primes of \(F^+\) above q which do not split in F. Then
where
If \(\varPi \) is an irreducible smooth representation of \(G_n({\mathbb {Q}}_q)\) then
We define \({{\text {BC}}}(\varPi )_{w_i}=\varPi _{w_i}\) and \({{\text {BC}}}(\varPi )_{c w_i}= \varPi _{w_i}^{\vee ,c}\). Note that this does not depend on the choice of primes \(w_i|u_i\). We will say that \(\varPi \) is unramified at \(w_i\) if \({{\text {BC}}}(\varPi )_{w_i}^{GL_{2n}({\mathcal {O}}_{F,w_i})} \ne (0)\). We will say that \(\varPi \) is unramified at \(v_i\) if \(v_i\) is unramified in F and
We will say that \(\varPi \) is unramified at q if \(\varPi \) is unramified at all primes above q and either q splits in \(F_0\) or q is unramified in F.
Suppose that \(\varPi \) is unramified at \(v_i\). Then there is a character \(\chi \) of the quotient \(T_n^1(F^+_{v_i})/T_n^1({\mathcal {O}}_{F^+,v_i})\) such that \(\varPi |_{G_n^1(F^+_{v_i})}\) and \({{\text {n-Ind}}}_{B_n^1(F^+_{v_i})}^{G_n^1(F^+_{v_i})} \chi \) share an irreducible subquotient with a \(G_n^1({\mathcal {O}}_{F^+,v_i})\)-fixed vector. Moreover this character \(\chi \) is unique modulo the action of the normalizer \(N_{G_n^1(F_{v_i}^+)}(T_n^1(F_{v_i}^+))/T_n^1(F_{v_i}^+)\). (If \(\pi \) and \(\pi '\) are two irreducible subquotients of \(\varPi _H|_{G_n^1(F_{v_i}^+)}\) then we must have \(\pi ' \cong \pi ^{\varsigma _{v_i}^{-1}}\) where
However
Let
Then we define \({{\text {BC}}}(\varPi )_{v_i}\) to be the unique subquotient of
with a \(GL_{2n}({\mathcal {O}}_{F,v_i})\)-fixed vector. The next lemma is easy to prove.
Lemma 1.1
Suppose that \(\psi \otimes \pi \) is an irreducible smooth representation of
-
(1)
If v is unramified over \(F^+\) and \(\pi _v\) is unramified then \({{\text {n-Ind}}}_{P_{n,(n)}({\mathbb {Q}}_q)}^{G_n({\mathbb {Q}}_q)} (\psi \otimes \pi )\) has a subquotient \(\varPi \) which is unramified at v. Moreover \({{\text {BC}}}(\varPi )_v\) is the unramified irreducible subquotient of \({{\text {n-Ind}}}_{Q_{2n,(n)}(F_v)}^{GL_{2n}(F_v)} (\pi ^{\vee ,c}_v \otimes \pi _v)\).
-
(2)
If v is split over \(F^+\) and \(\varPi \) is an irreducible subquotient of the normalized induction \({{\text {n-Ind}}}_{P_{n,(n)}({\mathbb {Q}}_q)}^{G_n({\mathbb {Q}}_q)} (\psi \otimes \pi )\), then \({{\text {BC}}}(\varPi )_v\) is an irreducible subquotient of \({{\text {n-Ind}}}_{Q_{2n,(n)}(F_v)}^{GL_{2n}(F_v)} ((\pi _{{}^cv})^{\vee ,c} \otimes \pi _v)\).
Note that in both cases \({{\text {BC}}}(\varPi _v)\) does not depend on \(\psi \).
In this paragraph let K be a number field, \(m\in {\mathbb {Z}}_{>0}\), and write \(U_{K,\infty }\) for a maximal compact subgroup of \(GL_m(K_\infty )\). We shall (slightly abusively) refer to an admissible
(resp.
resp.
module as an admissible \(G_n({\mathbb {A}})\)-module (resp. \(L_{n,(i)}({\mathbb {A}})\)-module, resp. \(GL_m({\mathbb {A}}_K)\)-module). By a square integrable automorphic representation of \(G_n({\mathbb {A}})\) (resp. \(L_{n,(i)}({\mathbb {A}})\), resp. \(GL_m({\mathbb {A}}_K)\)) we shall mean the twist by a character of an irreducible admissible \(G_n({\mathbb {A}})\)-module (resp. \(L_{n,(i)}({\mathbb {A}})\)-module, resp. \(GL_m({\mathbb {A}}_K)\)-module) that occurs discretely in the space of square integrable automorphic forms on the double coset space \(G_n({\mathbb {Q}}) \backslash G_n({\mathbb {A}})/A_n({\mathbb {R}})^0\) (respectively \(L_{n,(i)}({\mathbb {Q}})\backslash L_{n,(i)}({\mathbb {A}})/A_{n,(i)}({\mathbb {R}})^0\) or \(GL_m(K)\backslash GL_m({\mathbb {A}}_K)/{\mathbb {R}}^\times _{>0}\)). By a cuspidal automorphic representation of \(G_n({\mathbb {A}})\) (resp. \(L_{n,(i)}({\mathbb {A}})\), resp. \(GL_m({\mathbb {A}}_K)\)) we shall mean an irreducible admissible \(G_n({\mathbb {A}})\)-submodule (resp. \(L_{n,(i)}({\mathbb {A}})\)-submodule, resp. \(GL_m({\mathbb {A}}_K)\)-submodule) of the space of cuspidal automorphic forms on \(G_n({\mathbb {A}})\) (resp. \(L_{n,(i)}({\mathbb {A}})\), resp. \(GL_m({\mathbb {A}}_K)\)).
Proposition 1.2
Suppose that \(\varPi \) is a square integrable automorphic representation of \(G_n({\mathbb {A}})\) and that \(\varPi _\infty \) is cohomological. Then there is an expression
with \(m_i,n_i \in {\mathbb {Z}}_{>0}\) and cuspidal automorphic representations \(\widetilde{\varPi }_i\) of \(GL_{m_i}({\mathbb {A}}_F)\) such that
-
\(\widetilde{\varPi }_i^\vee \cong \widetilde{\varPi }_i^c\);
-
\(\widetilde{\varPi }_i ||\det ||^{(m_i+n_i-1)/2}\) is cohomological;
-
if v is a prime of F above a rational prime q such that
-
either q splits in \(F_0\),
-
or F and \(\varPi \) are unramified above q,
then
$$\begin{aligned} {{\text {BC}}}(\varPi _q)_v=\boxplus _{i=1}^r \boxplus _{j=0}^{n_i-1} \widetilde{\varPi }_{i,v} |\det |_v^{(n_i-1)/2-j}. \end{aligned}$$ -
Proof
This follows from the main theorem of [52] and the classification of square integrable automorphic representations of \(GL_m({\mathbb {A}}_F)\) in [48]. (Here we are using our assumption that F contains an imaginary quadratic field.) \(\square \)
Corollary 1.3
Keep the assumptions of the proposition. Then there is a continuous, semi-simple, algebraic (i.e. unramified almost everywhere and de Rham above p) representation
with the following property: If v is a prime of F above a rational prime \(q \ne p\) such that
-
either q splits in \(F_0\),
-
or F and \(\varPi \) are unramified above q,
then
Proof
Combine the proposition with, for instance, theorem 1.2 of [11] and theorem A of [10]. (These results are due to many people and we simply choose these particular references for convenience.) \(\square \)
2.4 Spaces of hermitian forms
If \(R \subset {\mathbb {R}}\) then we will denote by \({{\text {Herm}}}_X^{>0}\) (resp. \({{\text {Herm}}}_X^{\ge 0}\)) the set of pairings \((\,\,\,,\,\,\,)\) in \({{\text {Herm}}}_X\) such that
(resp. \(\ge 0\)) for all \(x \in X-\{0\}\). We will denote by \(S(F^m)^{>0}\) (resp. \(S(F^m)^{\ge 0}\)) the set of elements \(a \in S(F^m)\) such that for each \(\tau : F \hookrightarrow {\mathbb {C}}\) the image of a under the map
is positive definite (resp. positive semi-definite), i.e. all the roots of its characteristic polynomial are strictly positive (resp. non-negative) real numbers. Then \(S(F^m)^{> 0}\) is the set of elements of \(S(F^m)\) whose pairing with every nonzero element of \({{\text {Herm}}}_{F^m}^{\ge 0}\) is strictly positive, and \({{\text {Herm}}}_{F^m}^{>0}\) is the set of elements of \({{\text {Herm}}}_{F^m}\) whose pairing with every nonzero element of \(S(F^m)^{\ge 0}\) is strictly positive. We will also write
We will next turn to the study of certain spaces which play a key role in the definition of the auxiliary data controlling toroidal compactifications.
Suppose that \(W \subset V_n\) is an isotropic F-direct summand. We set
If \(m=0\) we will drop it from the notation. Note that we have a natural identification
There is also a natural map
Note that if \(f \in {{\text {Hom}}}_F(F^m,W)\) we can define \(f' \in {{\text {Hom}}}(F^m \otimes _{F,c} (V_n/W^\perp ),{\mathbb {Q}})\) by
This establishes an isomorphism
and hence an isomorphism
Thus
If \(g \in G_n({\mathbb {Q}})\) we define
where
We extend this to an action of \(G_n^{(m)}({\mathbb {Q}})\) as follows: If \(g \in {{\text {Hom}}}_F(F^m,V_n)\) then we set
where \(\theta _z: V_n \rightarrow W\) satisfies
for all \(x,y \in V_n\). If \(W' \subset W\) there is a natural embedding
We will write \({\mathfrak {C}}^{>0}(W)={{\text {Herm}}}_{V_n/W^\perp \otimes _{\mathbb {Q}}{\mathbb {R}}}^{>0}\) and \({\mathfrak {C}}^{\ge 0}(W)={{\text {Herm}}}_{V_n/W^\perp \otimes _{\mathbb {Q}}{\mathbb {R}}}^{\ge 0}\). We will also write \({\mathfrak {C}}^{(m),>0}(W)\) (resp. \({\mathfrak {C}}^{(m),\ge 0}(W)\)) for the pre-image of \({\mathfrak {C}}^{>0}(W)\) (resp. \({\mathfrak {C}}^{\ge 0}(W)\)) in \({\mathfrak {C}}^{(m)}(W)\). Moreover we will set
and \({\mathfrak {C}}^{\succ 0}(W)={\mathfrak {C}}^{(0),\succ 0}(W)\). Thus
Note that the natural map \({\mathfrak {C}}^{(m)}(W) \twoheadrightarrow {\mathfrak {C}}(W)\) gives rise to a surjection
and that the pre-image of a point in \({\mathfrak {C}}^{>0}(W')\) is \({{\text {Hom}}}_F(F^m,W')\) (and in particular the pre-image of (0) is (0)). Also note that if \(W' \subset W\) then there is a closed embedding
Finally note that the action of \(G_n^{(m)}({\mathbb {Q}})\) takes \({\mathfrak {C}}^{(m),\succ 0}(W)\) (resp. \({\mathfrak {C}}^{(m),> 0}(W)\), resp. \({\mathfrak {C}}^{(m),\ge 0}(W)\)) to \({\mathfrak {C}}^{(m),\succ 0}(gW)\) (resp. \({\mathfrak {C}}^{(m),> 0}(gW)\), resp. \({\mathfrak {C}}^{(m),\ge 0}(gW)\)).
Note that \(L_{n,(i)}^{(m)}({\mathbb {R}})\) acts on
and preserves
Moreover \(L_{n,(i)}^{(m)}({\mathbb {Q}})\) preserves
In fact, \(L_{n,(i)}^{(m)}({\mathbb {R}})\) acts transitively on \(\pi _0(L_{n,(i),{{\text {herm}}}}({\mathbb {R}})) \times {\mathfrak {C}}^{(m),>0}(V_{n,(i)})\). For this paragraph let \((\,\,\, ,\,\,\,)_0\in {\mathfrak {C}}^{>0}(V_{n,(i)})\) denote the pairing on \((V_n/V_{n,(i)}^\perp \otimes _{\mathbb {Q}}{\mathbb {R}})^2\) induced by \(\langle J_n\,\,\, ,\,\,\,\rangle _n\). Then the stabilizer of \(1 \times ((\,\,\, ,\,\,\,)_0,0)\) in \(L_{n,(i)}^{(m)}({\mathbb {R}})\) is
Thus we get an \(L_{n,(i)}^{(m)}({\mathbb {R}})\)-equivariant identification
We define \({\mathfrak {C}}^{(m)}\) to be the topological space
where \(\sim \) is the equivalence relation generated by the identification of \({\mathfrak {C}}^{(m),\succ 0}(W')\) with its image in \({\mathfrak {C}}^{(m),\succ 0}(W)\) whenever \(W' \subset W\). (This is sometimes referred to as the ‘conical complex’.) Thus as a set
We will let \({\mathfrak {C}}^{(m)}_{=i}\) denote
Note that \({\mathfrak {C}}^{(m)}_{=n}\) is a dense open subset of \({\mathfrak {C}}^{(m)}\). If \(m=0\) we drop it from the notation.
The space \({\mathfrak {C}}^{(m)}\) has a natural, continuous, left action of \(G^{(m)}_n({\mathbb {Q}}) \times {\mathbb {R}}^\times _{>0}\). (The second factor acts on each \({\mathfrak {C}}^{(m),\succ 0}(W)\) by scalar multiplication.)
We have homeomorphisms
(Use the fact, strong approximation for unipotent groups, that
for any open compact subgroup V of \(N_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )\).) If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )\) and if \(g^{-1}Ug \subset U'\) then the right translation map
corresponds to the coproduct of the right translation maps
where \(hg=g'h'u'\) with \(g' \in P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )\) and \(u' \in U'\).
When considering compactifications of just the ordinary locus we will need a variant of the above discussion.
We set
to be the subset of \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\) consisting of elements \((g,\delta ,x)\) such that for some W we have
and
It has a left action of \(G_n^{(m)}({\mathbb {Q}})\) and a right action of \(G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\times {\mathbb {R}}^\times _{>0}\). We define
similarly. We also set
(resp.
to be the image of \((G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)})^{{\text {ord}}}\) in
(resp.
Then as a set
In the case \(i=n\) we have a simpler description of
Lemma 1.4
Proof
There is a natural surjection. We must check that it is also injective. The right hand side equals
which is clearly isomorphic to the left hand side. \(\square \)
There does not seem to be such a simple description of
for \(i \ne n\). However the interested reader can see the end of this section for a partial result, with a very unpleasant proof.
We set
If U is a neat open compact subgroup of \(L_{n,(i)}^{(m)}({\mathbb {A}}^\infty )\), set
Corollary 1.5
If Y is a locally compact, Hausdorff topological space then we write \(H^i_{{\text {Int}}}(Y,{\mathbb {C}})\) for the image of
We define
a smooth \(G^{(m)}_n({\mathbb {A}}^\infty )^{{\text {ord}}}\)-module, and
a smooth \(L_{n,(n)}^{(m)}({\mathbb {A}}^\infty )\)-module. Note that
as N runs over positive integers and \(U^p\) runs over neat open compact subgroups of \(L_{n,(n)}^{(m)}({\mathbb {A}}^{\infty ,p})\). With these definitions we have the following corollary.
Corollary 1.6
There is a \(G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\)-equivariant isomorphism
Interior cohomology has the following property which will be key for us.
Lemma 1.7
Suppose that G is a locally compact, totally disconnected topological group. Suppose that for any sufficiently small open compact subgroup \(U \subset G\) we are given a compact Hausdorff space \(Z_U\) and an open subset \(Y_U \subset Z_U\). Suppose moreover that whenever U, \(U'\) are sufficiently small open compact subgroups of G and \(g \in G\) with \(g^{-1} U g \subset U'\), then there is a proper continuous map
with \(g Y_U \subset Y_{U'}\). Also suppose that \(g \circ h=hg\) whenever these maps are all defined and that if \(g\in U\) then the map \(g: Z_U \rightarrow Z_U\) is the identity.
If \(\varOmega \) is a field, set
and
These are both smooth G-modules. Moreover \(H^i_{{\text {Int}}}(Y,\varOmega )\) is a subquotient of \(H^i(Z,\varOmega )\) as G-modules.
Proof
Note that the diagram
is commutative. Set
and
Then
are G-invariant subspaces with
\(\square \)
We finish this section with our promised generalization of Lemma 1.4. This generalization is not needed for the proofs of the main results of this paper, but we include it for completeness sake. The reader may wish to skip the proof.
Lemma 1.8
There is a natural homeomorphism
where \(U^p(N_1) \subset G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\).
In particular
and
are independent of \(N_2 \ge N_1\).
Proof
Firstly we have that
We can replace the second \(P_{n,(i)}^{(m),+}({\mathbb {Q}})\) by \(P_{n,(i)}^{(m),+}({\mathbb {Q}}_p)\), and then, using in particular the Iwasawa decomposition for \(L_{n,(n)}({\mathbb {Q}}_p)\), replace \(P_{n,(n)}^{(m),+}({\mathbb {Q}}_p)\) by \(P_{n,(n)}^{(m),+}({\mathbb {Z}}_p)\). Next we can replace \(P_{n,(i)}^{(m),+}({\mathbb {Q}}_p)\) by \(P_{n,(i)}^{(m),+}({\mathbb {Z}}_p)\) as long as we also replace \(P_{n,(i)}^{(m),+}({\mathbb {Q}})\) by \(P_{n,(i)}^{(m),+}({\mathbb {Z}}_{(p)})\). This gives
Note that
[This follows from the fact that primes above p of \(F^+\) are unramified in F, which implies that
Thus
and
Moreover, by strong approximation, \(P_{n,(i)}^{(m),-}({\mathbb {Z}}_{(p)})\) (resp. \(L_{n,(i),{{\text {herm}}}}^-({\mathbb {Z}}_{(p)})\)) is dense in \(P_{n,(i)}^{(m),-}({\mathbb {A}}^{\infty ,p}\times {\mathbb {Z}}_{p})\) (resp. \(L_{n,(i),{{\text {herm}}}}^-({\mathbb {A}}^{\infty ,p}\times {\mathbb {Z}}_{p})\)). Thus
Next we claim that the natural map
is an isomorphism. It suffices to check this modulo \(p^{N_2}\), where the map becomes
which is clearly an isomorphism. Thus we have
where \(\gamma \in L_{n,(i)}^{(m)}({\mathbb {Z}}_{(p)})\) acts on \((P_{n,(i)}^{(m),-} \cap P_{n,(n)}^{(m),+}) ({\mathbb {Z}}_{p}) \backslash P_{n,(n)}^{(m),+}({\mathbb {Z}}_p)\) via an element of \(P_{n-i,(n-i)}^+({\mathbb {Z}}_p) \times L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_p)\) with the same image in \(C_{n-i}({\mathbb {Z}}_p) \times L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_p)\).
Note that
Also note that, if we set \(U_p=(U_p(N_1)_{n,(n)}^{(m)}{\mathbb {Z}}_p^\times N_{n,(n)}^{(m)}({\mathbb {Z}}_p))\), then
However as the primes above p split in \(F^+\) split in F we see that
and so
Thus we see that
As \(L_{n,(i),{{\text {herm}}}}^{-}({\mathbb {Z}}_{(p)})\) acts trivially on
and is dense in \(L_{n,(i),{{\text {herm}}}}^{-}({\mathbb {A}}^{\infty ,p})\), we further see that
as desired. \(\square \)
Abusing notation slightly, we will write
for
and
for
2.5 Locally symmetric spaces
In this section we will calculate \(H^i_{{\text {Int}}}({\mathfrak {T}}_{(n)}^{(m)},\overline{{{\mathbb {Q}}}}_p)\) in terms of automorphic forms on \(L_{n,(n)}({\mathbb {A}})\). Our main result will be the following, which will be an immediate consequence of Corollary 1.12 and Lemma 1.13 below.
Corollary 1.9
Suppose that \(n>1\) and that \(\rho \) is an irreducible algebraic representation of \(L_{n,(n),{{\text {lin}}}}\) on a finite dimensional \({\mathbb {C}}\)-vector space. Suppose also that \(\pi \) is a cuspidal automorphic representation of \(L_{n,(n),{{\text {lin}}}}({\mathbb {A}})\) such that \(\pi _\infty \) has the same infinitesimal character as \(\rho ^\vee \) and that \(\psi \) is a continuous character of \({\mathbb {Q}}^\times \backslash {\mathbb {A}}^\times / {\mathbb {R}}^\times _{>0}\). Then for all sufficiently large integers N there are integers \(m(N) \in {\mathbb {Z}}_{\ge 0}\) and \(i(N) \in {\mathbb {Z}}_{>0}\), and an \(L_{n,(n)}({\mathbb {A}}^\infty )\)-equivariant embedding
If \(m=0\) we will write \({\mathfrak {T}}_{(n)}\) for \({\mathfrak {T}}^{(0)}_{(n)}\). Let \(\varOmega \) denote an algebraically closed field of characteristic 0. If \(\rho \) is a finite dimensional algebraic representation of \(L_{n,(n)}\) on a \(\varOmega \)-vector space \(W_\rho \) then we define a locally constant sheaf \({\mathcal {L}}_{\rho ,U}/{\mathfrak {T}}_{(n),U}\) as
The system of sheaves \({\mathcal {L}}_{\rho ,U}\) has a right action of \(L_{n,(n)}({\mathbb {A}}^\infty )\). We define
smooth \(L_{n,(n)}({\mathbb {A}}^\infty )\)-module. Note that if \(\rho \) has a central character \(\chi _\rho \) then,
acts on \(H^i_{{\text {Int}}}({\mathfrak {T}}_{(n)},{\mathcal {L}}_\rho )\) via \(\chi _\rho (\alpha )^{-1}\). (Use the fact that \(Z(L_{n,(n)})({\mathbb {Q}})^+ \subset (L_{n,(n)}({\mathbb {R}}) \cap U_{n,\infty }^0)A_{n,(n)}({\mathbb {R}})^0\).)
The natural map \(L_{n,(n)}^{(m)} \rightarrow L_{n,(n)}\) gives rise to continuous maps
compatible with the action of \(L_{n,(n)}^{(m)}({\mathbb {A}}^\infty )\).
Lemma 1.10
-
(1)
The maps \(\pi ^{(m)}\) are real-torus bundles (i.e. \((S^1)^r\)-bundles for some r), and in particular are proper maps.
-
(2)
There are \(L^{(m)}_{n,(n)}({\mathbb {A}}^\infty )\)-equivariant identifications
$$\begin{aligned} R^i\pi _{*}^{(m)} \varOmega \cong {\mathcal {L}}_{\wedge ^i \left( \bigoplus _{\tau : F \hookrightarrow \varOmega } {{\text {Std}}}_\tau ^{\oplus m}\right) ^\vee }. \end{aligned}$$In particular the action of \(L^{(m)}_{n,(n)}({\mathbb {A}}^\infty )\) on the relative cohomology sheaf \(R^i\pi _{*}^{(m)} \varOmega \) factors through \(L_{n,(n)}({\mathbb {A}}^\infty )\).
Proof
Recall that
Suppose that U is a neat open compact subgroup of \(L^{(m)}_{n,(n)}({\mathbb {A}}^\infty )\) with image \(U'\) in \(L_{n,(n)}({\mathbb {A}}^\infty )\). Then \(L_{n,(n)}({\mathbb {Q}}) \times U'\) acts freely on
Thus it suffices to prove that the map \({\widetilde{\pi }} ^{(m)}\)
is a real-torus bundle and that there are \(L_{n,(n)}({\mathbb {Q}}) \times L^{(m)}_{n,(n)}({\mathbb {A}}^\infty ) \)-equivariant isomorphisms
Using the identification of spaces (but not of groups) that comes from the group product
we see that \({\widetilde{\pi }} ^{(m)}\) can be identified with the map
or, using the equality \(N(L_{n,(n),{{\text {lin}}}}^{(m)})({\mathbb {A}}^\infty )=N(L_{n,(n),{{\text {lin}}}}^{(m)})({\mathbb {Q}}) (U \cap N(L_{n,(n),{{\text {lin}}}}^{(m)})({\mathbb {A}}^\infty ))\), even with
The right \(L_{n,(n)}^{(m)}({\mathbb {A}}^\infty )\)-action is by right translation on the second factor. The left action of \(L_{n,(n)}({\mathbb {Q}})\) is via conjugation on the first factor and left translation on the second.
The first part of the lemma follows, and we see that
is \(L_{n,(n)}({\mathbb {Q}}) \times L^{(m)}_{n,(n)}({\mathbb {A}}^\infty )\) equivariantly identified with the locally constant sheaf
The lemma follows. \(\square \)
Lemma 1.11
There is an \(L_{n,(n)}^{(m)}({\mathbb {A}}^\infty )\)-equivariant isomorphism
Proof
There is an \(L_{n,(n)}^{(m)}({\mathbb {A}}^\infty )\)-equivariant spectral sequence
If \(\alpha \in {\mathbb {Q}}^\times _{>0} \subset Z(L_{n,(n),{{\text {lin}}}})({\mathbb {A}}^\infty )\), then \(\alpha \) acts on \(E_{2}^{i,j}\) via \(\alpha ^j\). We deduce that all the differentials (on the second and any later page) vanish, and so the spectral sequence degenerates on the second page. Moreover the \(\alpha \mapsto \alpha ^j\) eigenspace in \(H^{i+j}({\mathfrak {T}}^{(m)}_{(n)},\varOmega )\) is naturally identified with \(H^i({\mathfrak {T}}_{(n)}, {\mathcal {L}}_{\wedge ^j(\bigoplus _\tau {{\text {Std}}}_\tau ^{\oplus m})^\vee })\). (This standard argument is sometimes referred to as ‘Lieberman’s trick’.)
As the maps \(\pi ^{(m)}\) are proper, there is also an \(L_{n,(n)}^{(m)}({\mathbb {A}}^\infty )\)-equivariant spectral sequence
and \(\alpha \in {\mathbb {Q}}^\times _{>0} \subset Z(L_{n,(n),{{\text {lin}}}})({\mathbb {A}}^\infty )\) acts on \(E_{c,2}^{i,j}\) via \(\alpha ^j\). Again we see that the spectral sequence degenerates on the second page and that the \(\alpha \mapsto \alpha ^j\) eigenspace in \(H^{i+j}_c({\mathfrak {T}}^{(m)}_{(n)},\varOmega )\) is naturally identified with \(H^i_c({\mathfrak {T}}_{(n)}, {\mathcal {L}}_{\wedge ^j(\bigoplus _\tau {{\text {Std}}}_\tau ^{\oplus m})^\vee })\).
The lemma follows. \(\square \)
Corollary 1.12
Suppose that \(\rho \) is an irreducible representation of \(L_{n,(n),{{\text {lin}}}}\) over \(\varOmega \), which we extend to a representation of \(L_{n,(n)}\) by letting it be trivial on \(L_{n,(n),{{\text {herm}}}}\). Let \(d={\text{ N }}_{F/{\mathbb {Q}}}\circ \det : L_{n,(n),{{\text {lin}}}} \rightarrow {\mathbb {G}}_m\). Then for all N sufficiently large there are \(j(N),\, m(N) \in {\mathbb {Z}}_{\ge 0}\) such that, for all i,
is an \(L_{n,(n)}({\mathbb {A}}^\infty )\)-direct summand of
Proof
It follows from Weyl’s construction of the irreducible representations of \(GL_n\) that, for N sufficiently large, \(\rho \otimes d^{-N}\) is a direct summand of
for certain non-negative integers \(m_\tau (N)\). Hence for N sufficiently large and \(m(N)=\max \{m_\tau (N)\}\) the representation \(\rho \otimes d^{-N}\) is also a direct summand of
\(\square \)
Lemma 1.13
Suppose that \(\rho \) is an irreducible algebraic representation of \(L_{n,(n)}\) on a finite dimensional \({\mathbb {C}}\)-vector space.
-
(1)
Then
$$\bigoplus _\varPi \varPi ^\infty \otimes H^i \left( {{\text {Lie}}\,}L_{n,(n)}, \left( L_{n,(n)}({\mathbb {R}}) \cap U_{n,\infty }^0\right) A_{n,(n)}({\mathbb {R}})^0, \varPi _\infty \otimes \rho \right) \hookrightarrow H^i_{{\text {Int}}}({\mathfrak {T}}_{(n)}, {\mathcal {L}}_\rho ),$$where \(\varPi \) runs over cuspidal automorphic representations of \(L_{n,(n)}({\mathbb {A}})\).
-
(2)
If \(n>1\) and if \(\varPi \) is a cuspidal automorphic representation of \(L_{n,(n)}({\mathbb {A}})\) such that \(\varPi _\infty \) has the same infinitesimal character as \(\rho ^\vee \), then
$$H^i\left( {{\text {Lie}}\,}L_{n,(n)}, \left( L_{n,(n)}({\mathbb {R}}) \cap U_{n,\infty }^0\right) A_{n,(n)}({\mathbb {R}})^0, \varPi _\infty \otimes \rho \right) \ne (0)$$for some \(i>0\).
Proof
The first part results from [13], more precisely from combining theorem 5.2, the discussion in section 5.4 and corollary 5.5 of that paper. The second part results from [20], see the proof of theorem 3.13, and in particular lemma 3.14, of that paper. \(\square \)
We are now in a position to deduce Corollary 1.9, which we stated at the start of this section.
3 Tori, torsors and torus embeddings
The main aim of this section is to recall some basic facts about relative torus embeddings of tori torsors, which will provide local models for the boundary of toroidal compactifications of Shimura and Kuga–Sato varieties. Much of this material is in some sense standard, but we need to work with infinite cone decompositions, which are not treated in much of the literature. It will also be convenient to use a notation which emphasizes the boundary of the torus embedding and the completion of the torus embedding along the boundary. These seem to be more naturally parameterized by certain partial fans rather than fans. In Sect. 2.4 we compute certain cohomology groups. For finite fans (or partial fans) such results are fairly standard, but we found it quite tricky to formulate and prove the results we need in the presence of infinitely many cones. Maybe this is just our incompetence.
Throughout this section let \(R_0\) denote an irreducible noetherian ring (i.e. a noetherian ring with a unique minimal prime ideal). In the applications of this section elsewhere in this paper it will be either \({\mathbb {Q}}\) or \({\mathbb {Z}}_{(p)}\) or \({\mathbb {Z}}/p^r{\mathbb {Z}}\) for some r. We will consider \(R_0\) endowed with the discrete topology so that \({{\text {Spf}}\,}R_0 \cong {{\text {Spec}}\,}R_0\).
3.1 Tori and torsors
If S / Y is a torus (i.e. a group scheme etale locally on Y isomorphic to \({\mathbb {G}}_m^N\) for some N) then we can define its sheaf of characters \(X^*(S)={{\text {Hom}}}(S,{\mathbb {G}}_m)\) and its sheaf of cocharacters \(X_*(S)={{\text {Hom}}}({\mathbb {G}}_m,S)\). These are locally constant sheaves of free \({\mathbb {Z}}\)-modules in the etale topology on Y. They are naturally \({\mathbb {Z}}\)-dual to each other. More generally if \(S_1/Y\) and \(S_2/Y\) are two tori then \({{\text {Hom}}}(S_1,S_2)\) is a locally constant sheaf of free \({\mathbb {Z}}\)-modules in the etale topology on Y. In fact,
By a quasi-isogeny (resp. isogeny) from \(S_1\) to \(S_2\) we shall mean a global section of the sheaf \({{\text {Hom}}}(S_1,S_2)_{\mathbb {Q}}\) (resp. \({{\text {Hom}}}(S_1,S_2)\)) with an inverse in \({{\text {Hom}}}(S_2,S_1)_{\mathbb {Q}}\). We will write \([S]_{{\text {isog}}}\) for the category whose objects are tori over Y quasi-isogenous to S and whose morphisms are isogenies. The sheaves \(X_*(S)_{\mathbb {Q}}\) and \(X^*(S)_{\mathbb {Q}}\) only depend on the quasi-isogeny class of S so we will write \(X_*([S]_{{\text {isog}}})_{\mathbb {Q}}\) and \(X^*([S]_{{\text {isog}}})_{\mathbb {Q}}\).
If \({\overline{{y}}}\) is a geometric point of Y then we define
and
with the transition map from MN to N being multiplication by M. (The Tate modules of S.) Also define
and
If Y is a scheme over \({{\text {Spec}}\,}{\mathbb {Q}}\) then
If Y is a scheme over \({{\text {Spec}}\,}{\mathbb {Z}}_{(p)}\) then
Now suppose that S is split, i.e. isomorphic to \({\mathbb {G}}_m^N\) for some N. By an S-torsor T / Y we mean a scheme T / Y with an action of S, which locally in the Zariski topology on Y is isomorphic to S. By a rigidification of T along \(e: Y' \rightarrow Y\) we mean an isomorphism of S-torsors \(e^* T \cong S\) over \(Y'\). If U is a connected open subset of Y then
where \({\mathcal {L}}_T(\chi )\) is a line bundle on U. If Z is any open subset of Y and if \(\chi \in X^*(S)(Z)\) then there is a unique line bundle \({\mathcal {L}}_T(\chi )\) on Z whose restriction to any connected open subset \(U \subset Z\) is \({\mathcal {L}}_T(\chi |_U)\). Multiplication gives isomorphisms
The map
gives a bijection between isomorphism classes of \({\mathbb {G}}_m\)-torsors and isomorphism classes of line bundles on Y. The inverse map sends \({\mathcal {L}}\) to
If \(\alpha : S \rightarrow S'\) is a morphism of split tori and if T / Y is an S-torsor we can form a push-out \(\alpha _*T\), an \(S'\) torsor on Y defined as the quotient
where S acts by
There is a natural map \(T \rightarrow \alpha _* T\) compatible with \(\alpha : S \rightarrow S'\). If \(\alpha \) is an isogeny then
If \(T_1\) and \(T_2\) are S-torsors over Y we define
to be the S-torsor
where S acts by
If T is an S-torsor on Y we define an S-torsor \(T^\vee /Y\) by taking \(T^\vee =T\) as schemes but defining an S action . on \(T^\vee \) by
i.e. \(T^\vee =[-1]_{S,*} T\). Then
via the map that sends \((t_1,t_2)\) to the unique section s of S with \(st_1=t_2\).
3.2 Log structures
We will call a formal scheme
suitable if it has a cover by affine opens \({\mathfrak {U}}_i={{\text {Spf}}\,}(A_i)^\wedge _{I_i}\), where \(A_i\) is a finitely generated \(R_0\)-algebra and \(I_i\) is an ideal of \(A_i\) whose inverse image in \(R_0\) is (0).
By a log structure on a scheme X (resp. formal scheme \({\mathfrak {X}}\)) we mean a sheaf of monoids \({\mathcal {M}}\) on X (resp. \({\mathfrak {X}}\)) together with a morphism
(resp.
such that the induced map
(resp.
is an isomorphism. We will refer to a scheme (resp. formal scheme) endowed with a log structure as a log scheme (resp. log formal scheme). By a morphism of log schemes (resp. morphism of log formal schemes)
(resp.
we shall mean a morphism \(\phi : X \rightarrow Y\) (resp. \(\phi : {\mathfrak {X}}\rightarrow {\mathfrak {Y}}\)) and a map
such that \(\phi ^* \circ \phi ^{-1}(\beta )=\alpha \circ \psi \). We will consider \(R_0\) endowed with the trivial log structure \(({\mathcal {O}}_{{{\text {Spec}}\,}R_0}^\times , 1)\) (resp. \(({\mathcal {O}}_{{{\text {Spf}}\,}R_0}^\times , 1)\)). We will call a log formal scheme \(({\mathfrak {X}},{\mathcal {M}},\alpha )/{{\text {Spf}}\,}R_0\) suitable if \({\mathfrak {X}}/{{\text {Spf}}\,}R_0\) is suitable and if, locally in the Zariski topology, \({\mathcal {M}}/\alpha ^{-1} {\mathcal {O}}_{\mathfrak {X}}^\times \) is finitely generated. (In the case of schemes these definitions are well known: See, for example, [34]. We have not attempted to optimize the definition in the case of formal schemes. We are simply making a definition which works for the limited purposes of this article. The reader might like to compare our definitions with those in [5].)
If D is a closed subscheme of X we define a log structure \({\mathcal {M}}(D)\) on X by setting
If \(X/{{\text {Spec}}\,}R_0\) is a scheme of finite type and if \(Z \subset X\) is a closed subscheme which is flat over \({{\text {Spec}}\,}R_0\), then the formal completion \(X_Z^\wedge \) is a suitable formal scheme. Let \(i^\wedge \) denote the map of ringed spaces \(X_Z^\wedge \rightarrow X\). If \(({\mathcal {M}},\alpha )\) is a log structure on X, then we get a map
It induces a log structure \(({\mathcal {M}}^\wedge ,\alpha ^\wedge )\) on \(X_Z^\wedge \), where \({\mathcal {M}}^\wedge \) denotes the push-out
If
is a morphism of schemes with log structures over \({{\text {Spec}}\,}R_0\) then there is a right exact sequence
of sheaves of log differentials. If the map \((\phi ,\psi )\) is log smooth then this sequence is also left exact and the sheaf \(\varOmega ^1_{X/Y}(\log {\mathcal {M}}/{\mathcal {N}})\) is locally free. (See, e.g., [34].) As usual, we write \(\varOmega ^i_X(\log {\mathcal {M}})=\wedge ^i \varOmega ^1_X(\log {\mathcal {M}})\) and \(\varOmega ^i_{X/Y}(\log {\mathcal {M}}/{\mathcal {N}})=\wedge ^i\varOmega ^1_{X/Y}(\log {\mathcal {M}}/{\mathcal {N}})\).
By a coherent sheaf of differentials on a formal scheme \({\mathfrak {X}}/{{\text {Spf}}\,}R_0\) we will mean a coherent sheaf \(\varOmega /{\mathfrak {X}}\) together with a differential \(d: {\mathcal {O}}_{\mathfrak {X}}\rightarrow \varOmega \) which vanishes on \(R_0\). By a coherent sheaf of log differentials on a log formal scheme \(({\mathfrak {X}},{\mathcal {M}},\alpha )/{{\text {Spf}}\,}R_0\) we shall mean a coherent sheaf \(\varOmega /{\mathfrak {X}}\) together with a differential, which vanishes on \(R_0\),
and a homomorphism
such that
By a universal coherent sheaf of differentials (resp. universal coherent sheaf of log differentials) we shall mean a coherent sheaf of differentials \((\varOmega ,d)\) (resp. a coherent sheaf of log differentials \((\varOmega ,d,{{\text {dlog}}})\)) such that for any other coherent sheaf of differentials \((\varOmega ',d')\) (resp. a coherent sheaf of log differentials \((\varOmega ',d',{{\text {dlog}}}')\)) there is a unique map \(f: \varOmega \rightarrow \varOmega '\) such that \(f \circ d=d'\) (resp. \(f \circ d=d'\) and \(f \circ {{\text {dlog}}}={{\text {dlog}}}'\)).
Note that if a universal coherent sheaf of differentials (resp. universal coherent sheaf of log differentials) exists, it is unique up to unique isomorphism.
Lemma 2.1
Suppose that \(R_0\) is a discrete, noetherian topological ring.
-
(1)
A universal sheaf of coherent differentials \(\varOmega ^1_{{\mathfrak {X}}/{{\text {Spf}}\,}R_0}\) exists for any suitable formal scheme \({\mathfrak {X}}/{{\text {Spf}}\,}R_0\).
-
(2)
If \(X/{{\text {Spec}}\,}R_0\) is a scheme of finite type and if \(Z \subset X\) is flat over \(R_0\) then
$$\varOmega _{X_Z^\wedge /{{\text {Spf}}\,}R_0}^1 \cong \left( \varOmega ^1_{X/{{\text {Spec}}\,}R_0}\right) ^\wedge .$$ -
(3)
A universal sheaf of coherent log differentials \(\varOmega ^1_{{\mathfrak {X}}/{{\text {Spf}}\,}R_0}(\log {\mathcal {M}})\) exists for any suitable log formal scheme \(({\mathfrak {X}},{\mathcal {M}},\alpha )/{{\text {Spf}}\,}R_0\).
-
(4)
Suppose that \(X/{{\text {Spec}}\,}R_0\) is a scheme of finite type, that \(Z \subset X\) is flat over \(R_0\) and that \(({\mathcal {M}},\alpha )\) is a log structure on X such that Zariski locally \({\mathcal {M}}/\alpha ^{-1} {\mathcal {O}}_X^\times \) is finitely generated. Then
$$\varOmega _{X_Z^\wedge /{{\text {Spf}}\,}R_0}^1(\log {\mathcal {M}}^\wedge ) \cong \left( \varOmega ^1_{X/{{\text {Spec}}\,}R_0}(\log {\mathcal {M}})\right) ^\wedge .$$
Proof
Consider the first part. Suppose that \({\mathfrak {U}}={{\text {Spf}}\,}A_I^\wedge \) is an affine open in \({\mathfrak {X}}\), where A is a finitely generated \(R_0\)-algebra and I is an ideal of A with inverse image (0) in \(R_0\). Then there exists a universal finite module of differentials \(\varOmega ^1_{{\mathfrak {U}}}\) for \({\mathfrak {U}}\), namely the coherent sheaf of \({\mathcal {O}}_{\mathfrak {U}}\)-modules associated to \((\varOmega ^1_{A/R_0})_I^\wedge \). (See sections 11.5 and 12.6 of [40].) We must show that if \({\mathfrak {U}}' \subset {\mathfrak {U}}\) is open then \(\varOmega ^1_{{\mathfrak {U}}}|_{{\mathfrak {U}}'}\) is a universal finite module of differentials for \({\mathfrak {U}}'\). For then uniqueness will allow us to glue the coherent sheaves \(\varOmega ^1_{{\mathfrak {U}}}\) to form \(\varOmega ^1_{{\mathfrak {X}}}\).
So suppose that \((\varOmega ',d')\) is a finite module of differentials for \({\mathfrak {U}}'\). We must show that there is a unique map of \({\mathcal {O}}_{{\mathfrak {U}}'}\)-modules
such that \(d'=f \circ d\). We may cover \({\mathfrak {U}}'\) by affine opens of the form \({{\text {Spf}}\,}(A_g)^\wedge _I\) and it will suffice to find, for each g, a unique
with \(d'=f_g \circ d\). Thus we may assume that \({\mathfrak {U}}'={{\text {Spf}}\,}(A_g)^\wedge _I\). But in this case we know \(\varOmega ^1_{{\mathfrak {U}}'}\) exists, and is the coherent sheaf associated to
On the other hand \(\varOmega ^1_{{\mathfrak {U}}}|_{{\mathfrak {U}}'}\) is the coherent sheaf associated to
Thus
and the first part follows. The second part also follows from the proof of the first part.
For the third part, because of uniqueness, it suffices to work locally. Thus we may assume that there are finitely many sections \(m_1,\ldots ,m_r \in {\mathcal {M}}({\mathfrak {X}})\), which together with \(\alpha ^{-1} {\mathcal {O}}_{\mathfrak {X}}^\times \) generate \({\mathcal {M}}\). Then we define \(\varOmega ^1_{({\mathfrak {X}},{\mathcal {M}},\alpha )}\) to be the cokernel of the map
It is elementary to check that this has the desired universal property. The fourth part is also elementary to check. \(\square \)
If
is a map of suitable log formal schemes over \({{\text {Spf}}\,}R_0\) then we set
We also set
and
and
Corollary 2.2
Suppose that \(R_0\) is a discrete, noetherian topological ring; that
is a map of log schemes over \({{\text {Spec}}\,}R_0\); and that \(Z \subset X\) and \(W \subset Y\) are closed subschemes flat over \({{\text {Spec}}\,}R_0\) which map to each other under \(X \rightarrow Y\). Suppose moreover that X and Y have finite type over \({{\text {Spec}}\,}R_0\) and that \({\mathcal {M}}/\alpha ^{-1} {\mathcal {O}}_{\mathfrak {X}}^\times \) and \({\mathcal {N}}/\beta ^{-1} {\mathcal {O}}_{\mathfrak {Y}}^\times \) are locally (in the Zariski topology) finitely generated. Then
Proof
This follows from the lemma and from the exactness of completion. \(\square \)
If Y is a scheme we will let
denote affine n-space over Y and
denote the union of the coordinate hyperplanes in \({{\text {Aff}}}^n_Y\). Now suppose that \(X \rightarrow Y\) is a smooth map of schemes of relative dimension n. By a simple normal crossings divisor in X relative to Y we shall mean a closed subscheme \(D \subset X\) such that X has an affine Zariski-open cover \(\{ U_i\}\) such that each \(U_i\) admits an etale map \(f_i: U_i \rightarrow {{\text {Aff}}}^n_Y\) so that \(D|_{U_i}\) is the (scheme-theoretic) pre-image of \({{\text {Coord}}}^n_Y\). In the case that Y is just the spectrum of a field we will refer simply to a simple normal crossings divisor in X.
Suppose that Y is locally noetherian and separated and that the connected components of Y are irreducible. If S is a finite set of irreducible components of D we will set
It is smooth over Y. We will also set
If E is an irreducible component of \(D^{(s)}\) then the set S(E) of irreducible components of D containing E has cardinality s. If \(\ge \) is a total order on the set of irreducible components of D, we can define a delta set \({\mathcal {S}}(D, \ge )\), or simply \({\mathcal {S}}(D)\), as follows. (For the definition of ‘delta set’, see, for instance, [25]. We can, if we prefer to be more abstract, replace \({\mathcal {S}}(D,\ge )\) by the associated simplicial set.) The n cells consist of all irreducible components of \(D^{(n+1)}\). If E is such an irreducible component and if \(i\in \{0,\ldots ,n\}\) then the image of E under the face map \(d_i\) is the unique irreducible component of
which contains E. Here \(S(E)_i\) equals S(E) with its \((i+1)\)th smallest element removed. The topological realization \(|{\mathcal {S}}(D,\ge )|\) does not depend on the total order \(\ge \), so we will often write \(|{\mathcal {S}}(D)|\).
We record a general observation about log de Rham complexes and divisors with simple normal crossings, which is probably well known. We include a proof because it is of crucial importance for our argument.
Lemma 2.3
Suppose that Y is a smooth scheme of finite type over a field k and that \(Z \subset Y\) is a divisor with simple normal crossings. Let \(Z_1,\ldots ,Z_m\) denote the distinct irreducible components of Z and set
(in particular \(Z_\emptyset =Y\)), and
Let \(i_S\) (resp. \(i^{(s)}\)) denote the natural maps \(Z_S \rightarrow Y\) (resp. \(Z^{(s)} \rightarrow Y\)). Also let \({\mathcal {I}}_Z\) denote the ideal of definition of Z.
There is a double complex
with maps
and
being the sum of the maps
which are
-
0 if \( S \not \subset S'\),
-
and \((-1)^{\# \{ i \in S: i<j\} }\) times the natural pull-back if \(S \cup \{ j\} =S'\).
The natural inclusions
give rise to a map of complexes
For fixed r the simple complexes
are exact.
Proof
Only the last assertion is not immediate. So consider the last assertion. We can work Zariski locally, so we may assume that the complex is pulled back from the corresponding complex for the case \(Y={{\text {Spec}}\,}k[X_1,\ldots ,X_d]\) and Z is given by \(X_1X_2\ldots X_m=0\). In this case we take \(Z_j\) to be the scheme \(X_j=0\), for \(j=1,\ldots ,m\). In this case
where T runs over r element subsets of \(\{1,\ldots ,d\}\). On the other hand
where T runs over r element subsets of \(\{ 1,\ldots ,d\}-S\). Thus it suffices to show that, for each subset \(T \subset \{ 1,\ldots ,d\}\) the sequence
is exact, where \(S \subset \{ 1,\ldots ,m\}\). The sequence for \(T \subset \{1,\ldots ,d\}\) is obtained from the sequence for \(\emptyset \subset \{1,\ldots ,m\}-T\) by tensoring over k with \(k[X_j]_{j \in T \cup \{ m+1,\ldots ,d\} }\), and so we only need treat the case \(m=d\) and \(T=\emptyset \).
If \(\mu \) is a monomial in the variables \(X_1,\ldots ,X_m\), let \(R(\mu )\) denote the subset of \(\{ 1,\ldots ,m\}\) consisting of the indices j for which \(X_j\) does not occur in \(\mu \). Then our complex is the direct sum over \(\mu \) of the complexes
where \(A_\mu =k\) if \(R(\mu )=\emptyset \) and \(=(0)\) otherwise. So it suffices to prove this latter complex exact for all \(\mu \). If \(R(\mu )=\emptyset \) then it becomes
which is clearly exact. If \(R(\mu ) \ne \emptyset \), our complex becomes
If we suppress the first k, this is the complex that computes the simplicial cohomology with k-coefficients of the simplex with \(\# R(\mu )\) vertices. Thus it is exact everywhere except \(\bigoplus _{S \subset R(\mu ),\,\, \# S=1} k\) and the kernel of
is just k. The desired exactness follows. \(\square \)
3.3 Torus embeddings
We will now discuss relative torus embeddings, crucially in the context of infinite fans.
If W is a real vector space with dual \(W^\vee \) and if \(A \subset W\) is a subspace we set
and
and
We will suppose that \(Y/{{\text {Spec}}\,}R_0\) is flat and locally of finite type. To simplify the notation, for now we will restrict to the case of a split torus S / Y with Y connected. We will record the (trivial) generalization to the case of a disconnected base below. Thus we can think of \(X^*(S)\) and \(X_*(S)\) as abelian groups, rather than as locally constant sheaves on Y, i.e. we replace the sheaf by its global sections over Y. We will let T / Y denote an S-torsor.
By a rational polyhedral cone \(\sigma \subset X_*(S)_{\mathbb {R}}\) we mean a non-empty subset consisting of all \({\mathbb {R}}_{\ge 0}\)-linear combinations of a finite set of elements of \(X_*(S)\), but which contains no complete line through 0. (We include the case \(\sigma =\{0\}\). The notion we define here is sometimes called a ‘non-degenerate rational polyhedral cone’.) By the interior \(\sigma ^0\) of \(\sigma \) we shall mean the complement in \(\sigma \) of all its proper faces. (We consider \(\sigma \) as a face of \(\sigma \), but not a proper face.) We call \(\sigma \) smooth if it consists of all \({\mathbb {R}}_{\ge 0}\)-linear combinations of a subset of a \({\mathbb {Z}}\)-basis of \(X_*(S)\). Note that any face of a smooth cone is smooth. Moreover we set
Then \(T_\sigma \) is a scheme over Y with an action of S and there is a natural S-equivariant dense open embedding \(T \hookrightarrow T_\sigma \). If \(\sigma ' \subset \sigma \) there is a natural map \(T_{\sigma '} \rightarrow T_\sigma \) compatible with the embeddings of T. If \(f: Y' \rightarrow Y\) then \(T_\sigma /Y\) pulls back under f to \((f^*T)_\sigma /Y'\) compatibly with the maps \(T_{\sigma '}\hookrightarrow T_\sigma \) for \(\sigma '\subset \sigma \).
Suppose that \(\varSigma _0\) is a set of faces of \(\sigma \) such that
-
\(\{0\} \not \in \varSigma _0\),
-
and, if \(\tau ' \supset \tau \in \varSigma _0\), then \(\tau ' \in \varSigma _0\).
In this case define
Thus
Then we define \(\partial _{\varSigma _0} T_\sigma \subset T_\sigma \) to be the closed subscheme defined by the sheaf of ideals
If \(\varSigma _0\) contains all the faces of \(\sigma \) other than \(\{0\}\) we will write \(\partial T_\sigma \) for \(\partial _{\varSigma _0} T_\sigma \). Note that \(\partial _\emptyset T_\sigma =\emptyset \). If \(\sigma '\) is a face of \(\sigma \) then under the open embedding
\(\partial _{\varSigma _0}T_\sigma \) pulls back to \(\partial _{\{ \tau \in \varSigma _0: \tau \subset \sigma '\}} T_{\sigma '}\).
By a fan in \(X_*(S)_{\mathbb {R}}\) we shall mean a non-empty collection \(\varSigma \) of rational polyhedral cones \(\sigma \subset X_*(S)_{\mathbb {R}}\) which satisfy
-
if \(\sigma \in \varSigma \), so is each face of \(\sigma \),
-
if \(\sigma , \sigma ' \in \varSigma \) then \(\sigma \cap \sigma '\) is a face of \(\sigma \) and of \(\sigma '\).
Note that unless otherwise stated we will not assume that \(\varSigma \) is finite. We call \(\varSigma \) smooth if each \(\sigma \in \varSigma \) is smooth. We will call \(\varSigma \) full if every element of \(\varSigma \) is contained in an element of \(\varSigma \) with the same dimension as \(X_*(S)_{\mathbb {R}}\). Define
We call \(\varSigma '\) a refinement of \(\varSigma \) if each \(\sigma ' \in \varSigma '\) is a subset of some element of \(\varSigma \) and each element \(\sigma \in \varSigma \) is a finite union of elements of \(\varSigma '\).
Lemma 2.4
-
(1)
If \(\varSigma \) is a fan and \(\varSigma ' \subset \varSigma \) is a finite cardinality subfan then there is a refinement \(\widetilde{\varSigma }\) of \(\varSigma \) with the following properties:
-
any element of \(\varSigma \) which is smooth also lies in \(\widetilde{\varSigma }\);
-
any element of \(\widetilde{\varSigma }\) contained in an element of \(\varSigma '\) is smooth;
-
and if \(\sigma ' \in \varSigma -\widetilde{\varSigma }\) then \(\sigma '\) has a non-smooth face lying in \(\varSigma '\).
-
-
(2)
Any fan \(\varSigma \) has a smooth refinement \(\varSigma '\) such that every smooth cone \(\sigma \in \varSigma \) also lies in \(\varSigma '\).
Proof
The first part is proved just as for finite fans by making a finite series of elementary subdivisions by 1 cones that lie in some element \(\sigma ' \in \varSigma '\) but not in any of its smooth faces. See, for instance, section 2.6 of [26].
For the second part, consider the set \({\mathcal {S}}\) of pairs \((\widetilde{\varSigma },\varDelta )\) where \(\widetilde{\varSigma }\) is a refinement of \(\varSigma \) and \(\varDelta \) is a subfan of \(\varSigma \) such that
-
every smooth element of \(\varSigma \) lies in \(\widetilde{\varSigma }\);
-
and if \(\sigma \in \widetilde{\varSigma }\) is contained in an element of \(\varDelta \) then \(\sigma \) is smooth.
It suffices to show that \({\mathcal {S}}\) contains an element \((\widetilde{\varSigma },\varDelta )\) with \(\varDelta =\varSigma \).
If \((\widetilde{\varSigma },\varDelta ) \in {\mathcal {S}}\) and \(\sigma \in \varSigma \) we define \(\widetilde{\varSigma }(\sigma )\) to be the set of elements of \(\widetilde{\varSigma }\) contained in \(\sigma \). We define a partial order on \({\mathcal {S}}\) by decreeing that \((\widetilde{\varSigma },\varDelta ) \ge (\widetilde{\varSigma }',\varDelta ')\) if and only if the following conditions are satisfied:
-
\(\widetilde{\varSigma }\) refines \(\widetilde{\varSigma }'\);
-
\(\varDelta \supset \varDelta '\);
-
\(\widetilde{\varSigma }'(\sigma ) =\widetilde{\varSigma }(\sigma )\) unless \(\sigma \) has a face that is contained in an element of \(\varDelta \) but in no element of \(\varDelta '\).
Suppose that \({\mathcal {S}}' \subset {\mathcal {S}}\) is totally ordered. Set
and let \(\widetilde{\varSigma }\) denote the set of cones \(\sigma '\) which lie in \(\widetilde{\varSigma }'\) for all sufficiently large elements of \((\widetilde{\varSigma }',\varDelta ') \in {\mathcal {S}}'\). If \(\sigma \in \varSigma \) then we can choose \((\widetilde{\varSigma }',\varDelta ') \in {\mathcal {S}}'\) so that the number of faces of \(\sigma \) in \(\varDelta '\) is maximal. If \((\widetilde{\varSigma }',\varDelta ') \le (\widetilde{\varSigma }'',\varDelta '') \in {\mathcal {S}}'\) then \(\widetilde{\varSigma }'(\sigma )=\widetilde{\varSigma }''(\sigma )\). Thus \(\widetilde{\varSigma }(\sigma )=\widetilde{\varSigma }'(\sigma )\). We conclude that \(\widetilde{\varSigma }\) is a refinement of \(\varSigma \). Thus \((\widetilde{\varSigma },\varDelta ) \in {\mathcal {S}}\) and it is an upper bound for \({\mathcal {S}}'\).
By Zorn’s lemma \({\mathcal {S}}\) has a maximal element \((\widetilde{\varSigma },\varDelta )\). We will show that \(\varDelta =\varSigma \), which will complete the proof of the lemma. Suppose not. Choose \(\sigma \in \varSigma -\varDelta \). Set \(\varDelta '\) to be the union of \(\varDelta \) and the faces of \(\sigma \). Let \(\widetilde{\varSigma }'\) be a refinement of \(\widetilde{\varSigma }\) such that
-
any element of \(\widetilde{\varSigma }\) which is smooth also lies in \(\widetilde{\varSigma }'\);
-
any element of \(\widetilde{\varSigma }'\) contained in \(\sigma \) is smooth;
-
and if \(\sigma ' \in \widetilde{\varSigma }-\widetilde{\varSigma }'\) then \(\sigma '\) has a non-smooth face contained in \(\sigma \).
Then \((\widetilde{\varSigma }',\varDelta ') \in {\mathcal {S}}\) and \((\widetilde{\varSigma }',\varDelta ') > (\widetilde{\varSigma },\varDelta )\), a contradiction. \(\square \)
To a fan \(\varSigma \) one can attach a connected scheme \(T_\varSigma \) that is separated, locally (on \(T_\varSigma \)) of finite type and flat over Y of relative dimension \(\dim _{\mathbb {R}}X_*(S)_{\mathbb {R}}\), together with an action of S and an S-equivariant dense open embedding \(T \hookrightarrow T_\varSigma \) over Y. The scheme \(T_\varSigma \) has an open cover by the \(T_\sigma \) for \(\sigma \in \varSigma \) such that \(T_{\sigma '} \subset T_{\sigma }\) if and only if \(\sigma ' \subset \sigma \). We write \({\mathcal {O}}_{T_\varSigma }\) for the structure sheaf of \(T_\varSigma \). If \(\varSigma \) is smooth then \(T_\varSigma /Y\) is smooth. If \(\varSigma \) is finite and \(|\varSigma |=X_*(S)_{\mathbb {R}}\), then \(T_\varSigma /Y\) is proper. If \(\varSigma ' \subset \varSigma \) then \(T_{\varSigma '}\) can be identified with an open subscheme of \(T_\varSigma \). If \(\varSigma '\) refines \(\varSigma \) then there is an S-equivariant proper map
which restricts to the identity on T: Its restriction to \(T_{\sigma '}\) equals the map
where \(\sigma ' \subset \sigma \in \varSigma \).
By boundary data for \(\varSigma \) we shall mean a proper subset \(\varSigma _0 \subset \varSigma \) such that \(\varSigma - \varSigma _0\) is a fan. (Note that \(\varSigma _0\) may not be closed under taking faces.) If \(\varSigma _0\) is boundary data we define \(\partial _{\varSigma _0}T_\varSigma \) to be the closed subscheme of \(T_\varSigma \) with
Note that
Thus \(\partial _{\varSigma _0} T_\varSigma \) has an open cover by the sets
as \(\sigma \) runs over \(\varSigma _0\). We write \({\mathcal {I}}_{\partial _{\varSigma _0} T_\varSigma }\) for the ideal sheaf in \({\mathcal {O}}_{T_\varSigma }\) defining \(\partial _{\varSigma _0} T_\varSigma \). If \(\varSigma _0 \subset \varSigma ' \subset \varSigma \) then
Note that \({\mathcal {I}}_{\partial _{\varSigma _0}T_\varSigma }|_{T_\sigma }\) corresponds to the ideal
of
where
and \(\tau ^\perp \) denotes the annihilator of \(\tau \) in \(X^*(S)_{\mathbb {R}}\). If we let \({\mathfrak {X}}_{\varSigma _0,\sigma ,m}\) denote the set of sums of m elements of \({\mathfrak {X}}_{\varSigma _0,\sigma ,1}\), then \({\mathcal {I}}_{\partial _{\varSigma _0}T_\varSigma }^m|_{T_\sigma }\) corresponds to the ideal
If \(\sigma \not \in \varSigma _0\) then
for all m. If on the other hand \(\sigma \in \varSigma _0\) then
(For if \(\chi \in \sigma ^0 \cap X_*(S)\) then \(\chi \ge m\) on \({\mathfrak {X}}_{\varSigma _0,\sigma ,m}\).)
In the special case \(\varSigma _0=\varSigma -\{ \{0\}\}\) we will write \(\partial T_\varSigma \) for \(\partial _{\varSigma _0} T_\varSigma \) and \({\mathcal {I}}_{\partial T_\varSigma }\) for \({\mathcal {I}}_{\partial _{\varSigma _0} T_\varSigma }\). Then
We will write \({\mathcal {M}}_\varSigma \rightarrow {\mathcal {O}}_{T_\varSigma }\) for the log structure corresponding to the closed embedding \(\partial T_\varSigma \hookrightarrow T_\varSigma \). We will write \(\varOmega ^1_{T_\varSigma /{{\text {Spec}}\,}R_0}(\log \infty )\) for the log differentials previously denoted \(\varOmega ^1_{T_\varSigma /{{\text {Spec}}\,}R_0}(\log {\mathcal {M}}_\varSigma )\).
If \(\varSigma \) is smooth then \(\partial T_\varSigma \) is a simple normal crossings divisor on \(T_\varSigma \) relative to Y.
If \(\varSigma _0\) is boundary data for \(\varSigma \) we will set
and
We will call \(\varSigma _0\)
-
open if \(|\varSigma _0|^0\) is open in \(X_*(S)_{\mathbb {R}}\);
-
finite if it has finite cardinality;
-
locally finite if for every rational polyhedral cone \(\tau \subset |\varSigma _0|\) (not necessarily in \(\varSigma _0\)) the intersection \(\tau \cap |\varSigma _0|^0\) meets only finitely many elements of \(\varSigma _0\). (We remark that although this condition may be intuitive in the case \(|\varSigma _0|^0=|\varSigma _0|\), in other cases it may be less so.)
Let \(\varSigma \) continue to denote a fan and \(\varSigma _0\) boundary data for \(\varSigma \). If \(\sigma \in \varSigma \) we write
If \(\sigma \ne \{0\}\), then this is an example of boundary data for \(\varSigma \). If \(\sigma \in \varSigma _0\) then
and we will sometimes denote it \(\varSigma _0(\sigma )\). If \(\varSigma _0\) is locally finite then \(\varSigma _0(\sigma )\) is finite for all \(\sigma \in \varSigma _0\). If \(\{0\} \ne \sigma \in \varSigma \) we write
and
Sometimes we also write
If \(\varSigma _0\) is locally finite then the \(\partial _\sigma T_\varSigma \) for \(\sigma \in \varSigma _0\) form a locally finite closed cover of \(\partial _{\varSigma _0} T_\varSigma \). Set theoretically we have
and
and
and
If \(\dim \sigma =1\) then \(\partial _\sigma ^0 T_\varSigma = \partial T_\sigma \).
Keep the notation of the previous paragraph. We define \(S(\sigma )\) to be the split torus with cocharacter group \(X_*(S)\) divided by the subgroup generated by \(\sigma \cap X_*(S)\), and \(T(\sigma )\) to be the push-out of T to \(S(\sigma )\). We also define \(\overline{\varSigma }(\sigma )\) to be the set of images in \(X_*(S(\sigma ))_{\mathbb {R}}\) of elements of \(\varSigma (\sigma )\). It is a fan for \(X_*(S)_{\mathbb {R}}/\langle \sigma \rangle _{\mathbb {R}}\). [The main point to check is that if \(\tau , \tau ' \in \varSigma (\sigma )\) then \((\tau \cap \tau ') +\langle \sigma \rangle _{\mathbb {R}}= (\tau +\langle \sigma \rangle _{\mathbb {R}}) \cap (\tau ' + \langle \sigma \rangle _{\mathbb {R}})\). To verify this suppose that \(x \in \tau \) and \(y \in \tau '\) with \(x-y \in \langle \sigma \rangle _{\mathbb {R}}\). Then \(x-y=z-w\) with \(z ,w \in \sigma \). Thus \(x+w=y+z \in \tau \cap \tau '\) and \(x+\langle \sigma \rangle _{\mathbb {R}}=(x+w) +\langle \sigma \rangle _{\mathbb {R}}\).] If \(\sigma \in \varSigma _0\) we will sometimes write \(\overline{\varSigma }_0(\sigma )\) for \(\overline{\varSigma }(\sigma )\), as it depends only on \(\varSigma _0\) and not on \(\varSigma \). Then
Thus \(\partial _\sigma T_\varSigma \) is separated, locally (on the source) of finite type and flat over Y. The closed subscheme \(\partial _\sigma T_\varSigma \) has codimension in \(T_\varSigma \) equal to the dimension of \(\sigma \). If \(\varSigma (\sigma )\) is smooth then \(\partial _\sigma T_\varSigma \) is smooth over Y.
If \(\varSigma (\sigma )\) is open then \(\partial _\sigma T_\varSigma \) satisfies the valuative criterion of properness over Y. If in addition \(\varSigma (\sigma )\) is finite then \(\partial _\sigma T_\varSigma \) is proper over Y. If \(\varSigma _0\) is open, then \(\partial _{\varSigma _0} T_\varSigma \) satisfies the valuative criterion of properness over Y. If in addition \(\varSigma _0\) is finite then \(\partial _{\varSigma _0}T_\varSigma \) is proper over Y.
The schemes \(\partial _{\sigma _1} T_\varSigma ,\ldots ,\partial _{\sigma _s} T_\varSigma \) intersect if and only if \(\sigma _1, \ldots , \sigma _s\) are all contained in some \(\sigma \in \varSigma \). In this case the intersection equals \(\partial _\sigma T_\varSigma \) for the smallest such \(\sigma \). We set
If Y is irreducible then \(T_\varSigma \) and each \(\partial _\sigma T_\varSigma \) is irreducible. Moreover the irreducible components of \(\partial T_\varSigma \) are the \(\partial _\sigma T_\varSigma \) as \(\sigma \) runs over one-dimensional elements of \(\varSigma \). If \(\varSigma \) is smooth then we see that \({\mathcal {S}}(\partial T_\varSigma )\) is the delta set with cells in bijection with the elements of \(\varSigma -\{ \{0\}\}\) and with the same ‘face relations’. In particular it is in fact a simplicial complex and
We say that \((\varSigma ',\varSigma _0')\) refines \((\varSigma ,\varSigma _0)\) if \(\varSigma '\) refines \(\varSigma \) and \(\varSigma '-\varSigma _0'\) is the set of elements of \(\varSigma '\) contained in some element of \(\varSigma -\varSigma _0\). In this case \(\partial _{\varSigma _0'}T_{\varSigma '}\) maps to \(\partial _{\varSigma _0} T_{\varSigma }\), and in fact set theoretically \(\partial _{\varSigma _0'}T_{\varSigma '}\) is the pre-image of \(\partial _{\varSigma _0} T_{\varSigma }\) in \(T_{\varSigma '}\).
If \(\varSigma \) is a fan, then by line bundle data for \(\varSigma \) we mean a continuous function \(\psi : |\varSigma | \rightarrow {\mathbb {R}}\), such that for each cone \(\sigma \in \varSigma \), the restriction \(\psi |_{\sigma }\) equals some \(\psi _{\sigma } \in X^*(S)\). To \(\psi \) we can attach a line bundle \({\mathcal {L}}_\psi \) on \(T_\varSigma \): On \(T_{\sigma }\) (with \(\sigma \in \varSigma \)) it corresponds to the \(\bigoplus _{\chi \in \sigma ^\vee \cap X^*(S)} {\mathcal {L}}_T(\chi )\)-module
Note that there are natural isomorphisms
and that
We have the following examples of line bundle data.
-
(1)
\({\mathcal {O}}_{T_\varSigma }\) is the line bundle associated to \(\psi \equiv 0\).
-
(2)
If \(\varSigma \) is smooth then \({\mathcal {I}}_{\partial T_\varSigma }\) is the line bundle associated to the unique such function \(\psi _\varSigma \) which for every one-dimensional cone \(\sigma \in \varSigma \) satisfies
$$\begin{aligned} \psi _{\varSigma }(X_*(S) \cap \sigma )={\mathbb {Z}}_{\ge 0}. \end{aligned}$$
Suppose that \(\alpha : S \twoheadrightarrow S'\) is a surjective map of split tori over Y. Then \(X^*(\alpha ): X^*(S') \hookrightarrow X^*(S)\) and \(X_*(\alpha ): X_*(S) \rightarrow X_*(S')\), the latter with finite cokernel. We call fans \(\varSigma \) for \(X_*(S)\) and \(\varSigma '\) for \(X_*(S')\) compatible if for all \(\sigma \in \varSigma \) the image \(X_*(\alpha ) \sigma \) is contained in some element of \(\varSigma '\). In this case the map \(\alpha : T \rightarrow \alpha _* T\) extends to an S-equivariant map
We will write
If for all \(\sigma ' \in \varSigma '\) the pre-image \(X_*(\alpha )^{-1}(\sigma ')\) is a finite union of elements of \(\varSigma \), then \(\alpha : T_\varSigma \rightarrow (\alpha _*T)_{\varSigma '}\) is proper.
If \(\alpha \) is an isogeny, if \(\varSigma \) and \(\varSigma '\) are compatible, and if every element of \(\varSigma '\) is a finite union of elements of \(\varSigma \), then we call \(\varSigma \) a quasi-refinement of \(\varSigma '\). In that case the map \(\alpha : T_{\varSigma } \rightarrow (\alpha _*T)_{\varSigma '}\) is proper.
Lemma 2.5
If \(\alpha \) is surjective and \(\# {{\text {coker}}\,}X_*(\alpha )\) is invertible on Y then
is log smooth, and there is a natural isomorphism
Proof
We can work Zariski locally on \(T_\varSigma \). Thus we may replace \(T_\varSigma \) by \(T_\sigma \) and \((\alpha _* T)_{\varSigma '}\) by \((\alpha _*T)_{\sigma '}\) for cones \(\sigma \) and \(\sigma '\) with \(X_*(\alpha )\sigma \subset \sigma '\). We may also replace Y by an affine open subset U such that \(T|_U\) is trivial, i.e. each \({\mathcal {L}}_T(\chi ) \cong {\mathcal {O}}_Y\) compatibly with \({\mathcal {L}}_T(\chi ) \otimes {\mathcal {L}}_T(\chi ') \mathop {\rightarrow }\limits ^{\sim }{\mathcal {L}}_T(\chi +\chi ')\). Then the log structure on \(T_\sigma \) has a chart \({\mathbb {Z}}[\sigma ^\vee \cap X^*(S)] \rightarrow {\mathcal {O}}_{T_{\sigma }}\) sending \(\chi \) to
Similarly the log structure on \((\alpha _*T)_{\sigma '}\) has a chart \({\mathbb {Z}}[(\sigma ')^\vee \cap X^*(S')] \rightarrow {\mathcal {O}}_{(\alpha _*T)_{\sigma '}}\) sending \(\chi \) to
The lemma follows because
is injective and the torsion subgroup of the cokernel is finite with order invertible on Y. \(\square \)
We will call pairs \((\varSigma ,\varSigma _0)\) and \((\varSigma ',\varSigma _0')\) of fans and boundary data for S and \(S'\), respectively, compatible if \(\varSigma \) and \(\varSigma '\) are compatible and if no cone of \(\varSigma _0\) maps into any cone of \(\varSigma '-\varSigma _0'\). In this case
We will call them strictly compatible if they are compatible and \(\varSigma -\varSigma _0\) is the set of cones in \(\varSigma \) mapping into some element of \(\varSigma '-\varSigma _0'\).
Lemma 2.6
Suppose that \(\alpha : S \twoheadrightarrow S'\) is a surjective map of split tori, that T / Y is an S-torsor and that \((\varSigma ,\varSigma _0)\) and \((\varSigma ',\varSigma _0')\) are strictly compatible fans with boundary data for S and \(S'\), respectively. Then locally on \(T_\varSigma \) there is a strictly positive integer m such that
and
Proof
We may work locally on Y and so we may suppose that \(Y={{\text {Spec}}\,}A\) is affine and that each \({\mathcal {L}}_T(\chi )\) is trivial. It also suffices to check the lemma locally on \(T_{\varSigma }\). Thus we may suppose that \(\varSigma \) consists of a cone \(\sigma \) and all its faces. Let \(\sigma '\) denote the smallest element of \(\varSigma '\) containing the image of \(\sigma \). Then we may further suppose that \(\varSigma '\) consists of \(\sigma '\) and all its faces. We may further suppose that \(\sigma \in \varSigma _0\) and \(\sigma ' \in \varSigma _0'\), else there is nothing to prove.
Then
and \(\partial _{\varSigma _0} T_\varSigma \) is defined by
Moreover \(T_\varSigma \times _{(\alpha _* T)_{\varSigma '}} \partial _{\varSigma _0'} (\alpha _*T)_{\varSigma '}\) is defined by
Thus it suffices to show that for some positive integer m we have
This is equivalent to
Suppose that \(\chi _1 \in |\varSigma _0'|^{0,\vee ,0}\) and \(\chi _2 \in \sigma ^\vee \). Then
and so
Thus
Conversely suppose that \(\chi \in |\varSigma _0|^{0,\vee ,0}\). Let \(\tau \) denote the face of \(\sigma \), where \(\chi =0\). Then \(\tau \in \varSigma -\varSigma _0\). Let \(\tau '\) denote the smallest face of \(\sigma '\) containing \(X_*(\alpha ) \tau \). Then \(\tau ' \in \varSigma '-\varSigma _0'\). We can find \(\chi _1 \in |\varSigma _0'|^{0,\vee ,0}\) with \(\chi _1(\tau ')=\{0\}\). Note that if \(a \in \sigma \) and \(\chi (a)=0\) then \((X^*(\alpha )(\chi _1))(a)=0\). Thus we can find \(\epsilon >0\) such that
It follows that
The lemma follows. \(\square \)
Suppose that \((\varSigma ,\varSigma _0)\) and \((\varSigma ',\varSigma _0')\) are strictly compatible. We will say that
-
\(\varSigma _0\) is open over \(\varSigma _0'\) if \(|\varSigma _0|^0\) is open in \(X_*(\alpha )^{-1} |\varSigma _0'|^0\);
-
and that \(\varSigma _0\) is finite over \(\varSigma _0'\) if only finitely many elements of \(\varSigma _0\) map into any element of \(\varSigma _0'\).
If \(\alpha \) is an isogeny, if \(\varSigma \) is a quasi-refinement of \(\varSigma '\) and if \((\varSigma ,\varSigma _0)\) and \((\varSigma ',\varSigma _0')\) are strictly compatible, then we call \((\varSigma ,\varSigma _0)\) a quasi-refinement of \((\varSigma ',\varSigma _0')\). In this case \(\varSigma _0\) is open and finite over \(\varSigma _0'\).
Lemma 2.7
Suppose that \(\alpha : S \twoheadrightarrow S'\) is a surjective map of split tori, that T / Y is an S-torsor, and that \((\varSigma ,\varSigma _0)\) and \((\varSigma ',\varSigma _0')\) are strictly compatible fans with boundary data for S and \(S'\), respectively. If \(\varSigma _0\) is locally finite and \(\varSigma _0\) is open over \(\varSigma _0'\) then
satisfies the valuative criterion of properness. If in addition \(\varSigma _0\) is finite over \(\varSigma _0'\) then this morphism is proper.
Proof
It suffices to show that if \(\sigma \in \varSigma _0\) and if \(\sigma '\) is the smallest element of \(\varSigma '_0\) containing \(X_*(\alpha ) \sigma \), then
satisfies the valuative criterion of properness. However this is the map of toric varieties
As \(\overline{\varSigma }_0(\sigma )\) is finite, it suffices to check that
Choose a point \(P \in \sigma ^0\) such that
Then
[To see this choose nonzero vectors \(v_i\) in each one-dimensional face of \(\sigma '\). Then we can write \(X_*(\alpha )(P)=\sum _i \mu _i v_i\) with each \(\mu _i > 0\). If \(\lambda _i \in {\mathbb {R}}\), then for \(\lambda \) sufficiently large \(\lambda _i+\lambda \mu _i \in {\mathbb {R}}_{>0}\) for all i, and so
Thus
Hence for all \(\tau ' \in \varSigma _0'\) with \(\tau ' \supset \sigma '\), we have
and so
We deduce that it suffices to check that
The left hand side certainly contains the right hand side, so it suffices to prove that for all \(\tau ' \in \varSigma _0'\) with \(\tau ' \supset X_*(\alpha )\sigma \) we have
Let \(\pi \) denote the map
Because \(X_*(\alpha )^{-1} \tau '\) and \(\bigcup _{\begin{array}{c} \tau \supset \sigma \\ \tau \in \varSigma _0 \end{array}} \tau ^0\) are invariant under the action of \({\mathbb {R}}^\times _{>0}\) it suffices to find an open set \(U \subset X_*(S)_{\mathbb {R}}\) containing P such that
or equivalently such that
Thus it suffices to find an open set \(U \subset X_*(S)_{\mathbb {R}}\) containing P such that
-
(1)
\(U \cap X_*(\alpha )^{-1}|\varSigma _0'|^0 \subset \bigcup _{\begin{array}{c} \tau \supset \sigma \\ \tau \in \varSigma _0 \end{array}} \tau ^0\);
-
(2)
\(U \cap X_*(\alpha )^{-1} \tau ' \subset X_*(\alpha )^{-1} |\varSigma _0'|^0\);
-
(3)
and for all open \(U' \subset U\) containing P we have \(U' \cap (\langle \sigma \rangle _{\mathbb {R}}+ X_*(\alpha )^{-1} \tau ')=U' \cap X_*(\alpha )^{-1} \tau '\).
Moreover in order to find such a \(U \ni P\) it suffices to find one satisfying each property independently and take their intersection.
One can find an open set \(U \ni P\) satisfying the first property because
are both open inclusions.
To find \(U \ni P\) satisfying the second condition we just need to avoid the faces of \(X_*(\alpha )^{-1}\tau '\) which do not contain P.
It remains to check that we can find an open \(U \ni P\) satisfying the last condition. Suppose that \(X_*(\alpha )^{-1}\tau '\) is defined by inequalities \(\chi _i \ge 0\) for \(i=1,\ldots ,r\) with \(\chi _i \in X^*(S)_{\mathbb {R}}\). Suppose that \(\chi _i=0\) on \(\sigma \) for \(i=1,\ldots ,s\), but that \(\chi _i(P)>0\) for \(i=s+1,\ldots ,r\). It suffices to choose U so that \(\chi _i>0\) on U for \(i=s+1,\ldots ,r\). For then if \(x \in X_*(\alpha )^{-1} \tau '\) and \(y \in \langle \sigma \rangle _{\mathbb {R}}\) with \(x+y \in U\) we see that
for \(i=1,\ldots ,s\), while \(\chi _i(x+y)>0\) for \(i=s+1,\ldots ,r\). Thus for \(U' \subset U\) we have
as desired. \(\square \)
By a partial fan we will mean a collection \(\varSigma _0\) of rational polyhedral cones satisfying
-
\((0) \not \in \varSigma _0\);
-
if \(\sigma _1, \sigma _2 \in \varSigma _0\), then \(\sigma _1 \cap \sigma _2\) is a face of \(\sigma _1\) and of \(\sigma _2\);
-
if \(\sigma _1, \sigma _2 \in \varSigma _0\), and if \(\sigma \supset \sigma _2\) is a face of \(\sigma _1\), then \(\sigma \in \varSigma _0\).
(Again note that \(\varSigma _0\) may not be closed under taking faces.) In this case we will let \(\widetilde{\varSigma }_0\) denote the set of faces of elements of \(\varSigma _0\) together with \(\{ 0\}\). Then \(\widetilde{\varSigma }_0\) and \(\widetilde{\varSigma }_0-\varSigma _0\) are fans, and \(\varSigma _0\) is boundary data for \(\widetilde{\varSigma }_0\). [To see this suppose that \(\tau _i\) is a face of \(\sigma _i \in \varSigma _0\) for \(i=1,2\). Then \(\sigma _1 \cap \sigma _2\) is a face of \(\sigma _1\) and so \(\tau _1 \cap \sigma _2 =\tau _1 \cap (\sigma _1 \cap \sigma _2)\) is a face of \(\sigma _1 \cap \sigma _2\) and hence of \(\sigma _2\). Thus \(\tau _1 \cap \tau _2=\tau _2 \cap (\tau _1 \cap \sigma _2)\) is a face of \(\tau _2\).] If \(\varSigma \) is a fan and \(\varSigma _0\) is boundary data for \(\varSigma \), then \(\varSigma _0\) is a partial fan, and \(\varSigma \supset \widetilde{\varSigma }_0\). Thus
If \(\varSigma _0\) and \(\varSigma _0'\) are partial fans we will say that \(\varSigma _0\) refines \(\varSigma _0'\) if every element of \(\varSigma _0\) is contained in an element of \(\varSigma _0'\) and if every element of \(\varSigma _0'\) is a finite union of elements of \(\varSigma _0\). In this case \(\widetilde{\varSigma }_0\) also refines \(\widetilde{\varSigma }_0'\).
If \(\varSigma _0\) is a partial fan we will set
and
We will call \(\varSigma _0\)
-
smooth if each \(\sigma \in \varSigma _0\) is smooth;
-
full if every element of \(\varSigma _0\) which is not a face of any other element of \(\varSigma _0\), has the same dimension as S;
-
open if \(|\varSigma _0|^0\) is open in \(X_*(S)_{\mathbb {R}}\);
-
finite if it has finite cardinality;
-
locally finite if for every rational polyhedral cone \(\tau \subset |\varSigma _0|\) (not necessarily in \(\varSigma _0\)) the intersection \(\tau \cap |\varSigma _0|^0\) meets only finitely many elements of \(\varSigma _0\).
If \(\varSigma _0\) is smooth, so is \(\widetilde{\varSigma }_0\).
Suppose that \(\varSigma _0\) is a partial fan. If \(\varSigma \supset \widetilde{\varSigma }_0\) is a fan then the natural maps
and
are isomorphisms, and we will denote these schemes/formal schemes \(\partial _{\varSigma _0}T\) and \(T_{\varSigma _0}^\wedge \), respectively. Moreover the log structures induced on \(T_{\varSigma _0}^\wedge \) by \({\mathcal {M}}_{\widetilde{\varSigma }_0}\) and by \({\mathcal {M}}_\varSigma \) are the same and we will denote them \({\mathcal {M}}^\wedge _{\varSigma _0}\). If \(\varSigma _0' \subset \varSigma _0\) is also a partial fan, then \(T_{\varSigma _0'}^\wedge \) can be identified with the completion of \(T_{\varSigma _0}^\wedge \) along \(\partial _{\varSigma _0'}T\), and \({\mathcal {M}}_{\varSigma _0}^\wedge \) induces \({\mathcal {M}}_{\varSigma _0'}^\wedge \). If \(\sigma \in \widetilde{\varSigma }_0\) then we will let
denote the restriction of \(T_{\varSigma _0}^\wedge \) to the topological space \((\partial _{\varSigma _0} T_{\widetilde{\varSigma }_0})_\sigma \). Thus the \((T_{\varSigma _0}^\wedge )_\sigma \) for \(\sigma \in \varSigma _0\) form an affine open cover of \(T_{\varSigma _0}^\wedge \). We have
and
If \(\varSigma _0'\) refines \(\varSigma _0\) then there is an induced map
Continue to suppose that \(\varSigma _0\) is a partial fan. We will call \(\varSigma _1 \subset \varSigma _0\) boundary data if, whenever \(\sigma \in \varSigma _0\) contains \(\sigma ' \in \varSigma _1\), then \(\sigma \in \varSigma _1\). In this case \(\varSigma _1\) is a partial fan and \(T_{\varSigma _1}^\wedge \) is canonically identified with the completion of \(T_{\varSigma _0}^\wedge \) along \(\partial _{\varSigma _1} T_{\widetilde{\varSigma }_0}\).
We will also use the following notation.
-
\({\mathcal {O}}_{T_{\varSigma _0}^\wedge }\) will denote the structure sheaf of \(T_{\varSigma _0}^\wedge \).
-
\({\mathcal {I}}_{T_{\varSigma _0}^\wedge }\) will denote the completion of \({\mathcal {I}}_{\partial _{\varSigma _0} T_{\widetilde{\varSigma }_0}}\), an ideal of definition for \(T_{\varSigma _0}^\wedge \).
-
\({\mathcal {I}}_{\partial , \varSigma _0}^\wedge \) will denote the completion of \({\mathcal {I}}_{\partial T_{\widetilde{\varSigma }_0}}\). Thus \( {\mathcal {I}}_{T_{\varSigma _0}^\wedge } \supset {\mathcal {I}}_{\partial ,\varSigma _0}^\wedge \).
-
\(\varOmega ^1_{T_{\varSigma _0}^\wedge /{{\text {Spf}}\,}R_0}(\log \infty )\) will denote \(\varOmega ^1_{T_{\varSigma _0}^\wedge /{{\text {Spf}}\,}R_0}(\log {\mathcal {M}}_\varSigma ^\wedge )\), which is isomorphic to the completion of \(\varOmega ^1_{T_{\widetilde{\varSigma }_0}/{{\text {Spec}}\,}R_0}(\log \infty )\).
For \(\sigma \in \widetilde{\varSigma }_0\) recall that \({\mathcal {I}}_{\partial _{\varSigma _0}T_{\widetilde{\varSigma }_0}}^m|_{T_\sigma }\) corresponds to the ideal
of
Also recall that if \(\sigma \not \in \varSigma _0\) then
for all m, while if \(\sigma \in \varSigma _0\) then
By line bundle data for \(\varSigma _0\) we mean a continuous function \(\psi : |\varSigma _0| \rightarrow {\mathbb {R}}\), such that for each cone \(\sigma \in \widetilde{\varSigma }_0\), the restriction \(\psi |_{\sigma }\) equals some \(\psi _{\sigma } \in X^*(S)\). This is the same as line bundle data for the fan \(\widetilde{\varSigma }_0\), and we will write \({\mathcal {L}}_\psi ^\wedge \) for the line bundle on \(T_{\varSigma _0}^\wedge \), which is the completion of \({\mathcal {L}}_\psi /T_{\widetilde{\varSigma }_0}\). Note that
and that
We have the following examples of line bundle data.
-
(1)
\({\mathcal {O}}_{T_{\varSigma _0}^\wedge }\) is the line bundle associated to \(\psi \equiv 0\).
-
(2)
If \(\varSigma _0\) is smooth then \({\mathcal {I}}_{\partial , \varSigma _0}^\wedge \) is the line bundle associated to the unique such function \(\psi _{\widetilde{\varSigma }_0}\) which for every one-dimensional cone \(\sigma \in \widetilde{\varSigma }_0\) satisfies
$$\begin{aligned} \psi _{\widetilde{\varSigma }_0}(X_*(S) \cap \sigma )={\mathbb {Z}}_{\ge 0}. \end{aligned}$$
Suppose that \(\alpha : S \twoheadrightarrow S'\) is a surjective map of tori, and that \(\varSigma _0\) (resp. \(\varSigma _0'\)) is a partial fan for S (resp. \(S'\)). We call \(\varSigma _0\) and \(\varSigma _0'\) compatible if for every \(\sigma \in \varSigma _0\) the image \(X_*(\alpha ) \sigma \) is contained in some element of \(\varSigma _0'\) but in no element of \(\widetilde{\varSigma }_0'-\varSigma _0'\). In this case \((\widetilde{\varSigma }_0,\varSigma _0)\) and \((\widetilde{\varSigma }_0',\varSigma _0')\) are compatible, and there is a natural morphism
We will write
The following lemma follows immediately from Lemma 2.5.
Lemma 2.8
If \(\alpha \) is surjective and \(\# {{\text {coker}}\,}X_*(\alpha )\) is invertible on Y then there is a natural isomorphism
We will call \(\varSigma _0\) and \(\varSigma _0'\) strictly compatible if they are compatible and if an element of \(\widetilde{\varSigma }_0\) lies in \(\varSigma _0\) if and only if it maps to no element of \(\widetilde{\varSigma }_0'-\varSigma _0'\). In this case \((\widetilde{\varSigma }_0,\varSigma _0)\) and \((\widetilde{\varSigma }_0',\varSigma _0')\) are strictly compatible. We will say that
-
\(\varSigma _0\) is open over \(\varSigma _0'\) if \(|\varSigma _0|^0\) is open in \(X_*(\alpha )^{-1} |\varSigma _0'|^0\);
-
and that \(\varSigma _0\) is finite over \(\varSigma _0'\) if only finitely many elements of \(\varSigma _0\) map into any element of \(\varSigma _0'\).
If \(\alpha \) is an isogeny, if \(\varSigma _0\) and \(\varSigma _0'\) are strictly compatible and if every element of \(\varSigma _0'\) is a finite union of elements of \(\varSigma _0\), then we call \(\varSigma _0\) a quasi-refinement of \(\varSigma _0'\). In this case \(\varSigma _0\) is open and finite over \(\varSigma _0'\). The next lemma follows immediately from Lemmas 2.6 and 2.7.
Lemma 2.9
Suppose that \(\varSigma _0'\) and \(\varSigma _0\) are strictly compatible.
-
(1)
\(T_{\varSigma _0}^\wedge \) is the formal completion of \(T_{\widetilde{\varSigma }_0}\) along \(\partial _{\varSigma _0'}(\alpha _* T)\), and \(T_{\varSigma _0}^\wedge \) is locally (on the source) topologically of finite type over \((\alpha _*T)^\wedge _{\varSigma _0'}\).
-
(2)
If \(\varSigma _0\) is locally finite and if it is open and finite over \(\varSigma _0'\) then \(T_{\varSigma _0}^\wedge \) is proper over \((\alpha _* T)_{\varSigma _0'}^\wedge \).
Corollary 2.10
If \(\alpha \) is an isogeny, if \(\varSigma _0\) is locally finite and if \(\varSigma _0\) is a quasi-refinement of \(\varSigma _0'\) then \(T_{\varSigma _0}^\wedge \) is proper over \((\alpha _* T)_{\varSigma _0'}^\wedge \).
If \(\varSigma _0\) and \(\varSigma _0'\) are compatible partial fans and if \(\varSigma _1' \subset \varSigma _0'\) is boundary data then \(\varSigma _0(\varSigma _1')\) will denote the set of elements \(\sigma \in \varSigma _0\) such that \(X_*(\alpha )\sigma \) is contained in no element of \(\varSigma _0'-\varSigma _1'\). It is boundary data for \(\varSigma _0\). Moreover the formal completion of \(T_{\varSigma _0}^\wedge \) along the reduced subscheme of \((\alpha _*T)_{\varSigma _1'}^\wedge \) is canonically identified with \(T^\wedge _{\varSigma _0(\varSigma _1')}\). If \(\varSigma _1'=\{ \sigma '\}\) is a singleton we will write \(\varSigma _0(\sigma ')\) for \(\varSigma _0(\{ \sigma '\})\).
3.4 Cohomology of line bundles
In this section we will compute the cohomology of line bundles on formal completions of torus embeddings. We will work throughout over a base scheme Y which is connected, separated and flat and locally of finite type over \({{\text {Spec}}\,}R_0\).
We start with some definitions. We continue to assume that S / Y is a split torus, that T / Y is an S-torsor, that \(\varSigma _0\) is a partial fan and that \(\psi \) is line bundle data for \(\varSigma _0\). If \(\sigma \in \widetilde{\varSigma }_0\) then we set
For \(m>0\) we define \({\mathfrak {X}}_{\varSigma _0,\psi ,\sigma ,m}\) to be the set of sums of an element of \({\mathfrak {X}}_{\varSigma _0,\psi ,\sigma ,0}\) and an element of \({\mathfrak {X}}_{\varSigma _0,\sigma ,m}\). If \(\sigma \not \in \varSigma _0\) then
for all m, while if \(\sigma \in \varSigma _0\)
Further suppose that \(\chi \in X^*(S)\).
-
Set \(Y_\psi (\chi )=\{ x \in X^*(S)_{\mathbb {R}}: (\psi -\chi )(x)>0\} \).
-
If \(U \subset Y\) is open let \(H^j_{\varSigma _0,\psi ,m}(\chi )(U)\) denote the jth cohomology of the Cech complex with ith term
$$\prod _{\begin{array}{c} (\sigma _0,\ldots ,\sigma _i) \in \varSigma _0^{i+1} \\ \chi \in {\mathfrak {X}}_{\varSigma _0,\psi ,\sigma _0 \cap \cdots \cap \sigma _i,0} \\ \chi \not \in {\mathfrak {X}}_{\varSigma _0,\psi ,\sigma _0\cap \cdots \cap \sigma _i,m} \end{array}} {\mathcal {L}}_T(\chi )(U).$$
Note the examples:
-
(1)
\(Y_0(\chi )\cap |\varSigma _0|^0=\emptyset \) if and only if \(\chi \in |\varSigma _0|^\vee \).
-
(2)
\(Y_{\psi _{\widetilde{\varSigma }_0}}(\chi ) \cap |\varSigma _0|^0=\emptyset \) if and only if \(\chi \in |\varSigma _0|^{\vee ,0}\).
Also note that if \(\varSigma _0\) is finite then, for m large enough, \(H^j_{\varSigma _0,\psi ,m}(\chi )(U)\) does not depend on m. We will denote it simply \(H^j_{\varSigma _0,\psi }(\chi )(U)\). It equals the cohomology of the Cech complex
Lemma 2.11
If U is connected then
Proof
Write M for \({\mathcal {L}}_T(\chi )(U)\). We follow the argument of section 3.5 of [26]. As \(\sigma _0 \cap \cdots \cap \sigma _i \cap |\varSigma _0|^0\) and \(\sigma _0 \cap \cdots \cap \sigma _i \cap |\varSigma _0|^0 \cap Y_\psi (\chi )\) are convex, we see that
(See the first paragraph of section 3.5 of [26].) Thus the ith term of our Cech complex becomes
Thus it suffices to show that the Cech complex with ith term
computes
To this end choose an injective resolution
as sheaves of abelian groups on \(|\varSigma _0|^0\), and consider the double complex
We compute the cohomology of the corresponding total complex in two ways. Firstly the jth cohomology of the complex
equals
(See theorem 4.1, proposition 5.3 and theorem 5.5 of chapter II of [14].) This vanishes for \(j>0\), and so the cohomology of our total complex is the same as the cohomology of the Cech complex with ith term
Thus it suffices to identify the cohomology of our double complex with
For this it suffices to show that
is exact for all j. Let \({\widetilde{{\mathcal {I}}}}^j\) denote the sheaf of discontinuous sections of \({\mathcal {I}}^j\), i.e. \({\widetilde{{\mathcal {I}}}}^j(V)\) denotes the set of functions which assign to each point of \(x \in V\) an element of the stalk \({\mathcal {I}}^j_x\) of \({\mathcal {I}}^j\) at x. Then \({\mathcal {I}}^j\) is a direct summand of \({\widetilde{{\mathcal {I}}}}^j\) so it suffices to show that
is exact for all j. However \({\widetilde{{\mathcal {I}}}}^j\) is the direct product over x in \(|\varSigma _0|^0\) of the sky-scraper \(\overline{{{\mathcal {I}}}}^j_x\) sheaf at x with stalk \({\mathcal {I}}^j_x\). Thus it suffices to show that
is exact for all \(x \in |\varSigma _0|^0\) and for all j. If \(x \in Y_\psi (\chi )\cap |\varSigma _0|^0\) all the terms in this sequence are 0, so the sequence is certainly exact. If \(x \in |\varSigma _0|^0-Y_\psi (\chi )\), this sequence equals
A standard argument shows that this is indeed exact: Choose \(\sigma \in \varSigma _0\) with \(x \in \sigma \). Suppose
Define
by
If \(\partial a'\) denotes the image of \(a'\) in
then
i.e. \(a=(-1)^i \partial a'\). \(\square \)
In general we will let \(H^i_{|\varSigma _0|^0-Y_\psi (\chi )}(|\varSigma _0|^0,{\mathcal {L}}_T(\chi ))\) denote the sheaf of \({\mathcal {O}}_Y\)-modules on Y associated to the pre-sheaf
Lemma 2.12
Let Y be a connected, separated scheme which is flat and locally of finite type over an irreducible noetherian ring \(R_0\), let S / Y be a split torus, let T / Y be an S-torsor, let \(\varSigma _0\) be a partial fan for S, let \(\psi \) be line bundle data for \(\varSigma _0\), and let \(\pi _{\varSigma _0}^\wedge \) denote the map \(T^\wedge _{\varSigma _0} \rightarrow Y\). Suppose that \(\varSigma _0\) is finite, non-empty and open. Then
(Note that \(R^i\pi _{\varSigma _0,*}^\wedge {\mathcal {L}}_\psi ^\wedge \) may not be quasi-coherent on Y. Infinite products of quasi-coherent sheaves may not be quasi-coherent.)
Proof
The left hand side is the sheaf associated to the pre-sheaf
and the right hand side is the sheaf associated to the pre-sheaf
Thus it suffices to establish isomorphisms
compatibly with restriction, for \(U={{\text {Spec}}\,}A\), with A noetherian and \({{\text {Spec}}\,}A\) connected.
Write \(\partial _{\varSigma _0,m} T_{\widetilde{\varSigma }_0}\) for the closed subscheme of \(T_{\widetilde{\varSigma }_0}\) defined by \({\mathcal {I}}_{\partial _{\varSigma _0} T_{\widetilde{\varSigma }_0}}^m\). It has the same underlying topological space as \(\partial _{\varSigma _0}T_{\widetilde{\varSigma }_0}\). We will first compute
using the affine cover of \(\partial _{\varSigma _0,m} T_{\widetilde{\varSigma }_0}\) by the open sets \(T_\sigma \) for \(\sigma \in \varSigma _0\). This gives rise to a Cech complex with terms
As \(\varSigma _0\) is finite, we see that
Because A is noetherian, because \(\partial _{\varSigma _0,m} T_{\widetilde{\varSigma }_0}\) is proper over \({{\text {Spec}}\,}A\) and because \({\mathcal {L}}_\psi /{\mathcal {I}}_{\partial _{\varSigma _0} T_{\widetilde{\varSigma }_0}}^m {\mathcal {L}}_\psi \) is a coherent sheaf on \(\partial _{\varSigma _0,m} T_{\widetilde{\varSigma }_0}\), we see that the cohomology group \(H^i(\partial _{\varSigma _0,m} T_{\widetilde{\varSigma }_0}|_U,{\mathcal {L}}_\psi /{\mathcal {I}}_{\partial _{\varSigma _0} T_{\widetilde{\varSigma }_0}}^m {\mathcal {L}}_\psi )\) is a finitely generated A-module, and hence, for fixed m and i, we see that the groups \(H_{\varSigma _0,\psi ,m}^i(\chi )(U)=(0)\) for all but finitely many \(\chi \). In particular
Moreover, combining this observation with the fact that \(\{ H^i_{\varSigma _0,\psi ,m}(\chi )(U)\}\) satisfies the Mittag-Leffler condition, we see that the system
satisfies the Mittag-Leffler condition. Hence from proposition 0.13.3.1 of [23] we see that
and the present lemma follows from Lemma 2.11. \(\square \)
Lemma 2.13
Let Y be a connected, separated scheme which is flat and locally of finite type over an irreducible noetherian ring \(R_0\), S / Y be a split torus, let T / Y be an S-torsor, let \(\varSigma _\infty \) be a partial fan for S, let
be a nested sequence of partial fans with \(\varSigma _\infty =\bigcup _i \varSigma _i\) and let \(\psi \) be line bundle data for \(\varSigma _\infty \). For \(i=1,2,3,\ldots ,\infty \) let \(\pi _{\varSigma _i}^\wedge \) denote the map \(T_{\varSigma _i}^\wedge \rightarrow Y\).
Suppose that for \(i\in {\mathbb {Z}}_{>0}\) the partial fan \(\varSigma _i\) is finite, non-empty and open. Suppose also that for all \(i \in {\mathbb {Z}}_{\ge 0}\) and all connected, noetherian, affine open sets \(U \subset Y\), the inverse system
satisfies the Mittag-Leffler condition. Then
Proof
The left hand side is the sheaf associated to the pre-sheaf
and the right hand side is the sheaf associated to the pre-sheaf
Thus it suffices to establish isomorphisms
compatibly with restriction, for \(U={{\text {Spec}}\,}A\), with A noetherian and \({{\text {Spec}}\,}A\) connected.
We can compute \(H^i(T_{\varSigma _\infty }^\wedge |_U, {\mathcal {L}}_\psi ^\wedge )\) as the cohomology of the Cech complex with ith term
and we can compute \(H^i(T_{\varSigma _j}^\wedge |_U, {\mathcal {L}}_\psi ^\wedge )\) as the cohomology of the Cech complex with ith term
Note that as soon as the faces of \(\sigma \) in \(\varSigma _j\) equals the faces of \(\sigma \) in \(\varSigma _\infty \) then \((T_{\varSigma _\infty }^\wedge )_\sigma =(T_{\varSigma _j}^\wedge )_\sigma \). Thus
and
satisfies the Mittag-Leffler condition (with j varying but i fixed).
From theorem 3.5.8 of [57] we see that there is a short exact sequence
Applying Lemma 2.12 and the fact that \(\lim _{\leftarrow }\) and \({\lim _{\leftarrow }}^1\) in the category of abelian groups commute with arbitrary products, the present lemma follows. (It follows easily from definition 3.5.1 of [57] and the exactness of infinite products in the category of abelian groups that \(\lim _{\leftarrow }\) and \({\lim _{\leftarrow }}^1\) commute with arbitrary products in the category of abelian groups.) \(\square \)
We now turn to two specific line bundles: \({\mathcal {O}}_{T_{\varSigma _0}^\wedge }\) and, in the case that \(\varSigma _0\) is smooth, \({\mathcal {I}}_{\partial ,\varSigma _0}^\wedge \).
Lemma 2.14
Let Y be a connected, separated scheme which is flat and locally of finite type over an irreducible noetherian ring \(R_0\), let S / Y be a split torus, let T / Y be an S-torsor, let \(\varSigma _0\) be a partial fan for S, and let \(\pi _{\varSigma _0}^\wedge \) denote the map \(T^\wedge _{\varSigma _0} \rightarrow Y\). Suppose that \(\varSigma _0\) is non-empty, finite and open and that \(|\varSigma _0|^0\) is convex.
-
(1)
Then
$$R^i\pi ^\wedge _{\varSigma _0,*} {\mathcal {O}}_{T_{\varSigma _0}^\wedge }=\left\{ \begin{array}{ll} \prod _{\chi \in |\varSigma _0|^\vee } {\mathcal {L}}(\chi )&{} \quad \mathrm{if}\,\, i=0 \\ (0) &{} \quad \mathrm{otherwise.}\end{array} \right. $$ -
(2)
If in addition \(\varSigma _0\) is smooth then
$$R^i\pi _{\varSigma _0,*}^\wedge {\mathcal {I}}_{\partial , \varSigma _0}^\wedge =\left\{ \begin{array}{ll} \prod _{\chi \in |\varSigma _0|^{\vee ,0}} {\mathcal {L}}(\chi ) &{} \quad \mathrm{if}\,\, i=0 \\ (0) &{} \quad \mathrm{otherwise.}\end{array} \right. $$
Proof
The first part follows from Lemma 2.12 because \(Y_0(\chi )\cap |\varSigma _0|^0\) is empty if \(\chi \in |\varSigma _0|^\vee \) and otherwise, being the intersection of two convex sets, it is convex.
For the second part we have that \(Y_{\psi _{\widetilde{\varSigma }_0}}(\chi ) \cap |\varSigma _0|^0 =\emptyset \) if and only if \(\chi \in |\varSigma _0|^{\vee ,0}\). Thus it suffices to show that each \(Y_{\psi _{\widetilde{\varSigma }_0}}(\chi )\cap |\varSigma _0|^0\) is empty or contractible.
To this end, consider the sets
and
If \(\chi >0\) on \(\sigma -\{0\}\) then \(\chi \ge \psi _{\widetilde{\varSigma }_0}\) on \(\sigma \) so that \(\sigma \cap Y_{\psi _{\widetilde{\varSigma }_0}}(\chi )=\emptyset \). Thus
and
We will describe a deformation retraction
from \(Y''(\chi )\) to \(Y'(\chi )\), which restricts to deformation retractions
from \(Y_{\psi _{\widetilde{\varSigma }_0}}(\chi )\cap |\varSigma _0|^0\) to \(Y'(\chi )\cap |\varSigma _0|^0\), and
from \(\{ x \in |\varSigma _0|^0: \chi (x)\le 0\}\) to \(Y'(\chi )\cap |\varSigma _0|^0\). (Recall that in particular \(H|_{Y'(\chi ) \times [0,1]}\) is just projection to the first factor.) As \(\{ x \in |\varSigma _0|^0: \chi (x)\le 0\}\) is empty or convex, it would follow that \(Y_{\psi _{\widetilde{\varSigma }_0}}(\chi )\cap |\varSigma _0|^0\) is empty or contractible and the second part of the corollary would follow.
To define H it suffices to define, for each \(\sigma \in \widetilde{\varSigma }_0\) with \(\sigma \subset Y''(\chi )\), a deformation retraction
from \(\sigma \) to \(\sigma \cap Y'(\chi )\) with the following properties:
-
If \(\sigma ' \subset \sigma \) then \(H_\sigma |_{\sigma ' \times [0,1]}=H_{\sigma '}\).
-
\(H_\sigma |_{(\sigma \cap Y_{\psi _{\widetilde{\varSigma }_0}}(\chi )\cap |\varSigma _0|^0) \times [0,1]}\) is a deformation retraction from \(\sigma \cap Y_{\psi _{\widetilde{\varSigma }_0}}(\chi )\cap |\varSigma _0|^0\) to \(\sigma \cap Y'(\chi )\cap |\varSigma _0|^0\).
-
\(H_\sigma |_{(\sigma \cap \{ x \in |\varSigma _0|^0: \chi (x)\le 0\}) \times [0,1]}\) is a deformation retraction from \(\sigma \cap \{ x \in |\varSigma _0|^0: \chi (x)\le 0\}\) to \(\sigma \cap Y'(\chi )\cap |\varSigma _0|^0\).
To define \(H_\sigma \), let \(v_1,\ldots ,v_r,w_1,\ldots ,w_s\) denote the shortest nonzero vectors in \(X_*(S)\) on each of the one-dimensional faces of \(\sigma \) (so that \(r+s=\dim \sigma \)), with the notation chosen such that \(\chi (v_i)\le 0\) for all i and \(\chi (w_j)>0\) for all j. Note that \(1-\chi (v_i) >0\) for all i and \(1-\chi (w_j) \le 0\) for all j. We set
Because
and
are convex sets, and because
it is easy to check that it has the desired properties and the proof of the lemma is complete. \(\square \)
Lemma 2.15
Let Y be a connected, separated scheme which is flat and locally of finite type over an irreducible noetherian ring \(R_0\), let \(\alpha : S \rightarrow S'\) be an isogeny of split tori over Y, and let \(\varSigma _0'\) (resp. \(\varSigma _0\)) be a locally finite partial fan for \(S'\) (resp. S). Suppose that \(\varSigma _0'\) is full. Also suppose that \(\varSigma _0\) is a quasi-refinement of \(\varSigma _0'\), and let \(\pi ^\wedge \) denote the map \(T_{\varSigma _0}^\wedge \rightarrow (\alpha _*T)_{\varSigma _0'}^\wedge \).
Then for \(i>0\) we have
while
If moreover \(\varSigma _0\) and \(\varSigma _0'\) are smooth then, for \(i>0\) we have
while
Proof
We may reduce to the case that \(Y={{\text {Spec}}\,}A\) is affine. The map \(\pi ^\wedge \) is proper and hence
and
and
and
are coherent sheaves. Thus they have closed support. Their support is also S-invariant. Thus it suffices to show that for each maximal element \(\sigma ' \in \varSigma _0'\) the space \(\partial _{\sigma '}(\alpha _*T)_{\widetilde{\varSigma }_0'}\) does not lie in the support of these sheaves. Let \(\varSigma _0(\sigma ')\) denote the subset of elements \(\sigma \in \varSigma _0\) which lie in \(\sigma '\), but in no face of \(\sigma '\). Then \(\varSigma _0(\sigma ')\) is a partial fan and \(T^\wedge _{\varSigma _0(\sigma ')}\) equals the formal completion of \(T^\wedge _{\varSigma _0}\) along \(\partial _{\sigma '}T_{\widetilde{\varSigma }_0'}\). Thus the formal completion of the above four sheaves along \(\partial _{\sigma '}T_{\widetilde{\varSigma }_0'}\) equal the corresponding sheaf for the pair \(\varSigma _0(\sigma ')\) and \(\{ \sigma '\}\), so that we are reduced to the case that \(\varSigma _0'=\{ \sigma '\}\) is a singleton.
In the case that \(\varSigma _0'=\{\sigma '\}\) then \((\alpha _*T)_{\varSigma _0'}\) and Y have the same underlying topological space. Let \(\pi _1^\wedge \) denote the map of ringed spaces \(T^\wedge _{\varSigma _0} \rightarrow Y\). Then it suffices to show that for \(i>0\) we have
and
and that we have
and
This follows from Lemma 2.14. (Note that
where \(\ker \alpha \) acts on the \(\xi \) term via \(\xi \); and that
These assertions remain true with \(|\varSigma _0|^{\vee ,0}\) replacing \(|\varSigma _0|^\vee \) and \(|\{ \sigma '\}|^{\vee ,0}\) replacing \(|\{\sigma '\}|^{\vee }\).)
\(\square \)
Lemma 2.16
Let Y be a connected, separated scheme which is flat and locally of finite type over an irreducible noetherian ring \(R_0\), let S / Y be a split torus, let T / Y be an S-torsor, let \(\varSigma _0\) be a partial fan for S, and let \(\pi _{\varSigma _0}^\wedge \) denote the map of ringed spaces \(T^\wedge _{\varSigma _0} \rightarrow Y\). Suppose that \(\varSigma _0\) is non-empty, full, locally finite and open, and that \(|\varSigma _0|^0\) is convex.
-
(1)
Then
$$R^i\pi ^\wedge _{\varSigma _0,*} {\mathcal {O}}_{T_{\varSigma _0}^\wedge }=\left\{ \begin{array}{ll} \prod _{\chi \in |\varSigma _0|^\vee } {\mathcal {L}}(\chi )&{}\quad \mathrm{if}\,\, i=0 \\ (0) &{}\quad \mathrm{otherwise.}\end{array} \right. $$ -
(2)
If in addition \(\varSigma _0\) is smooth then
$$R^i\pi _{\varSigma _0,*}^\wedge {\mathcal {I}}_{\partial ,\varSigma _0}^\wedge =\left\{ \begin{array}{ll} \prod _{\chi \in |\varSigma _0|^{\vee ,0}} {\mathcal {L}}(\chi ) &{} \quad \mathrm{if}\,\, i=0 \\ (0) &{}\quad \mathrm{otherwise.}\end{array} \right. $$
Proof
Let \(\sigma _1,\sigma _2,\ldots \) be an enumeration of the 1 cones in \(\widetilde{\varSigma }_0\). Let \(\varDelta ^{(i)} \subset |\varSigma |\) denote the convex hull of \(\sigma _1,\ldots ,\sigma _i\). It is a rational polyhedral cone contained in \(|\varSigma _0|\), and there exists \(i_0\) such that for \(i \ge i_0\) the cone \(\varDelta ^{(i)}\) will have the same dimension as \(X_*(S)_{\mathbb {R}}\). Let \(\partial \varDelta ^{(i)}\) denote the union of the proper faces of \(\varDelta ^{(i)}\); and let \(\varDelta ^{(i),c}\) denote the closure of \(|\varSigma _0|-\varDelta ^{(i)}\) in \(|\varSigma _0|\).
Define recursively fans \(\varSigma ^{(i)}\) and boundary data \(\varSigma ^{(i)}_0\) as follows. We set \(\varSigma ^{(i_0-1)}=\widetilde{\varSigma }_0\) and \(\varSigma ^{(i_0-1)}_0=\varSigma _0\). For \(i\ge i_0\) set
Then \(\varSigma ^{(i)}\) refines \(\varSigma ^{(i-1)}\) and we choose \(\varSigma ^{(i)}_0\) to be the unique subset of \(\varSigma ^{(i)}\) such that \((\varSigma ^{(i)},\varSigma _0^{(i)})\) refines \((\varSigma ^{(i-1)},\varSigma _0^{(i-1)})\). Then \(\widetilde{\varSigma _0^{(i)}}=\varSigma ^{(i)}\). We also check by induction on i that
-
\(\varSigma ^{(i)} \cup \varSigma ^{(i-1)}-(\varSigma ^{(i)}\cap \varSigma ^{(i-1)})\) is finite;
-
and \(\varSigma ^{(i)}_0\) is locally finite.
(The point being that the local finiteness of \(\varSigma ^{(i-1)}_0\) implies that only finitely many elements of \(\varSigma ^{(i-1)}_0\), and hence of \(\varSigma ^{(i-1)}\), meet both \(\varDelta ^{(i)}-\partial \varDelta ^{(i)}\) and \(X_*(S)_{\mathbb {R}}-\varDelta ^{(i)}\).)
Now define \(\varSigma ^{(\infty )}\) (resp. \(\varSigma ^{(\infty )}_0\)) to be the set of cones that occur in \(\varSigma ^{(i)}\) (resp. \(\varSigma _0^{(i)}\)) for infinitely many i. Alternatively
Also let \(\varSigma ^{(\infty ),{{\text {sm}}}}\) denote a smooth refinement of \(\varSigma ^{(\infty )}\) and let \(\varSigma _0^{(\infty ),{{\text {sm}}}}\) denote the elements of \(\sigma \in \varSigma ^{(\infty ),{{\text {sm}}}}\) for which there exists \(\tau \in \varSigma _0\) such that \(\sigma \subset \tau \) and \(\sigma \cap \tau ^0 \ne \emptyset \). (See Lemma 2.4.) Then \(\varSigma ^{(\infty ),{{\text {sm}}}}\) is a fan, \(\varSigma _0^{(\infty ),{{\text {sm}}}}\) provides locally finite boundary data for \(\varSigma ^{(\infty ),{{\text {sm}}}}\), we have \(\widetilde{\varSigma _0^{(\infty ),{{\text {sm}}}}}=\varSigma ^{(\infty ),{{\text {sm}}}}\), and \((\varSigma ^{(\infty ),{{\text {sm}}}},\varSigma ^{(\infty ),{{\text {sm}}}}_0)\) refines \((\widetilde{\varSigma }_0,\varSigma _0)\). Moreover \(\varSigma ^{(\infty ),{{\text {sm}}}}_0\) is open. We also define \(\varSigma ^{(\infty ),{{\text {sm}}}}_i\) to be the set of \(\sigma \in \varSigma ^{(\infty ),{{\text {sm}}}}_0\) such that \(\sigma \subset \varDelta ^{(i)}\) but \(\sigma \not \subset \partial \varDelta ^{(i)}\). Note that:
-
\(\varSigma ^{(\infty ),{{\text {sm}}}}_i\) is finite and open;
-
\(\varSigma ^{(\infty ),{{\text {sm}}}}_i \supset \varSigma ^{(\infty ),{{\text {sm}}}}_{i-1}\);
-
\(|\varSigma ^{(\infty ),{{\text {sm}}}}_i|^0=\varDelta ^{(i)} - \partial \varDelta ^{(i)}\) is convex;
-
and \(\varSigma ^{(\infty ),{{\text {sm}}}}_0=\bigcup _{i>0} \varSigma ^{(\infty ),{{\text {sm}}}}_i\).
(For the last of these properties use the fact that \(\varSigma ^{(\infty ),{{\text {sm}}}}_0\) is open.)
By Lemma 2.15 it suffices to prove this lemma after replacing the pair \(\varSigma _0\) by \(\varSigma ^{(\infty ),{{\text {sm}}}}_0\). This lemma then follows from Lemmas 2.13 and 2.14. \(\square \)
3.5 The case of a disconnected base
Throughout this section we will continue to assume that Y is a separated scheme, flat and locally of finite type over \({{\text {Spec}}\,}R_0\). We prove nothing new in this section, we simply explain how to re-express the last two sections in a way that makes sense over a disconnected base, so that the results we have already established immediately extend.
Let S be a split torus over Y and let T / Y be an S-torsor. By a rational polyhedral cone \(\sigma \) in \(X_*(S)_{\mathbb {R}}\) we shall mean a locally constant sheaf of subsets \(\sigma \subset X_*(S)_{\mathbb {R}}\), such that
-
for each connected open \(U \subset Y\) the set \(\sigma (U) \subset X_*(S)_{\mathbb {R}}(U)\) is either empty or a rational polyhedral cone,
-
and the locus where \(\sigma \ne \emptyset \) is non-empty and connected. We call this locus the support of \(\sigma \).
We call \(\sigma '\) a face of \(\sigma \) if for each open connected U either \(\sigma (U)=\sigma '(U)=\emptyset \) or the cone \(\sigma '(U)\) is a face of \(\sigma (U)\). We call \(\sigma \) smooth if each \(\sigma (U)\) is smooth. By a fan \(\varSigma \) in \(X_*(S)_{\mathbb {R}}\) we mean a set of rational polyhedral cones in \(X_*(S)_{\mathbb {R}}\), such that
-
if \(\sigma \in \varSigma \) then so is any face \(\sigma '\) of \(\sigma \);
-
if \(\sigma ,\sigma ' \in \varSigma \) then \(\sigma \cap \sigma '\) is either empty or a face of \(\sigma \) and \(\sigma '\),
-
any connected component of Y arises as the support of some element of \(\varSigma \).
Thus to give a fan in \(X_*(S)_{\mathbb {R}}\) is the same as giving a fan in \(X_*(S)_{\mathbb {R}}(Z)\) for each connected component Z of Y. If U is a non-empty connected open in Y then we set
It is a fan for \(X_*(S)_{\mathbb {R}}(U)\).
We call \(\varSigma \) smooth (resp. full, resp. finite) if each \(\varSigma (U)\) is. We define a locally constant sheaf \(|\varSigma |\) of subsets of \(X_*(S)_{\mathbb {R}}\) by setting
(resp.
for U any connected open subset of Y. We will call \(|\varSigma |\) (resp. \(\vert \varSigma \vert ^*\)) convex if \(|\varSigma |(U)\) (resp. \(|\varSigma |^*(U)\)) is for each connected open \(U\subset Y\). We also define locally constant sheaves of subsets \(|\varSigma |^{\vee }\) and \(|\varSigma |^{\vee ,0}\) of \(X^*(S)_{\mathbb {R}}\) by setting
and
We call \(\varSigma '\) a refinement of \(\varSigma \) if each \(\varSigma '(U)\) is a refinement of \(\varSigma (U)\) for each open, connected U. Any fan \(\varSigma \) has a smooth refinement \(\varSigma '\) such that every smooth cone \(\sigma \in \varSigma \) also lies in \(\varSigma '\).
To a fan \(\varSigma \) one can attach a scheme \(T_\varSigma \) flat and separated over Y and locally (on \(T_\varSigma \)) of finite type over Y, together with an action of S and an S-equivariant embedding \(T \hookrightarrow T_\varSigma \). It has an open cover \(\{ T_\sigma \}_{\sigma \in \varSigma }\), with each \(T_\sigma \) relatively affine over Y. Over a connected open \(U \subset Y\) this restricts to the corresponding picture for \({\varSigma (U)}\). We write \({\mathcal {O}}_{T_\varSigma }\) for the structure sheaf of \(T_\varSigma \). If \(\varSigma \) is smooth then \(T_\varSigma /Y\) is smooth. If \(\varSigma \) is finite and \(|\varSigma |=X_*(S)_{\mathbb {R}}\), then \(T_\varSigma \) is proper over Y. If \(\varSigma '\) refines \(\varSigma \) then there is an S-equivariant proper map
which restricts to the identity on T.
By boundary data we shall mean a proper subset \(\varSigma _0 \subset \varSigma \) such that \(\varSigma - \varSigma _0\) is a fan. If \(U\subset Y\) is a connected open we set
If \(\varSigma _0\) is boundary data, then we can associate to it a closed subscheme \(\partial _{\varSigma _0} T_\varSigma \subset T_\varSigma \), which over a connected open \(U\subset Y\) restricts to \(\partial _{\varSigma _0(U)} (T|_U)_{\varSigma (U)} \subset (T|_U)_{\varSigma (U)}\).
In the case that \(\varSigma _0\) is the set of elements of \(\varSigma \) of dimension bigger than 0 we shall simply write \(\partial T_\varSigma \) for \(\partial _{\varSigma _0}T_\varSigma \). Thus \(T=T_\varSigma -\partial T_\varSigma \). We will write \({\mathcal {I}}_{\partial T_\varSigma }\) for the ideal of definition of \(\partial T_\varSigma \) in \({\mathcal {O}}_{T_\varSigma }\). We will also write \({\mathcal {M}}_\varSigma \rightarrow {\mathcal {O}}_{T_\varSigma }\) for the associated log structure and \(\varOmega ^1_{T_\varSigma /{{\text {Spec}}\,}R_0}(\log \infty )\) for the log differentials \(\varOmega ^1_{T_\varSigma /{{\text {Spec}}\,}R_0}(\log {\mathcal {M}}_\varSigma )\).
If \(\varSigma \) is smooth then \(\partial T_\varSigma \) is a simple normal crossings divisor on \(T_\varSigma \) relative to Y.
If \(\sigma \in \varSigma \) has positive dimension and if \(\varSigma _0\) denotes the set of elements of \(\varSigma \) which have \(\sigma \) for a face, then we will write \(\partial _\sigma T_\varSigma \) for \(\partial _{\varSigma _0} T_\varSigma \). It is connected and flat over Y, and, locally on Y, it has codimension in \(T_\varSigma \) equal to the dimension of \(\sigma \). If \(\varSigma \) is smooth then each \(\partial _\sigma T_\varSigma \) is smooth over Y. The schemes \(\partial _{\sigma _1} T_\varSigma ,\ldots ,\partial _{\sigma _s} T_\varSigma \) intersect if and only if \(\sigma _1, \ldots , \sigma _s\) are all contained in some \(\sigma \in \varSigma \). In this case the intersection equals \(\partial _\sigma T_\varSigma \) for the smallest such \(\sigma \). We set
If the connected components of Y are irreducible then each \(\partial _\sigma T_\varSigma \) is irreducible. Moreover the irreducible components of \(\partial T_\varSigma \) are the \(\partial _\sigma T_\varSigma \) as \(\sigma \) runs over one-dimensional elements of \(\varSigma \). If \(\varSigma \) is smooth then we see that \({\mathcal {S}}(\partial T_\varSigma )\) is the delta set with cells in bijection with the elements of \(\varSigma \) with dimension bigger than 0 and with the same ‘face relations’. In particular it is in fact a simplicial complex and
We will call \(\varSigma _0\) open (resp. finite, resp. locally finite) if \(\varSigma _0(U)\) is for each open connected \(U \subset Y\). If \(\varSigma _0\) is finite and open, then \(\partial _{\varSigma _0} T_\varSigma \) is proper over Y.
By a partial fan in \(X_*(S)\) we mean a collection \(\varSigma _0\) of rational polyhedral cones in \(X_*(S)\) such that
-
\(\varSigma _0\) does not contain \((0) \subset X_*(S)(U)_{\mathbb {R}}\) for any open connected U;
-
if \(\sigma _1, \sigma _2 \in \varSigma _0\) then \(\sigma _1 \cap \sigma _2\) is either empty or a face of \(\sigma _1\) and of \(\sigma _2\);
-
if \(\sigma _1, \sigma _2 \in \varSigma _0\) and if \(\sigma \supset \sigma _2\) is a face of \(\sigma _1\), then \(\sigma \in \varSigma _0\).
In this case we will let \(\widetilde{\varSigma }_0\) denote the set of faces of elements of \(\varSigma _0\) together with \(\{0\}\) supported on any connected component of Y. It is a fan, and \(\varSigma _0\) is boundary data for \(\widetilde{\varSigma }_0\). By boundary data \(\varSigma _1\) for \(\varSigma _0\) we shall mean a subset \(\varSigma _1 \subset \varSigma _0\) such that if \(\sigma \in \varSigma _0\) contains \(\sigma _1 \in \varSigma _1\), then \(\sigma \in \varSigma _1\). In this case \(\varSigma _1\) is again a partial fan and boundary data for \(\widetilde{\varSigma }_0\). We say that a partial fan \(\varSigma _0\) for \(X_*(S)\) refines a partial fan \(\varSigma _0'\) for \(X_*(S)\) if every element of \(\varSigma _0\) lies in an element of \(\varSigma _0'\) and if every element of \(\varSigma _0'\) is a finite union of elements of \(\varSigma _0\).
If \(\varSigma _0\) is a partial fan we define locally constant sheaves of subsets \(|\varSigma _0|\), \(|\varSigma _0|^*\), \(|\varSigma _0|^\vee \) and \(|\varSigma _0|^{\vee ,0}\) of \(X_*(S)_{\mathbb {R}}\) or \(X^*(S)_{\mathbb {R}}\) to be \(|\widetilde{\varSigma }_0|\), \(|\widetilde{\varSigma }_0|^*\), \(|\widetilde{\varSigma }_0|^\vee \) and \(|\widetilde{\varSigma }_0|^{\vee ,0}\), respectively. We also define a sheaf of subsets \(|\varSigma _0|^0\) by
for any connected open set \(U \subset Y\). We will call \(|\varSigma _0|\) (resp. \(|\varSigma _0|^0\)) convex if \(|\varSigma _0|(U)\) (resp. \(|\varSigma _0|^0(U)\)) is convex for all open connected subsets \(U \subset Y\).
We will call \(\varSigma _0\) smooth (resp. full, resp. open, resp. finite, resp. locally finite) if \(\varSigma _0(U)\) is for each \(U \subset Y\) open and connected. We will call \(\varSigma _0\) non-degenerate if for each non-empty connected open subset \(U \subset Y\) the set \(\varSigma _0(U)\) is non-empty.
If \(\varSigma _0\) is a partial fan we will write
for \(\partial _{\varSigma _0} T_{\widetilde{\varSigma }_0}\);
for the completion of \(T_{\widetilde{\varSigma }_0}\) along \(\partial _{\varSigma _0} T_{\widetilde{\varSigma }_0}\); and
for the log structure induced by \({\mathcal {M}}_{\widetilde{\varSigma }_0}\). We make the following definitions.
-
\({\mathcal {I}}_{T_{\varSigma _0}^\wedge }\) will denote the completion of \({\mathcal {I}}_{\partial _{\varSigma _0} T_{\widetilde{\varSigma }_0}}\), the sheaf of ideals defining \(\partial _{\varSigma _0} T_{\widetilde{\varSigma }_0}\). It is an ideal of definition for \(T_{\varSigma _0}^\wedge \).
-
\({\mathcal {I}}_{\partial , \varSigma _0}^\wedge \) will denote the completion of \({\mathcal {I}}_{\partial T_{\widetilde{\varSigma }_0}}\), the sheaf of ideals defining \(\partial _{\varSigma _0} T_{\widetilde{\varSigma }_0}\). Thus \({\mathcal {I}}_{\partial , \varSigma _0}^\wedge \subset {\mathcal {I}}_{T_{\varSigma _0}^\wedge }\).
-
\(\varOmega ^1_{T_{\varSigma _0}^\wedge /{{\text {Spf}}\,}R_0}(\log \infty )\) will denote \(\varOmega ^1_{T_{\varSigma _0}^\wedge /{{\text {Spf}}\,}R_0}(\log {\mathcal {M}}_\varSigma ^\wedge )\).
We will write
(resp.
for the sheaf (of abelian groups) on Y such that for any connected open subset \(U \subset Y\) we have
(resp.
Suppose that \(\alpha : S \rightarrow S'\) is a surjective map of split tori over Y. Then \(X^*(\alpha ): X^*(S') \hookrightarrow X^*(S)\) and \(X_*(\alpha ): X_*(S)_{\mathbb {R}}\twoheadrightarrow X_*(S')_{\mathbb {R}}\). We call fans \(\varSigma \) for \(X_*(S)\) and \(\varSigma '\) for \(X_*(S')\) compatible if for all \(\sigma \in \varSigma \) the image \(X_*(\alpha ) \sigma \) is contained in some element of \(\varSigma '\). In this case the map \(\alpha : T \rightarrow \alpha _* T\) extends to an S-equivariant map
We will write
The following lemma is an immediate consequence of Lemma 2.5.
Lemma 2.17
If \(\alpha \) is surjective and \(\# {{\text {coker}}\,}X_*(\alpha )\) is invertible on Y then \(\alpha : (T_\varSigma ,{\mathcal {M}}_\varSigma ) \rightarrow ((\alpha _*T)_{\varSigma '},{\mathcal {M}}_{\varSigma '})\) is log smooth, and there is a natural isomorphism
If \(\alpha \) is an isogeny, if \(\varSigma \) and \(\varSigma '\) are compatible, and if every element of \(\varSigma '\) is a finite union of elements of \(\varSigma \), then we call \(\varSigma \) a quasi-refinement of \(\varSigma '\). In that case the map \(\alpha : T_{\varSigma } \rightarrow (\alpha _*T)_{\varSigma '}\) is proper.
Suppose that \(\alpha : S \twoheadrightarrow S'\) is a surjective map of tori, and that \(\varSigma _0\) (resp. \(\varSigma _0'\)) is a partial fan for S (resp. \(S'\)). We call \(\varSigma _0\) and \(\varSigma _0'\) compatible if for every \(\sigma \in \varSigma _0\) the image \(X_*(\alpha ) \sigma \) is contained in some element of \(\varSigma _0'\) but in no element of \(\widetilde{\varSigma }_0'-\varSigma _0'\). In this case there is a natural morphism
We will write
The following lemma follows immediately from Lemma 2.8.
Lemma 2.18
If \(\alpha \) is surjective and \(\# {{\text {coker}}\,}X_*(\alpha )\) is invertible on Y then there is a natural isomorphism
We will call \(\varSigma _0\) and \(\varSigma _0'\) strictly compatible if they are compatible and if an element of \(\widetilde{\varSigma }_0\) lies in \(\varSigma _0\) if and only if it maps to no element of \(\widetilde{\varSigma }_0'-\varSigma _0'\). We will say that
-
\(\varSigma _0\) is open over \(\varSigma _0'\) if \(|\varSigma _0|^0(U)\) is open in \(X_*(\alpha )^{-1} |\varSigma _0'|^0(U)\) for all connected opens \(U \subset Y\);
-
and that \(\varSigma _0\) is finite over \(\varSigma _0'\) if only finitely many elements of \(\varSigma _0\) map into any element of \(\varSigma _0'\).
If \(\alpha \) is an isogeny, if \(\varSigma _0\) and \(\varSigma _0'\) are strictly compatible and if every element of \(\varSigma _0'\) is a finite union of elements of \(\varSigma _0\), then we call \(\varSigma _0\) a quasi-refinement of \(\varSigma _0'\). In this case \(\varSigma _0\) is open and finite over \(\varSigma _0'\). The next lemma follows immediately from Lemma 2.9.
Lemma 2.19
Suppose that \(\varSigma _0'\) and \(\varSigma _0\) are strictly compatible.
-
(1)
\(T_{\varSigma _0}^\wedge \) is the formal completion of \(T_{\widetilde{\varSigma }_0}\) along \(\partial _{\varSigma _0'}(\alpha _* T)\), and \(T_{\varSigma _0}^\wedge \) is locally (on the source) topologically of finite type over \((\alpha _*T)^\wedge _{\varSigma _0'}\).
-
(2)
If \(\varSigma _0\) is locally finite and if it is open and finite over \(\varSigma _0'\) then \(T_{\varSigma _0}^\wedge \) is proper over \((\alpha _* T)_{\varSigma _0'}^\wedge \).
Corollary 2.20
If \(\alpha \) is an isogeny, if \(\varSigma _0\) is locally finite and if \(\varSigma _0\) is a quasi-refinement of \(\varSigma _0'\) then \(T_{\varSigma _0}^\wedge \) is proper over \((\alpha _* T)_{\varSigma _0'}^\wedge \).
If \(\varSigma _0\) and \(\varSigma _0'\) are compatible partial fans and if \(\varSigma _1' \subset \varSigma _0'\) is boundary data then \(\varSigma _0(\varSigma _1')\) will denote the set of elements \(\sigma \in \varSigma _0\) such that \(X_*(\alpha )\sigma \) is contained in no element of \(\varSigma _0'-\varSigma _1'\). It is boundary data for \(\varSigma _0\). Moreover the formal completion of \(T_{\varSigma _0}^\wedge \) along the reduced subscheme of \((\alpha _*T)_{\varSigma _1'}^\wedge \) is canonically identified with \(T^\wedge _{\varSigma _0(\varSigma _1')}\). If \(\varSigma _1'=\{ \sigma '\}\) is a singleton we will write \(\varSigma _0(\sigma ')\) for \(\varSigma _0(\{ \sigma '\})\).
The next two lemmas follow immediately from Lemmas 2.15 and 2.16, respectively.
Lemma 2.21
Let Y be a separated scheme which is flat and locally of finite type over an irreducible noetherian ring \(R_0\), let \(\alpha : S \rightarrow S'\) be an isogeny of split tori over Y, and let \(\varSigma _0'\) (resp. \(\varSigma _0\)) be a locally finite partial fan for \(S'\) (resp. S). Suppose that Y is separated and locally noetherian, that \(\varSigma _0'\) is full and that \(\varSigma _0\) is locally finite. Also suppose that \(\varSigma _0\) is a quasi-refinement of \(\varSigma _0'\). Let \(\pi ^\wedge \) denote the map \(T_{\sigma _0}^\wedge \rightarrow (\alpha _*T)^\wedge _{\sigma _0'}\).
Then for \(i>0\) we have
while
If moreover \(\varSigma _0\) and \(\varSigma _0'\) are smooth then, for \(i>0\) we have
while
Lemma 2.22
Let Y be a separated scheme which is flat and locally of finite type over an irreducible noetherian ring \(R_0\), let S / Y be a split torus, let T / Y is an S-torsor, let \(\varSigma _0\) be a partial fan for S, and let \(\pi _{\varSigma _0}^\wedge \) denote the map \(T^\wedge _{\varSigma _0} \rightarrow Y\). Suppose that Y is separated and locally noetherian, that \(\varSigma _0\) is non-degenerate, full, locally finite and open and that \(|\varSigma _0|^0\) is convex.
-
(1)
Then
$$R^i\pi ^\wedge _{\varSigma _0,*} {\mathcal {O}}_{T_{\varSigma _0}^\wedge }=\left\{ \begin{array}{ll} \prod _{\chi \in |\varSigma _0|^\vee } {\mathcal {L}}(\chi )&{} \quad \mathrm{if}\,\, i=0 \\ (0) &{} \quad \mathrm{otherwise.}\end{array} \right. $$ -
(2)
If in addition \(\varSigma _0\) is smooth then
$$R^i\pi _{\varSigma _0,*}^\wedge {\mathcal {I}}_{\partial , \varSigma _0}^\wedge =\left\{ \begin{array}{ll} \prod _{\chi \in |\varSigma _0|^{\vee ,0}} {\mathcal {L}}(\chi ) &{} \quad \mathrm{if}\,\, i=0 \\ (0) &{} \quad \mathrm{otherwise.}\end{array} \right. $$
4 Shimura varieties
In this section we will describe the Shimura varieties associated to \(G_n\) and the mixed Shimura varieties associated to \(G_n^{(m)}\) and \(\widetilde{{G}}_n^{(m)}\). We assume that all schemes discussed in this section are locally noetherian.
4.1 Some Shimura varieties
4.1.1 Moduli problems
By a \(G_n\)-abelian scheme over a scheme \(Y/{\mathbb {Q}}\) we shall mean an abelian scheme A / Y of relative dimension \(n[F: {\mathbb {Q}}]\) together with an embedding
such that \({{\text {Lie}}\,}A\) is a locally free right \({\mathcal {O}}_Y \otimes _{\mathbb {Q}}F\)-module of rank n. By a morphism (resp. quasi-isogeny) of \(G_n\)-abelian schemes we mean a morphism (resp. quasi-isogeny) of abelian schemes which commutes with the F-action. If (A, i) is a \(G_n\)-abelian scheme then we give \(A^\vee \) the structure \((A^\vee ,i^\vee )\) of a \(G_n\)-abelian scheme by setting \(i^\vee (a)=i({}^ca)^\vee \). By a quasi-polarization of a \(G_n\)-abelian scheme (A, i) / Y we shall mean a quasi-isogeny \(\lambda : A \rightarrow A^\vee \) of \(G_n\)-abelian schemes, some \({\mathbb {Q}}^\times \)-multiple of which is a polarization. (Note that according to this convention, if \(\lambda \) is a polarization, then \(-\lambda \) is a quasi-polarization.) If \(Y={{\text {Spec}}\,}k\) with k a field, we will let \(\langle \,\,\,,\,\,\,\rangle _\lambda \) denote the Weil pairing induced on the adelic Tate module VA (see section 23 of [47]).
Lemma 3.1
If k is a field of characteristic 0 and if \((A,i,\lambda )/k\) is a quasi-polarized \(G_n\)-abelian scheme, then \(V_p(A \times \overline{{k}})\) is a free \(F_p\)-module of rank 2n.
Proof
We may suppose that k is a finitely generated field extension of \({\mathbb {Q}}\), which we may embed into \({\mathbb {C}}\). Then
so that \(V_p(A \times \overline{{k}}) \otimes _{{\mathbb {Q}}_p,\imath } {\mathbb {C}}\) is a free \(F \otimes _{\mathbb {Q}}{\mathbb {C}}\)-module. As \(F \otimes _{\mathbb {Q}}{\mathbb {C}}=F_p \otimes _{{\mathbb {Q}}_p, \imath } {\mathbb {C}}\) we deduce that \(V_p (A \times \overline{{k}})\) is a free \(F_p\)-module, as desired. \(\square \)
By an ordinary \(G_n\)-abelian scheme over a \({\mathbb {Z}}_{(p)}\)- scheme Y we shall mean an abelian scheme A / Y of relative dimension \(n[F: {\mathbb {Q}}]\), such that for each geometric point \({\overline{{y}}}\) of Y we have \(\# A[p](k({\overline{{y}}})) \ge p^{n[F: {\mathbb {Q}}]}\), together with an embedding
such that \({{\text {Lie}}\,}A\) is a locally free right \({\mathcal {O}}_Y \otimes _{{\mathbb {Z}}_{(p)}} {\mathcal {O}}_{F,(p)}\)-module of rank n. By a morphism of ordinary \(G_n\)-abelian schemes we mean a morphism of abelian schemes which commutes with the \({\mathcal {O}}_{F,(p)}\)-action. If (A, i) is an ordinary \(G_n\)-abelian scheme then we give \(A^\vee \) the structure, \((A^\vee ,i^\vee )\), of a \(G_n\)-abelian scheme by setting \(i^\vee (a)=i({}^ca)^\vee \). By a prime-to- p quasi-polarization of an ordinary \(G_n\)-abelian scheme (A, i) / Y we shall mean a prime-to-p quasi-isogeny \(\lambda : A \rightarrow A^\vee \) of ordinary \(G_n\)-abelian schemes, some \({\mathbb {Z}}^\times _{(p)}\)-multiple of which is a prime-to-p polarization.
If U is an open compact subgroup of \(G_n({\mathbb {A}}^{\infty })\) then by a U-level structure on a quasi-polarized \(G_n\)-abelian scheme \((A,i,\lambda )\) over a connected scheme \(Y/{{\text {Spec}}\,}{\mathbb {Q}}\) with a geometric point \({\overline{{y}}}\), we mean a \(\pi _1(Y,{\overline{{y}}})\)-invariant U-orbit \([\eta ]\) of pairs \((\eta _0,\eta _1)\) of \({\mathbb {A}}^\infty \)-linear isomorphisms
and
such that
for all \(a \in F\) and \(x \in V_n \otimes _{{\mathbb {Q}}} {\mathbb {A}}^{\infty }\), and such that
for all \(x,y \in V_n \otimes _{{\mathbb {Q}}} {\mathbb {A}}^{\infty }\). This definition is independent of the choice of geometric point \({\overline{{y}}}\) of Y. By a U-level structure on a quasi-polarized \(G_n\)-abelian scheme \((A,i,\lambda )\) over a general (locally noetherian) scheme \(Y/{{\text {Spec}}\,}{\mathbb {Q}}\), we mean the collection of a U-level structure over each connected component of Y. If \([(\eta _0,\eta _1)]\) is a level structure we define \(||\eta _0|| \in {\mathbb {Q}}^\times _{>0}\) by
Now suppose that \(U^p\) is an open compact subgroup of \(G_n({\mathbb {A}}^{\infty ,p})\) and that \(N_1 \le N_2\) are non-negative integers. By a \(U^p(N_1,N_2)\)-level structure on an ordinary, prime-to-p quasi-polarized, \(G_n\)-abelian scheme \((A,i,\lambda )\) over a connected scheme \(Y/{{\text {Spec}}\,}{\mathbb {Z}}_{(p)}\) with a geometric point \({\overline{{y}}}\), we mean a \(\pi _1(Y,{\overline{{y}}})\)-invariant \(U^p\)-orbit \([\eta ]\) of four-tuples \((\eta _0^p,\eta _1^p,C,\eta _p)\) consisting of
-
an \({\mathbb {A}}^{\infty ,p}\)-linear isomorphism \(\eta _0^p: {\mathbb {A}}^{\infty ,p}_{\overline{{y}}}\mathop {\longrightarrow }\limits ^{\sim }{\mathbb {A}}^{\infty ,p}(1)_{\overline{{y}}}=V^p{\mathbb {G}}_{m,{\overline{{y}}}}\);
-
an \({\mathbb {A}}_F^{\infty ,p}\)-linear isomorphism
$$\begin{aligned} \eta _1^p: V_n \otimes _{{\mathbb {Q}}} {\mathbb {A}}^{\infty ,p} \mathop {\longrightarrow }\limits ^{\sim }V^pA_{\overline{{y}}}\end{aligned}$$such that
$$\begin{aligned} \langle \eta _1^p x , \eta _1^p y \rangle _\lambda =\eta _0 \langle x,y \rangle _n \end{aligned}$$for all \(x,y \in V_n \otimes _{{\mathbb {Q}}} {\mathbb {A}}^{\infty ,p}\);
-
a locally free sub-\({\mathcal {O}}_{F,(p)}\)-module scheme \(C \subset A[p^{N_2}]\), such that for every geometric point \(\widetilde{{y}}\) of Y there is an \({\mathcal {O}}_{F,(p)}\)-invariant sub-Barsotti–Tate group \(\widetilde{{C}}_{\widetilde{{y}}} \subset A_{\widetilde{{y}}}[p^\infty ]\) with the following properties
-
\(C_{\widetilde{{y}}}=\widetilde{{C}}_{\widetilde{{y}}}[p^{N_2}]\),
-
for all N the subgroup scheme \(\widetilde{{C}}_{\widetilde{{y}}}[p^N]\) is isotropic in \(A[p^N]_{\widetilde{{y}}}\) for the \(\lambda \)-Weil pairing,
-
\(A_{\widetilde{{y}}}[p^\infty ]/\widetilde{{C}}_{\widetilde{{y}}}\) is ind-etale,
-
the Tate module \(T(A_{\widetilde{{y}}}[p^\infty ]/\widetilde{{C}}_{\widetilde{{y}}})\) is free over \({\mathcal {O}}_{F,p}\) of rank n;
-
-
and an isomorphism
$$\begin{aligned} \eta _p: p^{-N_1}\varLambda _n\big /\big (p^{-N_1}\varLambda _{n,(n)}+ \varLambda _n\big ) \mathop {\longrightarrow }\limits ^{\sim }A[p^{N_1}]/(A[p^{N_1}] \cap C) \end{aligned}$$such that
$$\begin{aligned} \eta _p(ax)=i(a) \eta _p(x) \end{aligned}$$for all \(a \in {\mathcal {O}}_{F,(p)}\) and \(x \in p^{-N_1}\varLambda _n/(p^{-N_1}\varLambda _{n,(n)}+ \varLambda _n)\).
This definition is independent of the choice of the geometric point \({\overline{{y}}}\in Y\). By a \(U^p(N_1,N_2)\)-level structure on an ordinary, prime-to-p quasi-polarized, \(G_n\)-abelian scheme \((A,i,\lambda )\) over a general (locally noetherian) scheme \(Y/{{\text {Spec}}\,}{\mathbb {Z}}_{(p)}\), we mean the collection of a \(U^p(N_1,N_2)\)-level structure over each connected component of Y. If \([(\eta _0^p,\eta _1^p,C,\eta _p)]\) is a level structure we define \(||\eta _0^p|| \in {\mathbb {Z}}^\times _{(p),>0}\) by
By a quasi-isogeny (resp. prime-to-p quasi-isogeny) between quasi-polarized, \(G_n\)-abelian schemes with U-level structures (resp. ordinary, prime-to-p quasi-polarized, \(G_n\)-abelian schemes with \(U^p(N_1,N_2)\)-level structures)
we mean a quasi-isogeny (resp. prime-to-p quasi-isogeny) of abelian schemes \(\beta \in {{\text {Hom}}}(A,A')_{\mathbb {Q}}\) (resp. \(\beta \in {{\text {Hom}}}(A,A')_{{\mathbb {Z}}_{(p)}}\)) and \(\delta \in {\mathbb {Q}}^\times \) (resp. \(\delta \in {\mathbb {Z}}_{(p)}^\times \)) such that
-
\(\beta \circ i(a)=i'(a) \circ \beta \) for all \(a \in F\) (resp. \({\mathcal {O}}_{F,(p)}\));
-
\(\delta \lambda =\beta ^\vee \circ \lambda ' \circ \beta \);
-
\([(\delta \eta _0,(V \beta ) \circ \eta _1)]=[\eta ']\) (resp. \([(\delta \eta _0^p,(V^p \beta ) \circ \eta _1^p, \beta C, \beta \circ \eta _p)]=[\eta ']\)).
Lemma 3.2
Suppose that T is an \({\mathcal {O}}_{F,p}\)-module, which is free over \({\mathcal {O}}_{F,p}\) of rank 2n, with a perfect alternating pairing
such that
for all \(x,y \in T\) and \(a \in {\mathcal {O}}_{F,p}\). Also suppose that \(\widetilde{{T}}\subset T\) is a sub-\({\mathcal {O}}_{F,p}\)-module which is isotropic for \(\langle \,\,\,,\,\,\,\rangle \) and such that \(T/\widetilde{{T}}\) is free of rank n over \({\mathcal {O}}_{F,p}\). Finally suppose that
is an \({\mathcal {O}}_{F,p}\)-module isomorphism.
Consider the set \([\eta ]\) of isomorphisms
such that
-
\(\eta (a x)=a \eta (x)\) for all \(a \in {\mathcal {O}}_{F,(p)}\);
-
there exists \(\delta \in {\mathbb {Z}}_p^\times \) such that
$$\begin{aligned} \langle \eta x, \eta y \rangle =\delta \langle x,y\rangle _n \end{aligned}$$for all \(x,y \in \varLambda _n \otimes {\mathbb {Z}}_p\);
-
\(\eta ((p^{-N_2} \varLambda _{n,(n)})\otimes {\mathbb {Z}}_p + \varLambda _{n}\otimes {\mathbb {Z}}_p)=p^{-N_2} \widetilde{{T}}+T\);
-
the map
$$\begin{aligned} p^{-N_1}\varLambda _n\big /\big (p^{-N_1}\varLambda _{n,(n)}+ \varLambda _n\big ) \mathop {\longrightarrow }\limits ^{\sim }p^{-N_1}T\big /\big (p^{-N_1}\widetilde{{T}}+T\big ) \end{aligned}$$induced by \(\eta \) equals \(\eta _p\).
Then \([\eta ]\) is non-empty and a single \(U_p(N_1,N_2)\)-orbit.
Proof
Let \(e_1,\ldots ,e_n\) denote a \({\mathcal {O}}_{F,p}\)-basis of \(T/\widetilde{{T}}\). Note that \(\langle \,\,\,,\,\,\,\rangle \) induces a perfect pairing between \(\widetilde{{T}}\) and \(T/\widetilde{{T}}\). We recursively lift the \(e_i\) to elements \(\widetilde{{e}}_i \in T\) with \(\widetilde{{e}}_i\) orthogonal to the \({\mathcal {O}}_{F,p}\) span of the \(\widetilde{{e}}_j\) for \(j< i\). Suppose that \(\widetilde{{e}}_1,\ldots ,\widetilde{{e}}_{i-1}\) have already been chosen. Choose some lift \(e_i'\) of \(e_i\). Then choose \(t \in \widetilde{{T}}\) such that
-
\(\langle t , x \rangle =\langle e_i',x \rangle \) for all \(x \in \bigoplus _{j=1}^{i-1} {\mathcal {O}}_{F,p} \widetilde{{e}}_j\),
-
and \(\langle t, \alpha e_i' \rangle =\langle e_i'/2,\alpha e_i'\rangle \) for all \(\alpha \in {\mathcal {O}}_{F,p}^{c=-1}\).
(If \(p=2\) some explanation is required as to why we can do this. The map
is of the form
for some \(\beta \in ({\mathcal {D}}_{F,p}^{-1})^{c=-1}\). Because \(p=2\) is unramified in \(F/F^+\), we can write \(\beta =\gamma -{}^c\gamma \) for some \(\gamma \in {\mathcal {D}}_{F,p}^{-1}\). Thus the second condition can be replaced by the condition
for all \(\alpha \in {\mathcal {O}}_{F,p}^{c=-1}\). Now it is clear that the required element t exists.) Then take \(\widetilde{{e}}_i=e_i'-t\). Then \(\widetilde{{e}}_i\) is orthogonal to \(\bigoplus _{j=1}^{i-1} {\mathcal {O}}_{F,p} \widetilde{{e}}_j\). Moreover for \(\alpha \in {\mathcal {O}}_{F,p}\) we have
Thus we can write
with \(\widetilde{{T}}'\) an isotropic \({\mathcal {O}}_{F,p}\)-subspace of T, which is free over \({\mathcal {O}}_{F,p}\) of rank n. We see that
The lemma now follows without difficulty. \(\square \)
Corollary 3.3
If Y is a \({\mathbb {Q}}\)-scheme with geometric point \({\overline{{y}}}\), if \((A,i,\lambda )/Y\) is an ordinary \(G_n\)-abelian scheme and if \([(\eta _0^p,\eta _1^p,C, \eta _p)]\) is a \(U^p(N_1,N_2)\)-level structure on \((A,i,\lambda )\), then there is a unique \(U_p(N_1,N_2)\)-orbit of pairs of isomorphisms
and
such that
-
\(\eta _{1,p}(a x)=a \eta _{1,p}(x)\) for all \(a \in {\mathcal {O}}_{F,(p)}\),
-
\(\langle \eta _{1,p}x,\eta _{1,p}y\rangle _\lambda =\eta _{0,p}\langle x,y \rangle _n\) for all \(x,y \in \varLambda _n \otimes {\mathbb {Z}}_p\),
-
\(\eta _{1,p} p^{-N_2} \varLambda _{n,(n)}/\varLambda _{n,(n)}=C\),
-
\(\eta _{1,p}\) induces \(\eta _p\).
Proof
This follows on combining Lemmas 3.1 and 3.2. \(\square \)
Corollary 3.4
Suppose that Y is a scheme over \({{\text {Spec}}\,}{\mathbb {Q}}\). There is a natural bijection between prime-to-p quasi-isogeny classes of ordinary, prime-to-p quasi-polarized \(G_n\)-abelian schemes with \(U^p(N_1,N_2)\)-level structure and quasi-isogeny classes of quasi-polarized \(G_n\)-abelian schemes with \(U^p(N_1,N_2)\)-level structure.
Proof
We may assume that Y is connected with geometric point \({\overline{{y}}}\). We will show both sets are in natural bijection with the set of prime-to-p quasi-isogeny classes of four-tuples \((A,i,\lambda ,[\eta ])\), where (A, i) is a \(G_n\)-abelian scheme, \(\lambda \) is a prime-to-p quasi-polarization of (A, i), and \([\eta ]\) is a \(\pi _1(Y,{\overline{{y}}})\)-invariant \(U^p(N_1,N_2)\)-orbit of pairs \((\eta _0,\eta _{1})\), where
-
\(\eta _0: {\mathbb {A}}^{\infty ,p}\times {\mathbb {Z}}_p \mathop {\rightarrow }\limits ^{\sim }{\mathbb {A}}^{\infty ,p}(1) \times {\mathbb {Z}}_p(1)\),
-
and \(\eta _1: \varLambda _n \otimes ({\mathbb {A}}^{\infty ,p} \times {\mathbb {Z}}_p) \mathop {\rightarrow }\limits ^{\sim }V^pA_{\overline{{y}}}\times T_pA_{\overline{{y}}}\) satisfies
$$\begin{aligned} \eta _0 \langle x,y \rangle _n=\langle \eta _1 x, \eta _1 y\rangle _\lambda . \end{aligned}$$
There is a natural map from this set to the set of quasi-isogeny classes of quasi-polarized \(G_n\)-abelian schemes with \(U^p(N_1,N_2)\)-level structure, which is easily checked to be a bijection. The bijection between this set and the set of prime-to-p quasi-isogeny classes of ordinary, prime-to-p quasi-polarized \(G_n\)-abelian schemes with a \(U^p(N_1,N_2)\)-level structure, follows by the usual arguments (see, for instance, section III.1 of [29]) from Corollary 3.3. \(\square \)
4.1.2 Hecke actions
If \((A,i,\lambda ,[\eta ])/Y\) is a quasi-polarized, \(G_n\)-abelian scheme with U-level structure and if \(g \in G_n({\mathbb {A}}^{\infty })\) with \(U' \supset g^{-1} U g\), then we can define a quasi-polarized, \(G_n\)-abelian scheme with \(U'\)-level structure \((A,i,\lambda ,[\eta ])g /Y\) by
This action takes one quasi-isogeny class to another.
If \((A,i,\lambda ,[\eta ])/Y\) is an ordinary, prime-to-p quasi-polarized, \(G_n\)-abelian scheme with \(U^p(N_1,N_2)\)-level structure and if \(g\in G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) with
(so that in particular \(N_i \ge N_i'\) for \(i=1,2\)), then we can define an ordinary, prime-to-p quasi-polarized, \(G_n\)-abelian scheme with \((U')^p(N_1',N_2')\)-level structure \((A,i,\lambda ,[\eta ])g /Y\) by
Recall the definition of \(\varsigma _p\) towards the end of Sect. 1.2. If \((U')^p(N_1',N_2') \supset \varsigma _p^{-1} U^p(N_1,N_2) \varsigma _p\) (so that in particular \(N_1 \ge N_1'\) and \(N_2> N_2'\)), then we can define an ordinary, prime-to-p quasi-polarized, \(G_n\)-abelian scheme with \((U')^p(N_1',N_2')\)-level structure \((A,i,\lambda ,[\eta ])\varsigma _p/Y\) by
where
with the last isomorphism being induced by the dual of the map \(A/C[p] \rightarrow A\) induced by multiplication by p on A; where \(F(\eta _1^p)\) is the composition of \(\eta _1^p\) with the natural map \(V^pA \mathop {\rightarrow }\limits ^{\sim }V^p(A/C[p])\); and where \(F(\eta _p)\) is the composition of \(\eta _p\) with the natural identification
If Y is an \({\mathbb {F}}_p\)-scheme then \(\varsigma _p\) is the composite of pull-back by absolute Frobenius followed by forgetting some of the structure.
Together these two definitions give an action of \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\). This action takes one prime-to-p quasi-isogeny class to another.
With these definitions the correspondence of Corollary 3.4 is \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\)-equivariant.
4.1.3 Representability
If U is a neat open compact subgroup of \(G_n({\mathbb {A}}^\infty )\) then the functor that sends a (locally noetherian) scheme \(S/{\mathbb {Q}}\) to the set of quasi-isogeny classes of polarized \(G_n\)-abelian schemes with U-level structures is represented by a quasi-projective scheme \(X_{n,U}\) which is smooth of relative dimension \(n^2[F^+: {\mathbb {Q}}]\) over \({\mathbb {Q}}\). Let
denote the universal equivalence class of polarized \(G_n\)-abelian schemes with U-level structure. If \(U' \supset g^{-1}Ug\) then there is a map \(g: X_{n,U} \rightarrow X_{n,U'}\) arising from \((A^{{\text {univ}}},i^{{\text {univ}}},\lambda ^{{\text {univ}}},[\eta ^{{\text {univ}}}])g/X_{n,U}\) and the universal property of \(X_{n,U'}\). This makes \(\{ X_{n,U}\}\) an inverse system of schemes with right \(G_n({\mathbb {A}}^{\infty })\)-action. The maps g are finite etale. If \(U_1 \subset U_2\) is a normal subgroup then \(X_{n,U_1}/X_{n,U_2}\) is Galois with group \(U_2/U_1\).
There are identifications of topological spaces:
compatible with the right action of \(G_n({\mathbb {A}}^{\infty })\). (See sections 7 and 8 of [39]. Note that \(\ker ^1({\mathbb {Q}},G_n)=(0)\), as is explained in section 7 of [39].) More precisely we associate to \((g,I) \in G_n({\mathbb {A}}^{\infty })/U \times {\mathfrak {H}}_n^+\) the torus \((\varLambda _n \otimes _{\mathbb {Z}}{\mathbb {R}})/\varLambda _n\) with complex structure coming from I; with polarization corresponding to the Riemann form given by \(\langle \,\,\, ,\,\,\,\rangle \); and with level structure coming from
and
where \(\zeta =\lim _{\leftarrow N} e^{2 \pi i/N} \in \widehat{{\mathbb {Z}}}(1)\). We deduce that
If \(U^p\) is neat then the functor that sends a scheme \(Y/{\mathbb {Z}}_{(p)}\) to the set of prime-to-p quasi-isogeny classes of ordinary, prime-to-p quasi-polarized, \(G_n\)-abelian schemes with \(U^p(N_1,N_2)\)-level structure is represented by a scheme \({\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}\) quasi-projective over \({\mathbb {Z}}_{(p)}\). (See theorems 3.4.1.9 and 3.4.2.5 in [44]. Note that, by theorem 3.4.1.9 in [44], the naive moduli problem there is already smooth, and hence the submoduli problem with the right Lie algebra condition agrees with the normalization in theorem 3.4.2.5 in [44].) Let
denote the universal equivalence class. If \(g \in G_n({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and \((U^p)'(N_1',N_2') \supset g^{-1}U^p(N_1,N_2)g\), then there is a quasi-finite map
arising from \(({\mathcal {A}}^{{\text {univ}}},i^{{\text {univ}}},\lambda ^{{\text {univ}}},[\eta ^{{\text {univ}}}])g/{\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}\) and the universal property of \({\mathcal {X}}_{n,(U^p)'(N_1',N_2')}^{{\text {ord}}}\). If \(g \in G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) then the map g is etale, and, if further \(N_2=N_2'\), then it is finite etale. If \(U^p(N_1,N_2)\) is a normal subgroup of \((U^p)'(N_1',N_2)\) then \({\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}/{\mathcal {X}}_{n,(U^p)'(N_1',N_2)}^{{\text {ord}}}\) is Galois with group \((U^p)'(N_1')/U^p(N_1)\). There are \(G_n({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-equivariant identifications
The scheme \({\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}\) is smooth over \({\mathbb {Z}}_{(p)}\) of relative dimension \(n^2[F^+: {\mathbb {Q}}]\). (By the Serre–Tate theorem (see [36]) the formal completion of \({\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}\) at a point x in the special fibre is isomorphic to
This is formally smooth as long as \(S(T_p{\mathcal {A}}^{{\text {univ}}}_x) \cong S({\mathcal {O}}_{F,p}^n)\) is torsion free. This module is torsion free because in the case \(p=2\) we are assuming that \(p=2\) is unramified in \(F/F^+\).) Suppose that \(g \in G_n({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and \((U^p)'(N_1',N_2') \supset g^{-1}U^p(N_1,N_2)g\), then the quasi-finite map
is in fact flat, because it is a quasi-finite map between locally noetherian regular schemes which are equidimensional of the same dimension. (See pages 507 and 508 of [37].)
On \({\mathbb {F}}_p\)-fibres the map
is the absolute Frobenius map composed with the forgetful map \(1: {\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2+1)} \rightarrow {\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}\) (for any \(N_2 \ge N_1 \ge 0\)). Thus if \(N_2>0\), then the quasi-finite, flat map
has all its fibres of degree \(p^{n^2[F^+: {\mathbb {Q}}]}\) and hence is finite flat of this degree. (A flat, quasi-finite morphism \(f: X \rightarrow Y\) between noetherian schemes with constant fibre degree is proper and hence, by theorem 8.11.1 of [24], finite. We give the argument for properness. By the valuative criterion we may reduce to the case \(Y={{\text {Spec}}\,}B\) for a DVR B with fraction field L. By theorem 8.12.6 of [24] X is a dense open subscheme of \({{\text {Spec}}\,}A\), for A a finite B algebra. Let I denote the ideal of A consisting of all \({\mathfrak {m}}_B\)-torsion elements. If \(f \in A\) and \({{\text {Spec}}\,}A_f \subset X\), then by flatness the map \(A \rightarrow A_f\) factors through A / I. Thus \(X \subset {{\text {Spec}}\,}A/I\) and in fact \(I=(0)\), so that A / B is finite flat. Because an open subscheme is determined by its points, we conclude that we must have \(X={{\text {Spec}}\,}A'\) for some \(A \subset A' \subset A \otimes _B L\). By the constancy of the fibre degree we conclude that \(A'\) is finite over B.) We deduce that for any \(g \in G_n({\mathbb {A}}^\infty )^{{{\text {ord}}}}\), if \(N_2'>0\) and \(p^{N_2-N_2'}\nu (g_p) \in {\mathbb {Z}}_p^\times \), then the map
is finite.
Lemma 3.5
Write \({\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},\wedge }\) for the completion of \({\mathcal {X}}_{U^p(N_1,N_2)}^{{{\text {ord}}}}\) along its \({\mathbb {F}}_p\)-fibre. If \(N_2'>N_2 \ge N_1\) then the map
is an isomorphism.
Proof
The map has an inverse which sends a tuple \([({\mathcal {A}}^{{\text {univ}}},i^{{\text {univ}}},\lambda ^{{\text {univ}}}, [\eta ^{{\text {univ}}}])]\) over \({\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},\wedge }\) to
over \({\mathcal {X}}_{n,U^p(N_1,N_2')}^{{{\text {ord}}},\wedge }\). \(\square \)
Thus we will denote \({\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},\wedge }\) simply
Then \(\{ {\mathfrak {X}}^{{\text {ord}}}_{n,U^p(N)} \}\) is a system of p-adic formal schemes with right \(G_n({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-action. We will write \(\overline{{X}}^{{\text {ord}}}_{n,U^p(N)}\) for the reduced subscheme of \({\mathfrak {X}}_{n,U^p(N)}^{{\text {ord}}}\).
Throughout the paper we will use usual Roman letters, such as X, for ‘Shimura-like’ varieties of finite type over \({\mathbb {Q}}\), cursive letters, such as \({\mathcal {X}}\), for models of them of finite type over \({\mathbb {Z}}_{(p)}\), over-lined usual Roman letters, such as \(\overline{{X}}\), for their \({\mathbb {F}}_p\)-fibre, and Gothic letters, such as \({\mathfrak {X}}\), for their formal completion along this special fibre.
4.2 Some Kuga–Sato varieties
Recall that a semi-abelian scheme is a smooth separated commutative group scheme such that each geometric fibre is the extension of an abelian scheme by a torus. To a semi-abelian scheme G / Y one can associate an etale constructible sheaf of abelian groups \(X^*(G)\), the ‘character group of the toric part of G’. See theorem I.2.10 of [17]. If \(X^*(G)\) is locally constant then G is an extension
of a uniquely determined abelian scheme \(A_G\) by a uniquely determined split torus \(S_G\) with character group \(X^*(G)\). By an isogeny (resp. prime-to-p isogeny) of semi-abelian schemes we mean a morphism which is quasi-finite and surjective (resp. quasi-finite and surjective and whose geometric fibres have orders relatively prime to p). If Y is locally noetherian, then by a quasi-isogeny (resp. prime-to- p quasi-isogeny) \(\alpha : G \rightarrow G'\) we mean an element of \({{\text {Hom}}}(G,G')_{{\mathbb {Q}}}\) (resp. \({{\text {Hom}}}(G,G')_{{\mathbb {Z}}_{(p)}}\)) with an inverse in \({{\text {Hom}}}(G',G)_{{\mathbb {Q}}}\) (resp. \({{\text {Hom}}}(G',G)_{{\mathbb {Z}}_{(p)}}\)).
Suppose that \(Y/{{\text {Spec}}\,}{\mathbb {Q}}\) is a locally noetherian scheme. By a \(G_n^{(m)}\)-semi-abelian scheme G over Y we mean a triple (G, i, j) where
-
G / Y is a semi-abelian scheme,
-
\(i: F \hookrightarrow {{\text {End}}}(G)_{\mathbb {Q}}\),
-
and \(j: F^m \mathop {\rightarrow }\limits ^{\sim }X^*(G)_{\mathbb {Q}}\) is an F-linear isomorphism;
-
such that \({{\text {Lie}}\,}A_G\) is a free \({\mathcal {O}}_Y \otimes _{\mathbb {Q}}F\) module of rank \(n[F: {\mathbb {Q}}]\).
Then \(A_G\) is a \(G_n\)-abelian scheme. By a quasi-isogeny of \(G_n^{(m)}\)-semi-abelian schemes we mean a quasi-isogeny of semi-abelian schemes
such that
for all \(a \in F\), and
Note that, if \({\overline{{y}}}\) is a geometric point of Y, then j induces a map
By a quasi-polarization of (G, i, j) we mean a quasi-polarization of \(A_G\).
If Y is connected and \({\overline{{y}}}\) is a geometric point of Y and if \(U \subset G_n^{(m)}({\mathbb {A}}^\infty )\) is a neat open compact subgroup then by a U level structure on a quasi-polarized \(G_n^{(m)}\)-semi-abelian scheme \((G,i,j,\lambda )\) we mean a \(\pi _1(Y,{\overline{{y}}})\)-invariant U-orbit of pairs \((\eta _0,\eta _1)\) where
is an \({\mathbb {A}}^\infty \)-linear map, and where
is an \({\mathbb {A}}_F^\infty \)-linear map such that
and
is a U-level structure on \(A_G\). This is canonically independent of \({\overline{{y}}}\). We define a U level structure on a \(G_n^{(m)}\)-semi-abelian scheme over a general locally noetherian scheme Y to be such a level structure over each connected component of Y. By a quasi-isogeny between two quasi-polarized, \(G_n^{(m)}\)-semi-abelian schemes with U-level structure
we mean a quasi-isogeny
and an element \(\delta \in {\mathbb {Q}}^\times \) such that
and
If \((G,i,j,\lambda ,[(\eta _0,\eta _1)])\) is a quasi-polarized, \(G_n^{(m)}\)-semi-abelian scheme with U-level structure, if \(g \in G_n^{(m)}({\mathbb {A}}^\infty )\) and if \(U'\supset g^{-1}Ug\) then we define a quasi-polarized, \(G_n^{(m)}\)-semi-abelian scheme with \(U'\)-level structure
The quasi-isogeny class of \((G,i,j,\lambda ,[(\eta _0,\eta _1)])g\) only depends on the quasi-isogeny class of \((G,i,j,\lambda ,[(\eta _0,\eta _1)])\). If \((G,i,j,\lambda ,[(\eta _0,\eta _1)])\) is a quasi-polarized, \(G_n^{(m)}\)-semi-abelian scheme with U-level structure, if \(\gamma \in GL_m(F)\) and \(U' \supset \gamma U\) then we define a quasi-polarized, \(G_n^{(m)}\)-semi-abelian scheme with \(U'\)-level structure
The quasi-isogeny class of \(\gamma (G,i,j,\lambda ,[(\eta _0,\eta _1)])\) only depends on the quasi-isogeny class of \((G,i,j,\lambda ,[(\eta _0,\eta _1)])\). We have \(\gamma \circ g=\gamma (g) \circ \gamma \). If \((G,i,j,\lambda ,[(\eta _0,\eta _1)])\) is a quasi-polarized, \(G_n^{(m)}\)-semi-abelian scheme with U-level structure, if \(m'\le m\) and if \(U'\supset i_{m',m}^*U\), then we define a quasi-polarized, \(G_n^{(m')}\)-semi-abelian scheme with \(U'\)-level structure
where \(S \subset S_G\) is the subtorus with
and where
The quasi-isogeny class of \(\pi _{m,m'}(G,i,j,\lambda ,[(\eta _0,\eta _1)])\) only depends on the quasi-isogeny class of \((G,i,j,\lambda ,[(\eta _0,\eta _1)])\). If \(\gamma \in Q_{m,m'}(F)\) then \(\pi _{m,m'} \circ \gamma =\overline{\gamma }\circ \pi _{m,m'}\), where \(\overline{\gamma }\) denotes the image of \(\gamma \) in \(GL_{m'}(F)\). If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )\) then \(\pi _{m,m'} \circ g=i_{m',m}^*(g)\circ \pi _{m,m'}\).
If U is a neat open compact subgroup of \(G_n^{(m)}({\mathbb {A}}^\infty )\) then the functor which sends a locally noetherian scheme \(Y/{\mathbb {Q}}\) to the set of quasi-isogeny classes of quasi-polarized \(G_n^{(m)}\)-semi-abelian schemes with U-level structure is represented by a quasi-projective scheme \(A_{n,U}^{(m)}\), which is smooth of dimension \(n(n+2m)[F^+: {\mathbb {Q}}]\). (See proposition 1.3.2.14 of [44].) We remark that according to our notational conventions we have
Let
denote the universal quasi-isogeny class of quasi-polarized \(G_n^{(m)}\)-semi-abelian schemes with U-level structure. If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )\) and \(U_1,U_2\) are neat open compact subgroups of \(G_n^{(m)}({\mathbb {A}}^\infty )\) with \(U_2 \supset g^{-1}U_1g\) then there is a map
arising from \((G^{{\text {univ}}},i^{{\text {univ}}},j^{{\text {univ}}},\lambda ^{{\text {univ}}},[\eta ^{{\text {univ}}}])g/A^{(m)}_{n,U_1}\) and from the universal property of \(A_{n,U_2}^{(m)}\). Similarly if \(\gamma \in GL_m(F)\) and \(U_1,U_2\) are neat open compact subgroups of \(G_n^{(m)}({\mathbb {A}}^\infty )\) with \(U_2 \supset \gamma U_1\) then there is a map
arising from \(\gamma (G^{{\text {univ}}},i^{{\text {univ}}},j^{{\text {univ}}},\lambda ^{{\text {univ}}},[\eta ^{{\text {univ}}}])/A^{(m)}_{n,U_1}\) and from the universal property of \(A_{n,U_2}^{(m)}\). Moreover if \(m' \le m\), if \(U \subset G_n^{(m)}({\mathbb {A}}^\infty )\) and if \(U'\) denotes the image of U in \(G_n^{(m')}({\mathbb {A}}^\infty )\), then there is a smooth projective map
arising from \(\pi _{m,m'} (G^{{\text {univ}}},i^{{\text {univ}}},j^{{\text {univ}}},\lambda ^{{\text {univ}}},[\eta ^{{\text {univ}}}])/A^{(m)}_{n,U}\) and the universal property of \(A_{n,U'}^{(m')}\). (We will sometimes write \(\pi _{A^{(m)}_n/X_n}\) for \(\pi _{A^{(m)}_n/A^{(0)}_n}\).) We see that these actions have the following properties.
-
\(g_1 \circ g_2=g_2g_1\) (i.e. this is a right action) and \(\gamma _1 \circ \gamma _2=\gamma _1 \gamma _2\) (i.e. this is a left action) and \(\gamma \circ g=\gamma (g) \circ \gamma \).
-
If \(\gamma \in Q_{m,m'}(F)\) then \(\pi _{A^{(m)}_n/A^{(m')}_n} \circ \gamma =\overline{\gamma }\circ \pi _{A^{(m)}_n/A^{(m')}_n}\), where \(\overline{\gamma }\) denotes the image of \(\gamma \) in \(GL_{m'}(F)\).
-
\(\pi _{A^{(m)}_n/A^{(m')}_n} \circ g=g'\circ \pi _{A^{(m)}_n/A^{(m')}_n}\), where \(g'\) denotes the image of g in \(G_n^{(m')}({\mathbb {A}}^\infty )\).
Moreover we have the following properties.
-
The maps g and \(\gamma \) are finite etale. The maps \(\pi _{m,m'}\) are smooth and projective.
-
If \(U_1 \subset U_2\) is an open normal subgroup of a neat open compact subgroup then \(A_{n,U_1}^{(m)}/A_{n,U_2}^{(m)}\) is Galois with group \(U_2/U_1\).
-
If \(U=U' \ltimes M\) with \(U' \subset G_n({\mathbb {A}}^\infty )\) and \(M \subset {{\text {Hom}}}_n^{(m)}({\mathbb {A}}^\infty )\) then \(A^{(m)}_{n,U}/X_{n,U'}\) is an abelian scheme of relative dimension \(mn[F: {\mathbb {Q}}]\).
-
In general \(A^{(m)}_{n,U}\) is a principal homogenous space for \(A^{(m)}_{n,U' \ltimes (U \cap {{\text {Hom}}}^{(m)}_n({\mathbb {A}}^\infty ))}\) over \(X_{n,U'}\), where \(U'\) denotes the image of U in \(G_n({\mathbb {A}}^\infty )\).
-
There are \(G^{(m)}_{n}({\mathbb {A}}^\infty )\) and \(GL_m(F)\) equivariant homeomorphisms
$$A^{(m)}_{n,U}({\mathbb {C}}) \cong G^{(m)}_{n}({\mathbb {Q}}) \big \backslash G^{(m)}_{n}({\mathbb {A}}) \big / \left( U \times U_{n,\infty }^0A_n({\mathbb {R}})^0\right) .$$
Moreover in the case \(U=U' \ltimes M\), if \(G^{{\text {univ}}}/A_{n,U}^{(m)}\) and \(A^{{\text {univ}}}/X_{n,U'}\) are chosen so that \(\pi _{A_n^{(m)}/X_{n}}^* A^{{\text {univ}}}\cong A_{G^{{\text {univ}}}}\), then there is a \({\mathbb {Q}}\)-linear map
with the following properties.
-
If \(a \in F\) then
$$\begin{aligned} i^{(m)}_{A^{{\text {univ}}}}(ax)=i^{{{\text {univ}}},\vee }({}^ca) \circ i^{(m)}_{A^{{\text {univ}}}}(x). \end{aligned}$$ -
If \((\beta ,\delta )\) is a quasi-isogeny
$$\left( G^{{\text {univ}}},i^{{\text {univ}}},j^{{\text {univ}}},\lambda ^{{\text {univ}}},[\eta ^{{\text {univ}}}]\right) \longrightarrow \left( G^{{{\text {univ}}},\prime },i^{{{\text {univ}}},\prime },j^{{{\text {univ}}},\prime },\lambda ^{{{\text {univ}}},\prime },[\eta ^{{{\text {univ}}},\prime }]\right) ,$$then
$$\begin{aligned} \beta ^\vee \circ i^{(m)}_{(A^{{\text {univ}}})'}(x)=i^{(m)}_{A^{{\text {univ}}}}(x). \end{aligned}$$In particular \(i^{(m)}_{A^{{\text {univ}}}}\) depends only on \(A^{{\text {univ}}}\) and not on \(G^{{\text {univ}}}\).
-
If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )\) and \(\gamma \in GL_m(F)\) then
$$\begin{aligned} i^{(m)}_{A^{{\text {univ}}}}(x) \circ g=i^{(m)}_{g^* A^{{\text {univ}}}}(x) \end{aligned}$$and
$$i^{(m)}_{A^{{\text {univ}}}}(x) \circ \gamma =i^{(m)}_{\gamma ^* A^{{\text {univ}}}}(\gamma ^{-1} x).$$ -
If \(e_1,\ldots ,e_m\) denotes the standard basis of \(F^m\) then
$$\begin{aligned}&i_{A^{{\text {univ}}}}=||\eta _0^{{\text {univ}}}||^{-1}\left( (\lambda ^{{\text {univ}}})^{-1}\circ i^{(m)}_{A^{{\text {univ}}}}(e_1),\ldots , (\lambda ^{{\text {univ}}})^{-1} \circ i^{(m)}_{A^{{\text {univ}}}}(e_m)\right) : A_{n,U}^{(m)} \longrightarrow (A^{{\text {univ}}})^m \end{aligned}$$is a quasi-isogeny. If \((\beta ,\delta )\) is a quasi-isogeny
$$\begin{aligned} (G^{{\text {univ}}},i^{{\text {univ}}},j^{{\text {univ}}},\lambda ^{{\text {univ}}},[\eta ^{{\text {univ}}}]) \longrightarrow (G^{{{\text {univ}}},\prime },i^{{{\text {univ}}},\prime },j^{{{\text {univ}}},\prime },\lambda ^{{{\text {univ}}},\prime },[\eta ^{{{\text {univ}}},\prime }]), \end{aligned}$$then
$$\begin{aligned} \beta ^{\oplus m} \circ i_{A^{{\text {univ}}}}=i_{(A^{{\text {univ}}})'}. \end{aligned}$$ -
The map
$$\begin{array}{rcccc} \eta _{n,U}^{(m)}: {{\text {Hom}}}_F(F^m,V_n) \otimes _{\mathbb {Q}}{\mathbb {A}}^\infty &{}\mathop {\rightarrow }\limits ^{\sim }&{} V(A^{{\text {univ}}})^m&{}\mathop {\rightarrow }\limits ^{\sim }&{} VA_{n,U}^{(m)} \\ f &{} \mapsto &{} (\eta _1^{{\text {univ}}}(f(e_1)),\ldots ,\eta _1^{{\text {univ}}}(f(e_m))) &{}&{}\\ &{}&{} x &{} \mapsto &{} V(i_{A^{{\text {univ}}}})^{-1} x \end{array} $$is an isomorphism, which does not depend on the choice of \(G^{{\text {univ}}}\). It satisfies
$$\begin{aligned} \eta _{n,U}^{(m)} M=TA_{n,U}^{(m)}. \end{aligned}$$
(See lemmas 1.3.2.7 and 1.3.2.50, propositions 1.3.2.14, 1.3.2.24 and 1.3.2.55, theorem 1.3.3.15, and remark 1.3.3.33 of [44]; and section 3.5 of [43].)
Note that
and
Define
by
This does not depend on the choice of \(A^{{\text {univ}}}\). We have
Moreover
If \(a \in (F^m \otimes _{F,c} F^m)^{{{\text {sw}}}=1}\) has image in \(S(F^m)\) lying in \(S(F^m)^{>0}\) then
for some matrix \(a' \in M_{m \times m}(F)^{t=c}\) all whose eigenvalues are positive real numbers. (See Sect. 1.1 for the definition of \({{\text {sw}}}\).) Thus \(i_\lambda ^{(m)}(a)\) is a quasi-polarization. (See the end of section 21 of [47].)
Now suppose that \(Y/{{\text {Spec}}\,}{\mathbb {Z}}_{(p)}\) is a locally noetherian scheme. By an ordinary \(G_n^{(m)}\)-emi-abelian scheme G over Y we mean a triple (G, i, j) where
-
G / Y is a semi-abelian scheme such that \(X^*(G)\) is locally constant over Y, and such that \(\# G[p](k({\overline{{y}}})) \ge p^{(n+m)[F: {\mathbb {Q}}]}\) for each geometric point \({\overline{{y}}}\) of Y,
-
\(i: {\mathcal {O}}_{F,(p)} \hookrightarrow {{\text {End}}}(G)_{{\mathbb {Z}}_{(p)}}\) such that \({{\text {Lie}}\,}A_G\) is a free \({\mathcal {O}}_Y \otimes _{{\mathbb {Z}}_{(p)}} {\mathcal {O}}_{F,(p)}\) module of rank \(n[F: {\mathbb {Q}}]\),
-
and \(j: {\mathcal {O}}_{F,(p)}^m \mathop {\rightarrow }\limits ^{\sim }X^*(G)_{{\mathbb {Z}}_{(p)}}\) is a \({\mathcal {O}}_{F,(p)}\)-linear isomorphism.
Then \(A_G\) is an ordinary \(G_n\)-abelian scheme. By a prime-to- p quasi-isogeny of ordinary \(G_n^{(m)}\)-semi-abelian schemes we mean a prime-to-p quasi-isogeny of semi-abelian schemes
such that
for all \(a \in {\mathcal {O}}_{F,(p)}\), and
Note that, if \({\overline{{y}}}\) is a geometric point of Y, then j induces a map
By a prime-to- p quasi-polarization of (G, i, j) we shall mean a prime-to-p quasi-polarization of \(A_G\).
If Y is connected and \({\overline{{y}}}\) is a geometric point of Y, if \(U^p \subset G_n^{(m)}({\mathbb {A}}^{\infty ,p})\) is a neat open compact subgroup and if \(N_2 \ge N_1 \ge 0\) then by a \(U^p(N_1,N_2)\) level structure on a prime-to-p quasi-polarized ordinary \(G_n^{(m)}\)-semi-abelian scheme \((G,i,j,\lambda )\) we mean a \(\pi _1(Y,{\overline{{y}}})\)-invariant \(U^p\)-orbit \([\eta ]\) of five-tuples \((\eta _0^p,\eta _1^p,C,D,\eta _p)\) consisting of
-
an \({\mathbb {A}}^{\infty ,p}\)-linear isomorphism \(\eta _0^p: {\mathbb {A}}^{\infty ,p} \mathop {\longrightarrow }\limits ^{\sim }{\mathbb {A}}^{\infty ,p}(1)_{\overline{{y}}}=V^p{\mathbb {G}}_{m,{\overline{{y}}}}\);
-
an \({\mathbb {A}}_F^{\infty ,p}\)-linear isomorphism
$$\begin{aligned} \eta _1^p: \varLambda _n^{(m)} \otimes _{{\mathbb {Z}}} {\mathbb {A}}^{\infty ,p} \mathop {\longrightarrow }\limits ^{\sim }V^pG_{\overline{{y}}}\end{aligned}$$such that \(\eta _1^p|_{{{\text {Hom}}}_{\mathbb {Z}}({\mathcal {O}}_F^m,{\mathbb {A}}^{\infty ,p})}=(j^*)^{-1} \circ {{\text {Hom}}}(1_{{\mathcal {O}}_F^m},\eta _0^p)\);
-
a locally free sub-\({\mathcal {O}}_{F,(p)}\)-module scheme \(C \subset G[p^{N_2}]\), such that for every geometric point \(\widetilde{{y}}\) of Y there is an \({\mathcal {O}}_{F,(p)}\)-invariant sub-Barsotti–Tate group \(\widetilde{{C}}_{\widetilde{{y}}} \subset G_{\widetilde{{y}}}[p^\infty ]\) with the following properties
-
\(C_{\widetilde{{y}}}=\widetilde{{C}}_{\widetilde{{y}}}[p^{N_2}]\),
-
\(\widetilde{{C}}_{\widetilde{{y}}} \supset S_{G,\widetilde{{y}}}[p^\infty ]\),
-
for all N the subgroup scheme \(\widetilde{{C}}_{\widetilde{{y}}}[p^N]/S_{G,\widetilde{{y}}}[p^N]\) is isotropic in \(A_G[p^N]_{\widetilde{{y}}}\) for the \(\lambda \)-Weil pairing,
-
\(G_{\widetilde{{y}}}[p^\infty ]/\widetilde{{C}}_{\widetilde{{y}}}\) is ind-etale,
-
the Tate module \(T(G_{\widetilde{{y}}}[p^\infty ]/\widetilde{{C}}_{\widetilde{{y}}})\) is free over \({\mathcal {O}}_{F,p}\) of rank n;
-
-
a locally free sub-\({\mathcal {O}}_{F,(p)}\)-module scheme \(D \subset C[p^{N_1}]\) such that
$$\begin{aligned} D \mathop {\rightarrow }\limits ^{\sim }C[p^{N_1}]/S_G[p^{N_1}]; \end{aligned}$$ -
and an isomorphism
$$\begin{aligned} \eta _p: p^{-N_1}\varLambda _n\Big /\left( p^{-N_1}\varLambda _{n,(n)}+ \varLambda _n\right) \mathop {\longrightarrow }\limits ^{\sim }G[p^{N_1}]/C[p^{N_1}] \end{aligned}$$such that
$$\begin{aligned} \eta _p(ax)=i(a) \eta _p(x) \end{aligned}$$for all \(a \in {\mathcal {O}}_{F,(p)}\) and \(x \in p^{-N_1}\varLambda _n/(p^{-N_1}\varLambda _{n,(n)}+ \varLambda _n)\);
such that
is a \(U^p(N_1,N_2)\)-level structure for \((A_G,i,\lambda )\). This definition is independent of the choice of geometric point \({\overline{{y}}}\) of Y. By a \(U^p(N_1,N_2)\)-level structure on an ordinary, prime-to-p quasi-polarized, \(G_n^{(m)}\)-semi-abelian scheme \((G,i,j,\lambda )\) over a general (locally noetherian) scheme \(Y/{{\text {Spec}}\,}{\mathbb {Z}}_{(p)}\), we mean the collection of a \(U^p(N_1,N_2)\)-level structure over each connected component of Y.
By a prime-to-p quasi-isogeny between two quasi-polarized, ordinary \(G_n^{(m)}\)-semi-abelian schemes with \(U^p(N_1,N_2)\)-level structure
we mean a prime-to-p quasi-isogeny
and an element \(\delta \in {\mathbb {Z}}_{(p)}^\times \) such that
and
If \((G,i,j,\lambda ,[(\eta _0^p,\eta _1^p,C,D,\eta _p)])\) is a prime-to-p quasi-polarized, ordinary \(G_n^{(m)}\)-semi-abelian scheme with \(U^p(N_1,N_2)\)-level structure, if \(g \in G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) and if
then we define a prime-to-p quasi-polarized, ordinary \(G_n^{(m)}\)-semi-abelian scheme with \((U^p)'(N_1',N_2')\)-level structure
The prime-to-p quasi-isogeny class of \((G,i,j,\lambda ,[(\eta _0^p,\eta _1^p,C,D,\eta _p)])g\) only depends on the prime-to-p quasi-isogeny class of \((G,i,j,\lambda ,[(\eta _0^p,\eta _1^p,C,D,\eta _p)])\). Similarly, if
is a prime-to-p quasi-polarized, ordinary \(G_n^{(m)}\)-semi-abelian scheme with \(U^p(N_1,N_2)\)-level structure and if
then we define a prime-to-p quasi-polarized, ordinary \(G_n^{(m)}\)-semi-abelian scheme with \((U^p)'(N_1',N_2')\)-level structure
where
with the latter isomorphism being induced by the dual of the map \(A_G/C[p] \rightarrow A_G\) induced by multiplication by p on \(A_G\); where \(F(\eta _1^p)\) is the composition of \(\eta _1^p\) with the natural map \(V^pG \mathop {\rightarrow }\limits ^{\sim }V^p(G/C[p])\); where \(D'\) denotes the pre-image of D under the multiplication by p map \(C \rightarrow C\); and where \(F(\eta _p)\) is the composition of \(\eta _p\) with the natural identification
Together these two definitions give an action of \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\).
If \((G,i,j,\lambda ,[(\eta ^p_0,\eta ^p_1,C,D,\eta _p)])\) is a prime-to-p quasi-polarized, ordinary \(G_n^{(m)}\)-semi-abelian scheme with \(U^p(N_1,N_2)\)-level structure, if \(\gamma \in GL_m({\mathcal {O}}_{F,(p)})\) and if
then we define a prime-to-p quasi-polarized, ordinary \(G_n^{(m)}\)-semi-abelian scheme with \((U^p)'(N_1',N_2')\)-level structure
The prime-to-p quasi-isogeny class of \(\gamma (G,i,j,\lambda ,[(\eta ^p_0,\eta ^p_1,C,D,\eta _p)])\) only depends on the quasi-isogeny class of \((G,i,j,\lambda ,[(\eta ^p_0,\eta ^p_1,C,D,\eta _p)])\). We have \(\gamma \circ g=\gamma (g) \circ \gamma \). If \((G,i,j,\lambda ,[(\eta _0^p,\eta _1^p,C,D,\eta _p)])\) is a prime-to-p quasi-polarized, ordinary \(G_n^{(m)}\)-semi-abelian scheme with \(U^p(N_1,N_2)\)-level structure, if \(m'\le m\) and if \((U^p)'(N_1',N_2')\supset i_{m',m}^*U^p(N_1,N_2)\), then we define a quasi-polarized, ordinary \(G_n^{(m')}\)-semi-abelian scheme with \((U^p)'(N_1',N_2')\)-level structure
where \(S \subset S_G\) is the subtorus with
and where
and \(C'\) (resp. \(D'\)) denotes the image of C (resp. D). The prime-to-p quasi-isogeny class of \(\pi _{m,m'}(G,i,j,\lambda ,[(\eta ^p_0,\eta ^p_1,C,D,\eta _p)])\) only depends on the quasi-isogeny class of \((G,i,j,\lambda ,[(\eta ^p_0,\eta ^p_1,C,D,\eta _p)])\). If \(\gamma \in Q_{m,m'}({\mathcal {O}}_{F,(p)})\) then \(\pi _{m,m'} \circ \gamma =\overline{\gamma }\circ \pi _{m,m'}\), where \(\overline{\gamma }\) denotes the image of \(\gamma \) in \(GL_{m'}({\mathcal {O}}_{F,(p)})\). If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )\) then \(\pi _{m,m'} \circ g=i_{m',m}^*(g)\circ \pi _{m,m'}\).
For each \(m \ge 0\) there is a system of \({\mathbb {Z}}_{(p)}\)-schemes \(\{ {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)} \}\) as \(U^p\) runs over neat open compact subgroups of \(G_n^{(m)}({\mathbb {A}}^{\infty ,p})\) and \(N_1,N_2\) run over integers with \(N_2 \ge N_1 \ge 0\), together with the following extra structures:
-
If \(g \in G_n^{(m)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and \(U_2^p(N_{21},N_{22}) \supset g^{-1}U_1^p(N_{11},N_{12})g\) then there is a quasi-finite, flat map
$$\begin{aligned} g: {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U_1^p(N_{11},N_{12})} \longrightarrow {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U_2^p(N_{21},N_{22})}. \end{aligned}$$ -
If \(m'\le m\) and if \((U^p)'\) denotes the image of \(U^p\) in \(G_n^{(m')}({\mathbb {A}}^{\infty ,p})\), then there is a smooth projective map with geometrically connected fibres
$$\begin{aligned} \pi _{{\mathcal {A}}^{(m),{{\text {ord}}}}_n/{\mathcal {A}}^{(m'),{{\text {ord}}}}_n}: {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)} \longrightarrow {\mathcal {A}}^{(m'),{{\text {ord}}}}_{n,(U^p)'(N_1,N_2)}. \end{aligned}$$ -
If \(\gamma \in GL_m({\mathcal {O}}_{F,(p)})\) and \(U^p_2 \supset \gamma U_1^p\) then there is a finite etale map
$$\begin{aligned} \gamma : {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U_1^p(N_1,N_2)} \longrightarrow {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U_2^p(N_1,N_2)}. \end{aligned}$$
Moreover there is a canonical prime-to-p quasi-isogeny class of ordinary \(G_n^{(m)}\)-semi-abelian schemes with \(U^p(N_1,N_2)\) level structure
These enjoy the following properties:
-
\({\mathcal {A}}^{(0),{{\text {ord}}}}_{n,U^p(N_1,N_2)}={\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}\). (We will sometimes write \(\pi _{{\mathcal {A}}^{(m),{{\text {ord}}}}_n/{\mathcal {X}}^{{\text {ord}}}_n}\) instead of \(\pi _{{\mathcal {A}}^{(m),{{\text {ord}}}}_n/A^{(0),{{\text {ord}}}}_n}\).) This identification is \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\) equivariant.
-
\(g_1 \circ g_2=g_2g_1\) (i.e. this is a right action) and \(\gamma _1 \circ \gamma _2=\gamma _1 \gamma _2\) (i.e. this is a left action) and \(\gamma \circ g=\gamma (g) \circ \gamma \).
-
If \(\gamma \in Q_{m,m'}({\mathcal {O}}_{F,(p)})\) then \(\pi _{{\mathcal {A}}^{(m),{{\text {ord}}}}_n/{\mathcal {A}}^{(m'),{{\text {ord}}}}_n} \circ \gamma =\overline{\gamma }\circ \pi _{{\mathcal {A}}^{(m),{{\text {ord}}}}_n/{\mathcal {A}}^{(m'),{{\text {ord}}}}_n}\), where \(\overline{\gamma }\) denotes the image of \(\gamma \) in \(GL_{m'}({\mathcal {O}}_{F,(p)})\).
-
\(\pi _{{\mathcal {A}}^{(m),{{\text {ord}}}}_n/{\mathcal {A}}_n^{(m'),{{\text {ord}}}}} \circ g=g'\circ \pi _{{\mathcal {A}}_n^{(m),{{\text {ord}}}}/{\mathcal {A}}_n^{(m'),{{\text {ord}}}}}\), where \(g'\) denotes the image of g in \(G_n^{(m')}({\mathbb {A}}^{\infty })^{{\text {ord}}}\).
-
If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\), then the induced map
$$\begin{aligned} g: {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U_1^p(N_{11},N_{12})} \longrightarrow g^* {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U_2^p(N_{21},N_{22})} \end{aligned}$$over \({\mathcal {X}}^{{\text {ord}}}_{n,U_1^p(N_{11},N_{12})}\) is finite flat of degree \(p^{nm[F: {\mathbb {Q}}]}\). If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\), then this map is also etale.
-
If \(U_1^p \subset U_2^p\) is an open normal subgroup of a neat open compact of \(G_n^{(m)}({\mathbb {A}}^{\infty ,p})\) and if \(N_{11}\ge N_{21}\), then \({\mathcal {A}}_{n,U_1^p(N_{11},N_2)}^{(m),{{\text {ord}}}}/{\mathcal {A}}_{n,U_2^p(N_{21},N_2)}^{(m),{{\text {ord}}}}\) is Galois with Galois group \(U_2^p(N_{21})/U^p_1(N_{11})\).
-
On \({\mathbb {F}}_p\)-fibres the map
$$\begin{aligned} \varsigma _p: {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2+1)} \times {{\text {Spec}}\,}{\mathbb {F}}_p \longrightarrow {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)} \times {{\text {Spec}}\,}{\mathbb {F}}_p \end{aligned}$$equals the composition of the absolute Frobenius map with the forgetful map (for any \(N_2 \ge N_1 \ge 0\)).
-
If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) and \(U_2^p(N_{21},N_{22}) \supset g^{-1}U^p_1(N_{11},N_{12})g\) then the pull-back
$$g^* \left( {\mathcal {G}}^{{\text {univ}}}_2,i_2^{{\text {univ}}}, j_2^{{\text {univ}}},\lambda _2^{{\text {univ}}}, [\eta _2^{{\text {univ}}}]\right) $$is prime-to-p quasi-isogenous to the tuple \(({\mathcal {G}}^{{\text {univ}}}_1,i_1^{{\text {univ}}}, j_1^{{\text {univ}}},\lambda _1^{{\text {univ}}}, [\eta _1^{{\text {univ}}}])g\).
-
If \(\gamma \in GL_m({\mathcal {O}}_{F,(p)})\) and \(U_2^p(N_{21},N_{22}) \supset \gamma U^p_1(N_{11},N_{12})\) then the pull-back
$$\gamma ^* \left( {\mathcal {G}}^{{\text {univ}}}_2,i_2^{{\text {univ}}}, j_2^{{\text {univ}}},\lambda _2^{{\text {univ}}}, \left[ \eta _2^{{\text {univ}}}\right] \right) $$is prime-to-p quasi-isogenous to the tuple \(\gamma ({\mathcal {G}}^{{\text {univ}}}_1,i_1^{{\text {univ}}}, j_1^{{\text {univ}}},\lambda _1^{{\text {univ}}}, [\eta _1^{{\text {univ}}}])\).
-
If \(m' \le m\) and if \(U_2^p(N_{21},N_{22}) \supset i_{m',m}^* U^p_1(N_{11},N_{12})\) then the pull-back \(\pi _{{\mathcal {A}}_n^{(m)}/{\mathcal {A}}_n^{(m')}}^* ({\mathcal {G}}^{{\text {univ}}}_2,i_2^{{\text {univ}}}, j_2^{{\text {univ}}},\lambda _2^{{\text {univ}}}, [\eta _2^{{\text {univ}}}])\) is prime-to-p quasi-isogenous to the tuple \(\pi _{m,m'} ({\mathcal {G}}^{{\text {univ}}}_1,i_1^{{\text {univ}}}, j_1^{{\text {univ}}},\lambda _1^{{\text {univ}}}, [\eta _1^{{\text {univ}}}])\).
-
If \(U^p=(U^p)' \ltimes M^p\) with \((U^p)' \subset G_n({\mathbb {A}}^{\infty ,p})\) and \(M^p \subset {{\text {Hom}}}_n^{(m)}({\mathbb {A}}^{\infty ,p})\) then
$$\begin{aligned} {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)}\big /{\mathcal {X}}^{{\text {ord}}}_{n,(U^p)'(N_1,N_2)} \end{aligned}$$is an abelian scheme of relative dimension \(mn[F: {\mathbb {Q}}]\).
-
In general \({\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)}\) is a principal homogenous space for the abelian scheme \({\mathcal {A}}^{(m),{{\text {ord}}}}_{n,((U^p)' \ltimes M^p)(N_1,N_2)}\) over \({\mathcal {X}}^{{\text {ord}}}_{n,(U^p)'(N_1,N_2)}\), where \((U^p)'\) denotes the image of \(U^p\) in \(G_n({\mathbb {A}}^{\infty ,p})\) and \(M^p=U^p \cap {{\text {Hom}}}^{(m)}_n({\mathbb {A}}^{\infty ,p})\).
-
There are natural identifications
$${\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)} \times {{\text {Spec}}\,}{\mathbb {Q}}\cong A^{(m)}_{n,U^p(N_1,N_2)}.$$These identifications are compatible with the identifications
$${\mathcal {X}}_{n,(U^p)'(N_1,N_2)}^{{{\text {ord}}}} \times {{\text {Spec}}\,}{\mathbb {Q}}\cong X_{n,(U^p)'(N_1,N_2)}$$and the maps \(\pi _{{\mathcal {A}}_n^{(m),{{\text {ord}}}}/{\mathcal {A}}_n^{(m'),{{\text {ord}}}}}\) and \(\pi _{A^{(m)}_n/A_n^{(m')}}\). They are also equivariant for the actions of the semi-group \(G_n^{(m)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and the group \(GL_m({\mathcal {O}}_{F,(p)})\).
Moreover in the case \(U^p=(U^p)' \ltimes M^p\), if \({\mathcal {G}}^{{\text {univ}}}/{\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)}\) and \({\mathcal {A}}^{{\text {univ}}}/{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}\) are chosen so that \(\pi ^*_{{\mathcal {A}}_n^{(m),{{\text {ord}}}}/{\mathcal {X}}_n^{{\text {ord}}}} {\mathcal {A}}^{{\text {univ}}}\cong A_{{\mathcal {G}}^{{\text {univ}}}}\), then there is a \({\mathbb {Z}}_{(p)}\)-linear map
with the following properties.
-
If \(a \in {\mathcal {O}}_{F,(p)}\) then
$$\begin{aligned} i^{(m)}_{{\mathcal {A}}^{{\text {univ}}}}(ax)=i^{{{\text {univ}}},\vee }({}^ca) \circ i^{(m)}_{{\mathcal {A}}^{{\text {univ}}}}(x). \end{aligned}$$ -
If \((\beta ,\delta )\) is a prime-to-p quasi-isogeny
$$\left( {\mathcal {G}}^{{\text {univ}}},i^{{\text {univ}}},j^{{\text {univ}}},\lambda ^{{\text {univ}}},[\eta ^{{\text {univ}}}]\right) \longrightarrow \left( {\mathcal {G}}^{{{\text {univ}}},\prime },i^{{{\text {univ}}},\prime },j^{{{\text {univ}}},\prime },\lambda ^{{{\text {univ}}},\prime },[\eta ^{{{\text {univ}}},\prime }]\right) ,$$then
$$\begin{aligned} \beta ^\vee \circ i^{(m)}_{({\mathcal {A}}^{{\text {univ}}})'}(x)=i^{(m)}_{{\mathcal {A}}^{{\text {univ}}}}(x). \end{aligned}$$In particular \(i^{(m)}_{{\mathcal {A}}^{{\text {univ}}}}\) depends only on \({\mathcal {A}}^{{\text {univ}}}\) and not on \({\mathcal {G}}^{{\text {univ}}}\).
-
If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) and \(\gamma \in GL_m({\mathcal {O}}_{F,(p)})\) then
$$\begin{aligned} i^{(m)}_{{\mathcal {A}}^{{\text {univ}}}}(x) \circ g=i^{(m)}_{g^* {\mathcal {A}}^{{\text {univ}}}}(x) \end{aligned}$$and
$$\begin{aligned} i^{(m)}_{{\mathcal {A}}^{{\text {univ}}}}(x) \circ \gamma =i^{(m)}_{\gamma ^* {\mathcal {A}}^{{\text {univ}}}}(\gamma ^{-1} x). \end{aligned}$$ -
If \(e_1,\ldots ,e_m\) denotes the standard basis of \({\mathcal {O}}_{F,(p)}^m\) then
$$i_{{\mathcal {A}}^{{\text {univ}}}}=\left| \left| \eta ^{p,{{\text {univ}}}}_0\right| \right| ^{-1}\left( \left( \lambda (N_1)^{{\text {univ}}}\right) ^{-1}\circ i^{(m)}_{{\mathcal {A}}^{{\text {univ}}}}(e_1),\ldots , \left( \lambda (N_1)^{{\text {univ}}}\right) ^{-1} \circ i^{(m)}_{{\mathcal {A}}^{{\text {univ}}}}(e_m) \right) $$is a prime-to-p quasi-isogeny
$${\mathcal {A}}_{n,U^p(N_1,N_2)}^{(m),{{\text {ord}}}} \longrightarrow ({\mathcal {A}}^{{\text {univ}}}/C^{{\text {univ}}}[p^{N_1}])^m.$$Here \(\lambda (N_1)^{{\text {univ}}}\) refers to the prime-to-p quasi-polarization \({\mathcal {A}}^{{\text {univ}}}/C[p^{N_1}] \rightarrow ({\mathcal {A}}^{{\text {univ}}}/C[p^{N_1}])^\vee \) for which the composite
$$\begin{aligned} {\mathcal {A}}^{{\text {univ}}}\longrightarrow {\mathcal {A}}^{{\text {univ}}}/C[p^{N_1}] \mathop {\longrightarrow }\limits ^{\lambda (N_1)^{{\text {univ}}}} ({\mathcal {A}}^{{\text {univ}}}/C[p^{N_1}])^\vee \longrightarrow {\mathcal {A}}^{{{\text {univ}}},\vee } \end{aligned}$$equals \(p^{N_1} \lambda ^{{\text {univ}}}\).
We have
$$\begin{aligned} \beta ^{\oplus m} \circ i_{{\mathcal {A}}^{{\text {univ}}}}=i_{({\mathcal {A}}^{{\text {univ}}})'}. \end{aligned}$$The composite map
$$\begin{aligned} \begin{array}{rcl} \eta _{n,U^p(N_1,N_2)}^{(m)}: {{\text {Hom}}}_{{\mathcal {O}}_{F}}({\mathcal {O}}_{F}^m,\varLambda _n) \otimes _{\mathbb {Z}}{\mathbb {A}}^{\infty ,p} &{}\longrightarrow &{} V^p({\mathcal {A}}^{{\text {univ}}})^m \\ &{} \mathop {\longrightarrow }\limits ^{p^{-N_1}} &{}V^p({\mathcal {A}}^{{\text {univ}}}/C^{{\text {univ}}}[p^{N_1}])^m \\ &{} \longrightarrow &{} V^p{\mathcal {A}}_{n,U^p(N_1,N_2)}^{(m),{{\text {ord}}}}, \end{array} \end{aligned}$$where the first maps sends
$$\begin{aligned} f \longmapsto \left( \eta _1^{p,{{\text {univ}}}}(f(e_1)),\ldots ,\eta _1^{p,{{\text {univ}}}}(f(e_m))\right) \end{aligned}$$and the third map sends
$$\begin{aligned} x\longmapsto V^p(i_{{\mathcal {A}}^{{\text {univ}}}})^{-1} x, \end{aligned}$$is an isomorphism, which does not depend on the choice of \({\mathcal {G}}^{{\text {univ}}}\). It satisfies
$$\begin{aligned} \eta _{n,U^p(N_1,N_2)}^{(m)} M^p=T^p{\mathcal {A}}_{n,U^p(N_1,N_2)}^{(m),{{\text {ord}}}}. \end{aligned}$$
(See lemmas 5.2.4.7 and 7.1.2.1, propositions 5.2.4.13, 5.2.4.25 and 7.1.2.5, remarks 7.1.2.38 and 7.1.4.27, and theorem 7.1.4.1 of [44].)
We deduce the following additional properties:
-
If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) then the map \(g: {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U_1^p(N_{11},N_{12})} \rightarrow {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U_2^p(N_{21},N_{22})}\) is etale. If further \(N_{12}=N_{22}\), then it is finite etale.
-
If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\), if \(N_{22}>0\) and if \(p^{N_{12}-N_{22}}\nu (g_p) \in {\mathbb {Z}}_p^\times \) then
$$\begin{aligned} g: {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U_1^p(N_{11},N_{12})} \longrightarrow {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U_2^p(N_{21},N_{22})} \end{aligned}$$is finite. If \(N_2>0\) then the finite flat map
$$\begin{aligned} \varsigma _p: {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2+1)} \longrightarrow {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)} \end{aligned}$$has degree \(p^{n(n+2m)[F^+: {\mathbb {Q}}]}\).
-
$$\begin{aligned} i_{{\mathcal {A}}^{{\text {univ}}}} \circ g=i_{g^* {\mathcal {A}}^{{\text {univ}}}} \end{aligned}$$
and
$$\begin{aligned} i_{{\mathcal {A}}^{{\text {univ}}}} \circ \gamma ={}^t\gamma ^{-1} \circ i_{\gamma ^* {\mathcal {A}}^{{\text {univ}}}}. \end{aligned}$$
Also in this case define
by
This does not depend on the choice of \({\mathcal {A}}^{{\text {univ}}}\). We have
Moreover
If \(a \in ({\mathcal {O}}_{F,(p)}^m \otimes _{{\mathcal {O}}_{F,(p)},c} {\mathcal {O}}_{F,(p)}^m)^{{{\text {sw}}}=1}\) has image in \(S({\mathcal {O}}_{F,(p)}^m)\) lying in \(S({\mathcal {O}}_{F,(p)}^m)^{>0}\) then
for some matrix \(a' \in M_{m \times m}({\mathcal {O}}_{F,(p)})^{t=c}\) all whose eigenvalues are positive real numbers. Thus \(i_\lambda ^{(m)}(a)\) is a quasi-polarization. (See the end of section 21 of [47].)
The completion of \({\mathcal {A}}_{U^p(N_1,N_2)}^{(m),{{\text {ord}}}}\) along its \({\mathbb {F}}_p\)-fibre does not depend on \(N_2\), so we will denote it
(See theorem 7.1.4.1 of [44].) Then \(\{ {\mathfrak {A}}^{(m),{{\text {ord}}}}_{U^p(N)}\}\) is a system of p-adic formal schemes with a right \(G_n^{(m)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-action and a left \(GL_m({\mathcal {O}}_{F,(p)})\)-action. There is an equivariant map
We will write \(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U^p(N)}\) for the reduced subscheme of \({\mathfrak {A}}_{n,U^p(N)}^{(m),{{\text {ord}}}}\).
4.3 Some mixed Shimura varieties
If \(\widetilde{{U}}\) (resp. \(\widetilde{{U}}^p\)) is a neat open compact subgroup of \(\widetilde{{G}}^{(m)}_n({\mathbb {A}}^\infty )\) (resp. \(\widetilde{{G}}^{(m)}_n({\mathbb {A}}^{\infty ,p})\)) we will denote by \(S_{n,\widetilde{{U}}}^{(m)}\) (resp. \({\mathcal {S}}_{n,\widetilde{{U}}^p}^{(m),{{\text {ord}}}}\)) the split torus over \({{\text {Spec}}\,}{\mathbb {Q}}\) (resp. \({{\text {Spec}}\,}{\mathbb {Z}}_{(p)}\)) with
(resp.
If \(g \in \widetilde{{G}}^{(m)}_n({\mathbb {A}}^{\infty })\) (resp. \(\widetilde{{G}}^{(m)}_n({\mathbb {A}}^{\infty })^{{\text {ord}}}\)) and \(\widetilde{{U}}_2 \supset g^{-1}\widetilde{{U}}_1g\) (resp. \(\widetilde{{U}}_2^p \supset g^{-1}\widetilde{{U}}_1^pg\)) we get a map
(resp.
corresponding to
(resp.
where we think of the domain and codomain both as subspaces of \({{\text {Herm}}}^{(m)}\). If \(\gamma \in GL_m({\mathbb {Q}})\) (resp. \(GL_m({\mathbb {Z}}_{(p)})\)) and \(\widetilde{{U}}_2 \supset \gamma \widetilde{{U}}_1\) (resp. \(\widetilde{{U}}_2^p \supset \gamma \widetilde{{U}}_1^p\)) we get a map
(resp.
corresponding to
(resp.
where again we think of the domain and codomain both as subspaces of \({{\text {Herm}}}^{(m)}\). If \(m_1 \ge m_2\) and if \(\widetilde{{U}}_2\) (resp. \(\widetilde{{U}}_2^p\)) is the image of \(\widetilde{{U}}_1\) (resp. \(\widetilde{{U}}_1^p\)) in \(\widetilde{{G}}_n^{(m_2)}({\mathbb {A}}^\infty )\) (resp. \(\widetilde{{G}}_n^{(m_2)}({\mathbb {A}}^{\infty ,p})\)), then our chosen map \({{\text {Herm}}}^{(m_1)} \rightarrow {{\text {Herm}}}^{(m_2)}\) induces a map
(resp.
As \(\widetilde{{U}}\) runs over neat open compact subgroups of \(\widetilde{{G}}^{(m)}_n({\mathbb {A}}^\infty )\), there is a system of \(S_{n,\widetilde{{U}}}^{(m)}\)-torsors
over \(A^{(m)}_{n,\widetilde{{U}}}\) together with the following extra structures:
-
If \(g \in \widetilde{{G}}_n^{(m)}({\mathbb {A}}^\infty )\) and \(\widetilde{{U}}_1,\widetilde{{U}}_2\) are neat open compact subgroups of \(\widetilde{{G}}_n^{(m)}({\mathbb {A}}^\infty )\) with \(\widetilde{{U}}_2 \supset g^{-1}\widetilde{{U}}_1g\) then there is a finite etale map
$$\begin{aligned} g: T^{(m)}_{n,\widetilde{{U}}_1} \longrightarrow T^{(m)}_{n,\widetilde{{U}}_2} \end{aligned}$$compatible with the maps \(g: A^{(m)}_{n,\widetilde{{U}}_1} \longrightarrow A^{(m)}_{n,\widetilde{{U}}_2}\) and \(g: S^{(m)}_{n,\widetilde{{U}}_1} \longrightarrow S^{(m)}_{n,\widetilde{{U}}_2}\).
-
If \(\gamma \in GL_m(F)\) and \(\widetilde{{U}}_1,\widetilde{{U}}_2\) are neat open compact subgroups of \(\widetilde{{G}}_n^{(m)}({\mathbb {A}}^\infty )\) with \(\widetilde{{U}}_2 \supset \gamma \widetilde{{U}}_1\) then there is a finite etale map
$$\begin{aligned} \gamma : T^{(m)}_{n,\widetilde{{U}}_1} \longrightarrow T^{(m)}_{n,\widetilde{{U}}_2}, \end{aligned}$$compatible with the maps \(\gamma : A^{(m)}_{n,\widetilde{{U}}_1} \longrightarrow A^{(m)}_{n,\widetilde{{U}}_2}\) and \(\gamma : S^{(m)}_{n,\widetilde{{U}}_1} \longrightarrow S^{(m)}_{n,\widetilde{{U}}_2}\).
-
If \(m_1 \ge m_2\) and \(\widetilde{{U}}_2\) is the image of \(\widetilde{{U}}_1\) in \(\widetilde{{G}}^{(m_2)}({\mathbb {A}}^\infty )\), then there is a map
$$\begin{aligned} T^{(m_1)}_{n,\widetilde{{U}}_1} \longrightarrow T^{(m_2)}_{n,\widetilde{{U}}_2} \end{aligned}$$compatible with the maps \(S^{(m_1)}_{n,\widetilde{{U}}_1} \longrightarrow S^{(m_2)}_{n,\widetilde{{U}}_2}\) and \(A^{(m_1)}_{n,\widetilde{{U}}_1} \longrightarrow A^{(m_2)}_{n,\widetilde{{U}}_2}\).
These enjoy the following properties:
-
\(g_1 \circ g_2=g_2g_1\) (i.e. this is a right action) and \(\gamma _1 \circ \gamma _2=\gamma _1 \gamma _2\) (i.e. this is a left action) and \(\gamma \circ g=\gamma (g) \circ \gamma \).
-
If \(\widetilde{{U}}_1 \subset \widetilde{{U}}_2\) is an open normal subgroup of a neat open compact subgroup of \(\widetilde{{G}}_n^{(m)}({\mathbb {A}}^\infty )\), then \(T_{n,\widetilde{{U}}_1}^{(m)}/T_{n,\widetilde{{U}}_2}^{(m)}\) is Galois with group \(\widetilde{{U}}_2/\widetilde{{U}}_1\).
-
The maps \(T^{(m_1)}_{n,\widetilde{{U}}_1} \longrightarrow T^{(m_2)}_{n,\widetilde{{U}}_2}\) are compatible with the actions of \(\widetilde{{G}}_n^{(m_1)}({\mathbb {A}}^\infty )\) and \(\widetilde{{G}}_n^{(m_2)}({\mathbb {A}}^\infty )\) and the map \(\widetilde{{G}}_n^{(m_1)}({\mathbb {A}}^\infty )\rightarrow \widetilde{{G}}_n^{(m_2)}({\mathbb {A}}^\infty )\), and also with the action of \(Q_{m_1,m_2}(F)\).
-
Suppose that \(\widetilde{{U}}=U' \ltimes M\) with \(U' \subset G_n({\mathbb {A}}^\infty )\) and \(M \subset N_n^{(m)}({\mathbb {A}}^\infty )\). Also suppose that
$$\begin{aligned} \chi \in X^*(S^{(m)}_{n,\widetilde{{U}}}) \subset S(F^m) \end{aligned}$$is sufficiently divisible. Then we can find \(a \in F^m \otimes _{F,c} F^m\) lifting \(\chi \) such that
$$\begin{aligned} i^{(m)}_\lambda (a): A^{(m)}_{n,\widetilde{{U}}} \longrightarrow \left( A^{(m)}_{n,\widetilde{{U}}}\right) ^\vee \end{aligned}$$is a homomorphism. For any such a
$$\begin{aligned} {\mathcal {L}}^{(m)}_{n,\widetilde{{U}}}(\chi )=\left( 1,i^{(m)}_\lambda (a)\right) ^* {\mathcal {P}}_{A^{(m)}_{n,\widetilde{{U}}}}. \end{aligned}$$ -
If \(\chi \in X^*(S^{(m)}_{n,\widetilde{{U}}}) \cap S(F^m)^{>0}\) then \({\mathcal {L}}_{n,\widetilde{{U}}}^{(m)}(\chi )\) is relatively ample for \(A^{(m)}_{n,\widetilde{{U}}}/X_{n,\widetilde{{U}}}\).
-
There are \(\widetilde{{G}}^{(m)}_{n}({\mathbb {A}}^\infty )\) and \(GL_m(F)\) equivariant homeomorphisms
$$T^{(m)}_{n,\widetilde{{U}}}({\mathbb {C}}) \cong \widetilde{{G}}^{(m)}_{n}({\mathbb {Q}}) \Big \backslash \widetilde{{G}}^{(m)}_{n}({\mathbb {A}}) {{\text {Herm}}}^{(m)}({\mathbb {C}})\Big /\left( \widetilde{{U}}\times U_{n,\infty }^0A_n({\mathbb {R}})^0\right) .$$
(See lemmas 1.3.2.25 and 1.3.2.72, and propositions 1.3.2.31, 1.3.2.45 and 1.3.2.90 of [44]; section 3.6 of [43]; and the second paragraph of Sect. 3.2 above.)
Similarly as \(\widetilde{{U}}^p\) runs over neat open compact subgroups of \(\widetilde{{G}}^{(m)}_n({\mathbb {A}}^{\infty ,p})\) and \(N_1,N_2\) run over integers with \(N_2 \ge N_1 \ge 0\), there is a system of \({\mathcal {S}}_{n,\widetilde{{U}}^p}^{(m),{{\text {ord}}}}\)-torsors
over \({\mathcal {A}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}^p(N_1,N_2)}\) together with the following extra structures:
-
If \(g \in \widetilde{{G}}_n^{(m)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and \(\widetilde{{U}}_2^p(N_{21},N_{22}) \supset g^{-1}\widetilde{{U}}_1^p(N_{11},N_{12})g\) then there is a quasi-finite, flat map
$$\begin{aligned} g: {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_1^p(N_{11},N_{12})} \longrightarrow {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_2^p(N_{21},N_{22})} \end{aligned}$$compatible with the map \(g: {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_1^p(N_{11},N_{12})} \longrightarrow {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_2^p(N_{21},N_{22})}\) and the map \(g: {\mathcal {S}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_1^p} \longrightarrow {\mathcal {S}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_2^p}\).
-
If \(\gamma \in GL_m({\mathcal {O}}_{F,(p)})\) and \(\widetilde{{U}}_2^p \supset \gamma \widetilde{{U}}_1^p\) then there is a finite etale map
$$\begin{aligned} \gamma : {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_1^p(N_1,N_2)} \longrightarrow {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_2^p(N_1,N_2)}, \end{aligned}$$compatible with the maps
$$\begin{aligned} \gamma : {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_1^p(N_1,N_2)} \longrightarrow {\mathcal {A}}^{(m)}_{n,\widetilde{{U}}_2^p(N_1,N_2)} \end{aligned}$$and
$$\begin{aligned} \gamma : {\mathcal {S}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_1^p} \longrightarrow {\mathcal {S}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_2^p}. \end{aligned}$$ -
If \(m_1 \ge m_2\) and \(\widetilde{{U}}_2^p\) is the image of \(\widetilde{{U}}_1^p\) in \(\widetilde{{G}}^{(m_2)}({\mathbb {A}}^{\infty ,p})\), then there is a map
$$\begin{aligned} {\mathcal {T}}^{(m_1),{{\text {ord}}}}_{n,\widetilde{{U}}_1^p(N_1,N_2)} \longrightarrow {\mathcal {T}}^{(m_2),{{\text {ord}}}}_{n,\widetilde{{U}}_2^p(N_1,N_2)} \end{aligned}$$compatible with the map \({\mathcal {S}}^{(m_1),{{\text {ord}}}}_{n,\widetilde{{U}}_1^p} \longrightarrow {\mathcal {S}}^{(m_2),{{\text {ord}}}}_{n,\widetilde{{U}}_2^p}\) and the map \({\mathcal {A}}^{(m_1),{{\text {ord}}}}_{n,\widetilde{{U}}_1^p(N_1,N_2)} \longrightarrow {\mathcal {A}}^{(m_2),{{\text {ord}}}}_{n,\widetilde{{U}}_2^p(N_1,N_2)}\).
These enjoy the following properties:
-
\(g_1 \circ g_2=g_2g_1\) (i.e. this is a right action) and \(\gamma _1 \circ \gamma _2=\gamma _1 \gamma _2\) (i.e. this is a left action) and \(\gamma \circ g=\gamma (g) \circ \gamma \).
-
If \(g \in \widetilde{{G}}_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) then the map \(g: {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_1^p(N_{11},N_{12})} \rightarrow {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_2^p(N_{21},N_{22})}\) is etale. If further \(N_{12}=N_{22}\), then it is finite etale.
-
The maps \({\mathcal {T}}^{(m_1),{{\text {ord}}}}_{n,\widetilde{{U}}_1^p(N_1,N_2)} \longrightarrow {\mathcal {T}}^{(m_2),{{\text {ord}}}}_{n,\widetilde{{U}}_2^p(N_1,N_2)}\) are compatible with the actions of \(\widetilde{{G}}_n^{(m_1)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) and \(\widetilde{{G}}_n^{(m_2)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) and the map \(\widetilde{{G}}_n^{(m_1)}({\mathbb {A}}^\infty )\rightarrow \widetilde{{G}}_n^{(m_2)}({\mathbb {A}}^\infty )\), and with the action of \(Q_{m_1,m_2}({\mathcal {O}}_{F,(p)})\).
-
If \(\widetilde{{U}}_1^p \subset \widetilde{{U}}_2^p\) is an open normal subgroup of a neat open compact of \(\widetilde{{G}}_n^{(m)}({\mathbb {A}}^{\infty ,p})\), and if \(N_{11} \ge N_{21}\) then \({\mathcal {T}}_{n,\widetilde{{U}}_1^p(N_{11},N_2)}^{(m),{{\text {ord}}}}/{\mathcal {T}}_{n,\widetilde{{U}}_2^p(N_{21},N_2)}^{(m),{{\text {ord}}}}\) is Galois with Galois group \(\widetilde{{U}}_2^p(N_{21})/\widetilde{{U}}_1^p(N_{11})\).
-
If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\), if \(N_{22}>0\) and if \(p^{N_{12}-N_{22}}\nu (g_p) \in {\mathbb {Z}}_p^\times \), then the map \(g: {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_1^p(N_{11},N_{12})} \rightarrow {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_2^p(N_{21},N_{22})}\) is finite. If \(N_{2}>0\) then the finite flat map
$$\begin{aligned} \varsigma _p: {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_1^p(N_{1},N_{2}+1)} \rightarrow {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}_2^p(N_{1},N_{2})} \end{aligned}$$has degree \(p^{(n+m)^2[F^+: {\mathbb {Q}}]}\).
-
On the \({\mathbb {F}}_p\)-fibre
$$\begin{aligned} \varsigma _p: {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}^p(N_1,N_2+1)}\times {{\text {Spec}}\,}{\mathbb {F}}_p \longrightarrow {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}^p(N_1,N_2)} \times {{\text {Spec}}\,}{\mathbb {F}}_p \end{aligned}$$equals the composition of the absolute Frobenius map with the forgetful map (for any \(N_2 \ge N_1 \ge 0\)).
-
Suppose that \(\widetilde{{U}}^p=(U^p)' \ltimes M^p\) with \((U^p)' \subset G_n({\mathbb {A}}^{\infty ,p})\) and \(M^p \subset N_n^{(m)}({\mathbb {A}}^{\infty ,p})\). Also suppose that
$$\begin{aligned} \chi \in X^*\left( {\mathcal {S}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}^p}\right) \subset S({\mathcal {O}}_{F,(p)}^m) \end{aligned}$$is sufficiently divisible. Then we can find \(a \in {\mathcal {O}}_{F,(p)}^m \otimes _{{\mathcal {O}}_{F,(p)}} {\mathcal {O}}_{F,(p)}^m\) lifting \(\chi \) such that
$$i^{(m)}_\lambda (a): {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}^p(N_1,N_2)} \longrightarrow \left( {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}^p(N_1,N_2)}\right) ^\vee $$is a homomorphism. For any such a
$$\begin{aligned} {\mathcal {L}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}^p(N_1,N_2)}(\chi )=\left( 1,i^{(m)}_\lambda (a)\right) ^* {\mathcal {P}}_{{\mathcal {A}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}^p(N_1,N_2)}}. \end{aligned}$$ -
If \(\chi \in X^*({\mathcal {S}}^{(m)}_{n,\widetilde{{U}}^p}) \cap S({\mathcal {O}}_{F,(p)}^m)^{>0}\) then \({\mathcal {L}}_{n,\widetilde{{U}}^p(N_1,N_2)}^{(m),{{\text {ord}}}}(\chi )\) is relatively ample for
$$\begin{aligned} {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}^p(N_1,N_2)}\big /{\mathcal {X}}_{n,\widetilde{{U}}^p(N_1,N_2)}^{{\text {ord}}}. \end{aligned}$$ -
There are natural identifications
$$\begin{aligned} {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}^p(N_1,N_2)} \times {{\text {Spec}}\,}{\mathbb {Q}}\cong T^{(m)}_{n,\widetilde{{U}}^p(N_1,N_2)}. \end{aligned}$$These identifications are compatible with the identifications
$$\begin{aligned} {\mathcal {A}}_{n,\widetilde{{U}}^p(N_1,N_2)}^{(m),{{\text {ord}}}} \times {{\text {Spec}}\,}{\mathbb {Q}}\cong A^{(m)}_{n,\widetilde{{U}}^p(N_1,N_2)} \end{aligned}$$and the maps
$$\begin{aligned} {\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}^p(N_1,N_2)} \longrightarrow {\mathcal {A}}_{n,\widetilde{{U}}^p(N_1,N_2)}^{(m),{{\text {ord}}}} \end{aligned}$$and
$$\begin{aligned} T^{(m)}_{n,\widetilde{{U}}^p(N_1,N_2)} \longrightarrow A^{(m)}_{n,\widetilde{{U}}^p(N_1,N_2)}. \end{aligned}$$The identifications are also equivariant for the actions of the semi-group \(\widetilde{{G}}_n^{(m)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and the group \(GL_m({\mathcal {O}}_{F,(p)})\).
(See lemmas 5.2.4.26 and 7.1.2.22, propositions 5.2.4.30, 5.2.4.41 and 7.1.2.36, and remark 7.1.2.38 of [44].)
4.4 Vector bundles
4.4.1 Vector bundles on Shimura varieties in characteristic zero
Suppose that U is a neat open compact subgroup of \(G_n({\mathbb {A}}^\infty )\). We will let \(\varOmega _{n,U}\) denote the pull-back by the identity section of the sheaf of relative differentials \(\varOmega ^1_{A^{{\text {univ}}}/X_{n,U}}\). This is a locally free sheaf of rank \(n[F: {\mathbb {Q}}]\). Up to unique isomorphism its definition does not depend on the choice of \(A^{{\text {univ}}}\). (Because, by the neatness of U, there is a unique quasi-isogeny between any two universal four-tuples \((A^{{\text {univ}}}, i^{{\text {univ}}}, \lambda ^{{\text {univ}}},[\eta ^{{\text {univ}}}])\).) The system of sheaves \(\{ \varOmega _{n,U}\}\) has an action of \(G_n({\mathbb {A}}^\infty )\). There is a natural isomorphism between \(\varOmega ^1_{A^{{\text {univ}}}/X_{n,U}}\) and the pull-back of \(\varOmega _{n,U}\) from \(X_{n,U}\) to \(A^{{\text {univ}}}\). We will write
Similarly, if \(\pi : A^{{\text {univ}}}\rightarrow X_{n,U}\) is the structural map, then the sheaf
is locally free and canonically independent of the choice of \(A^{{\text {univ}}}\). These sheaves again have an action of \(G_n({\mathbb {A}}^\infty )\).
We will also write \(\varXi _{n,U}={\mathcal {O}}_{X_{n,U}}(||\nu ||)\) for the sheaf \({\mathcal {O}}_{X_{n,U}}\) but with the \(G_n({\mathbb {A}}^\infty )\)-action multiplied by \(||\nu ||\).
For any \(m \in {\mathbb {Z}}\) such that \(m \lambda ^{{\text {univ}}}\) is a true isogeny we get a class
The class
is well defined independently of m. We obtain an embedding
sending 1 to \(||\eta ^{{\text {univ}}}|| [ (1,\lambda ^{{\text {univ}}})^* {\mathcal {P}}_{A^{{{\text {univ}}}}}]\). (See Sect. 3.1 for the definition of \(||\eta ^{{\text {univ}}}||\).) These maps are compatible with the isomorphisms
induced by the unique quasi-isogeny between two universal four-tuples. They are also \(G_n({\mathbb {A}}^\infty )\)-equivariant.
The composites of induced maps
are \(G_n({\mathbb {A}}^\infty )\)-equivariant isomorphisms, independent of the choice of \(A^{{\text {univ}}}\). Moreover the short exact sequence
gives rise to a map
and hence to a map
These maps do not depend on the choice of \(A^{{\text {univ}}}\) and are \(G_n({\mathbb {A}}^\infty )\)-equivariant. They further induce \(G_n({\mathbb {A}}^\infty )\)-equivariant isomorphisms
which again do not depend on the choice of \(A^{{\text {univ}}}\). (See, for instance, propositions 2.1.7.3 and 2.3.5.2 of [41]. This is referred to as the ‘Kodaira–Spencer isomorphism’.)
Let \({\mathcal {E}}_{U}\) denote the principal \(L_{n,(n)}\)-bundle on \(X_{n,U}\) in the Zariski topology defined by setting, for \(W \subset X_{n,U}\) a Zariski open, \({\mathcal {E}}_{U}(W)\) to be the set of pairs \((\xi _0,\xi _1)\), where
and
We define the \(L_{n,(n)}\)-action on \({\mathcal {E}}_{U}\) by
The inverse system \(\{ {\mathcal {E}}_{U} \}\) has an action of \(G_n({\mathbb {A}}^\infty )\).
Suppose that \(R_0\) is a \({\mathbb {Q}}\)-algebra and that \(\rho \) is a representation of \(L_{n,(n)}\) on a finite, locally free \(R_0\)-module \(W_\rho \). We define a locally free sheaf \({\mathcal {E}}_{U,\rho }\) over \(X_{n,U} \times {{\text {Spec}}\,}R_0\) by setting \({\mathcal {E}}_{U,\rho }(W)\) to be the set of \(L_{n,(n)}({\mathcal {O}}_W)\)-equivariant maps of Zariski sheaves of sets
Then \(\{ {\mathcal {E}}_{U,\rho } \}\) is a system of locally free sheaves with \(G_n({\mathbb {A}}^\infty )\)-action over the system of schemes \(\{ X_{n,U} \times {{\text {Spec}}\,}R_0\}\). If \(g \in G_n({\mathbb {A}}^\infty )\), then the natural map
is an isomorphism.
In the case \(R_0={\mathbb {C}}\), the holomorphic vector bundle on \(X_{n,U}({\mathbb {C}})\) associated to \({\mathcal {E}}_{U,\rho }\) is
over
(See Sect. 1.1 for the definition of the holomorphic vector bundle \({\mathfrak {E}}_\rho /{\mathfrak {H}}_n^{\pm }\).)
Note that
and
and
and
(See Sect. 1.2 for the definition of the representation \({{\text {KS}}}\).)
4.4.2 Vector bundles on Kuga–Sato varieties in characteristic zero
Suppose now that U is a neat open compact subgroup of \(G_n^{(m)}({\mathbb {A}}^\infty )\) with image \(U'\) in \(G_n({\mathbb {A}}^\infty )\). We will let \(\varOmega _{n,U}^{(m)}\) denote the pull-back by the identity section of the sheaf of relative differentials \(\varOmega ^1_{G^{{\text {univ}}}/A_{n,U}^{(m)}}\). This is a locally free sheaf of rank \((n+m)[F: {\mathbb {Q}}]\). Up to unique isomorphism its definition does not depend on the choice of \(G^{{\text {univ}}}\). The system of sheaves \(\{ \varOmega ^{(m)}_{n,U}\}\) has actions of \(G_n^{(m)}({\mathbb {A}}^\infty )\) and of \(GL_m(F)\). Moreover there is an exact sequence
which is equivariant for the actions of \(G_n^{(m)}({\mathbb {A}}^\infty )\) and \(GL_m(F)\).
Let \({\mathcal {E}}_{U}^{(m)}\) denote the principal \(R^{(m)}_{n,(n)}\)-bundle on \(A^{(m)}_{n,U}\) in the Zariski topology defined by setting, for \(W \subset A^{(m)}_{n,U}\) a Zariski open, \({\mathcal {E}}_{U}^{(m)}(W)\) to be the set of pairs \((\xi _0,\xi _1)\), where
and
satisfies
and induces the canonical isomorphism
We define the \(R^{(m)}_{n,(n)}\)-action on \({\mathcal {E}}_{U}^{(m)}\) by
The inverse system \(\{ {\mathcal {E}}_{U}^{(m)} \}\) has an action of \(G_n^{(m)}({\mathbb {A}}^\infty )\) and of \(GL_m(F)\).
Suppose that \(R_0\) is a \({\mathbb {Q}}\)-algebra and that \(\rho \) is a representation of \(R^{(m)}_{n,(n)}\) on a finite, locally free \(R_0\)-module \(W_\rho \). We define a locally free sheaf \({\mathcal {E}}^{(m)}_{U,\rho }\) over \(A^{(m)}_{n,U} \times {{\text {Spec}}\,}R_0\) by setting \({\mathcal {E}}^{(m)}_{U,\rho }(W)\) to be the set of \(R^{(m)}_{n,(n)}({\mathcal {O}}_W)\)-equivariant maps of Zariski sheaves of sets
Then \(\{ {\mathcal {E}}^{(m)}_{U,\rho } \}\) is a system of locally free sheaves with both \(G^{(m)}_n({\mathbb {A}}^\infty )\)-action and \(GL_m(F)\)-action over the system of schemes \(\{ A^{(m)}_{n,U} \times {{\text {Spec}}\,}R_0\}\). If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )\) and \(\gamma \in GL_m(F)\), then the natural maps
and
are isomorphisms. If \(\rho \) factors through \(R^{(m)}_{n,(n)} \twoheadrightarrow L_{n,(n)}\) then \({\mathcal {E}}^{(m)}_{U,\rho }\) is canonically isomorphic to the pull-back of \({\mathcal {E}}_{U,\rho }\) from \(X_{n,U}\). In general \(W_\rho \) has a filtration by \(R_{n,(n)}^{(m)}\)-invariant direct summands such that the action of \(R^{(m)}_{n,(n)}\) on each graded piece factors through \(L_{n,(n)}\). (To see this apply proposition 4.7.3 of exposé I of [53] to the action of \(L_{n,(n),{{\text {herm}}}}\) on \(W_\rho \).) Thus \({\mathcal {E}}^{(m)}_{U,\rho }\) has a \(G_n^{(m)}({\mathbb {A}}^\infty )\) and \(GL_m(F)\) invariant filtration by local direct summands such that each graded piece is the pull-back of some \({\mathcal {E}}_{U,\rho '}\) from \(X_{n,U}\).
4.4.3 Vector bundles on Shimura varieties in mixed characteristic
Similarly suppose that \(U^p\) is a neat open compact subgroup of \(G_n({\mathbb {A}}^{\infty ,p})\), and that \(N_2 \ge N_1 \ge 0\) are integers. We will let \(\varOmega _{n,U^p(N_1,N_2)}^{{\text {ord}}}\) denote the pull-back by the identity section of \(\varOmega ^1_{{\mathcal {A}}^{{\text {univ}}}/{\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}}\). This is a locally free sheaf of rank \(n[F: {\mathbb {Q}}]\). Up to unique isomorphism its definition does not depend on the choice of \({\mathcal {A}}^{{\text {univ}}}\). (Because, by the neatness of \(U^p\), there is a unique prime-to-p quasi-isogeny between any two universal four-tuples \(({\mathcal {A}}^{{\text {univ}}}, i^{{\text {univ}}}, \lambda ^{{\text {univ}}},[\eta ^{{\text {univ}}}])\).) The system of sheaves \(\{ \varOmega _{n,U^p(N_1,N_2)}^{{\text {ord}}}\}\) has an action of \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\). There is a natural isomorphism between \(\varOmega ^1_{{\mathcal {A}}^{{\text {univ}}}/{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}}\) and the pull-back of \(\varOmega _{n,U^p(N_1,N_2)}^{{\text {ord}}}\). We will write
We will also write \(\varXi _{n,U^p(N_1,N_2)}={\mathcal {O}}_{{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}}(||\nu ||)\) for the sheaf \({\mathcal {O}}_{{\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}}\) but with the \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\)-action multiplied by \(||\nu ||\).
For any \(m \in {\mathbb {Z}}\) such that \(p {\not | }m\) and \(m\lambda ^{{\text {univ}}}\) is a true isogeny we get a class
The class
is well defined independently of m. We obtain an embedding
sending 1 to \(||\eta ^{{\text {univ}}}|| [ (1,\lambda ^{{\text {univ}}})^* {\mathcal {P}}_{{\mathcal {A}}^{{{\text {univ}}}}}]\). These maps are compatible with the isomorphisms
induced by the unique prime-to-p quasi-isogeny between two universal four-tuples. They are also \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\)-equivariant.
The composites of induced maps
are \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\)-equivariant isomorphisms, independent of the choice of \({\mathcal {A}}^{{\text {univ}}}\). Moreover the short exact sequence
gives rise to a map
and hence to a map
These maps do not depend on the choice of \({\mathcal {A}}^{{\text {univ}}}\) and are \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\)-equivariant. They further induce \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\) isomorphisms
which again do not depend on the choice of \({\mathcal {A}}^{{\text {univ}}}\). (See, for instance, proposition 3.4.3.3 of [44].)
Let \({\mathcal {E}}_{U^p(N_1,N_2)}^{{{\text {ord}}}}\) denote the principal \(L_{n,(n)}\)-bundle on \({\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}\) in the Zariski topology defined by setting, for \(W \subset {\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}\) a Zariski open, \({\mathcal {E}}^{{{\text {ord}}}}_{U^p(N_1,N_2)}(W)\) to be the set of pairs \((\xi _0,\xi _1)\), where
and
We define the \(L_{n,(n)}\)-action on \({\mathcal {E}}_{U^p(N_1,N_2)}^{{{\text {ord}}}}\) by
The inverse system \(\{ {\mathcal {E}}_{U^p(N_1,N_2)}^{{{\text {ord}}}} \}\) has an action of \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\).
Suppose that \(R_0\) is a \({\mathbb {Z}}_{(p)}\)-algebra and that \(\rho \) is a representation of the algebraic group \(L_{n,(n)}\) on a finite, locally free \(R_0\)-module \(W_\rho \). We define a locally free sheaf \({\mathcal {E}}_{U^p(N_1,N_2),\rho }^{{{\text {ord}}}}\) over \({\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}\times {{\text {Spec}}\,}R_0\) by setting \({\mathcal {E}}_{U^p(N_1,N_2),\rho }^{{{\text {ord}}}}(W)\) to be the set of \(L_{n,(n)}({\mathcal {O}}_W)\)-equivariant maps of Zariski sheaves of sets
Then \(\{ {\mathcal {E}}^{{{\text {ord}}}}_{U^p(N_1,N_2),\rho } \}\) is a system of locally free sheaves with \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-action over the system of schemes \(\{ {\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)} \times {{\text {Spec}}\,}R_0 \} \). The maps
are isomorphisms. The pull-back of \({\mathcal {E}}^{{{\text {ord}}}}_{U^p(N_1,N_2),\rho }\) to
is canonically identified with the sheaf \({\mathcal {E}}_{U^p(N_1,N_2),\rho \otimes _{R_0} R_0[1/p]}\). This identification is \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-equivariant.
Note that
and
and
and
4.4.4 Vector bundles on Kuga–Sato varieties in mixed characteristic
Suppose now that \(U^p\) is a neat open compact subgroup of \(G_n^{(m)}({\mathbb {A}}^{\infty ,p})\) with image \((U^p)'\) in \(G_n({\mathbb {A}}^{\infty ,p})\). We will let \(\varOmega _{n,U^p(N_1,N_2)}^{(m),{{\text {ord}}}}\) denote the pull-back by the identity section of the sheaf of relative differentials \(\varOmega ^1_{{\mathcal {G}}^{{\text {univ}}}/{\mathcal {A}}_{n,U^p(N_1,N_2)}^{(m),{{\text {ord}}}}}\). This is a locally free sheaf of rank \((n+m)[F: {\mathbb {Q}}]\). Up to unique isomorphism its definition does not depend on the choice of \({\mathcal {G}}^{{\text {univ}}}\). The system of sheaves \(\{ \varOmega ^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)}\}\) has actions of \(G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) and of \(GL_m({\mathcal {O}}_{F,(p)})\). Moreover there is an exact sequence
which is equivariant for the actions of \(G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) and \(GL_m({\mathcal {O}}_{F,(p)})\).
Let \({\mathcal {E}}_{U^p(N_1,N_2)}^{(m),{{\text {ord}}}}\) denote the principal \(R^{(m)}_{n,(n)}\)-bundle on \({\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)}\) in the Zariski topology defined by setting, for \(W \subset {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)}\) a Zariski open, \({\mathcal {E}}_{U^p(N_1,N_2)}^{(m),{{\text {ord}}}}(W)\) to be the set of pairs \((\xi _0,\xi _1)\), where
and
satisfies
and induces the canonical isomorphism
We define the \(R^{(m)}_{n,(n)}\)-action on \({\mathcal {E}}_{U^p(N_1,N_2)}^{(m),{{\text {ord}}}}\) by
The inverse system \(\{ {\mathcal {E}}_{U^p(N_1,N_2)}^{(m),{{\text {ord}}}} \}\) has an action of \(G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) and of \(GL_m({\mathcal {O}}_{F,(p)})\).
Suppose that \(R_0\) is a \({\mathbb {Z}}_{(p)}\)-algebra and that \(\rho \) is a representation of \(R^{(m)}_{n,(n)}\) on a finite, locally free \(R_0\)-module \(W_\rho \). We define a locally free sheaf \({\mathcal {E}}^{(m),{{\text {ord}}}}_{U^p(N_1,N_2),\rho }\) over \({\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)} \times {{\text {Spec}}\,}R_0\) by setting \({\mathcal {E}}^{(m)}_{U,\rho }(W)\) to be the set of \(R^{(m)}_{n,(n)}({\mathcal {O}}_W)\)-equivariant maps of Zariski sheaves of sets
Then \(\{ {\mathcal {E}}^{(m),{{\text {ord}}}}_{U^p(N_1,N_2),\rho } \}\) is a system of locally free sheaves with \(G^{(m)}_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-action and \(GL_m({\mathcal {O}}_{F,(p)})\)-action over the system of schemes \(\{ {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)} \times {{\text {Spec}}\,}R_0\}\). If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) and \(\gamma \in GL_m({\mathcal {O}}_{F,(p)})\), then the natural maps
and
are isomorphisms. If \(\rho \) factors through \(R^{(m)}_{n,(n)} \twoheadrightarrow L_{n,(n)}\) then \({\mathcal {E}}^{(m),{{\text {ord}}}}_{U^p(N_1,N_2),\rho }\) is canonically isomorphic to the pull-back of \({\mathcal {E}}^{{\text {ord}}}_{U^p(N_1,N_2),\rho }\) from \({\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}\). In general \(W_\rho \) has a filtration by \(R_{n,(n)}^{(m)}\)-invariant local direct summands such that the action of \(R^{(m)}_{n,(n)}\) on each graded piece factors through \(L_{n,(n)}\). (To see this apply proposition 4.7.3 of exposé I of [53] to the action of \(L_{n,(n),{{\text {herm}}}}\) on \(W_\rho \).) Thus \({\mathcal {E}}^{(m),{{\text {ord}}}}_{U^p(N_1,N_2),\rho }\) has a \(G_n^{(m)}({\mathbb {A}}^\infty )\) and \(GL_m({\mathcal {O}}_{F,(p)})\) invariant filtration by local direct summands such that each graded piece is the pull-back of some \({\mathcal {E}}^{{\text {ord}}}_{U^p(N_1,N_2),\rho '}\) from \({\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}\).
4.4.5 Higher direct images from Kuga–Sato varieties to Shimura varieties, characteristic zero case
If \(m \ge m'\) and if U is a neat open compact subgroup of \(G^{(m)}_n({\mathbb {A}}^\infty )\) with image \(U'\) in \(G_n^{(m')}({\mathbb {A}}^\infty )\) then the sheaf
depends only on \(U'\) and not on U. We will denote it
If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )\) and \(g^{-1}U_1g \subset U_2\) then there is a natural isomorphism
where \(g'\) (resp. \(U_1'\), resp. \(U_2'\)) denotes the image of g (resp. \(U_1\), resp. \(U_2\)) in \(G_n^{(m')}({\mathbb {A}}^\infty )\). This isomorphism only depends on \(g'\), \(U_1'\) and \(U_2'\) and not on g, \(U_1\) and \(U_2\). This gives the system of sheaves \(\{ (R^j\pi _* \varOmega ^i_{A^{(m)}_n/A^{(m')}_n})_{U'} \}\) a left action of \(G_n^{(m')}({\mathbb {A}}^\infty )\). Also if \(\gamma \in Q_{m,m'}(F)\) then \(\gamma : A^{(m)}_{n,U} \rightarrow A^{(m)}_{n,\gamma U}\) gives a natural isomorphism
which depends only on \(U'\) and not on U. This gives the system of sheaves
a right action of \(Q_{m,m'}(F)\). We have \(\gamma \circ g=\gamma (g) \circ \gamma \).
If \(U_1' \supset U_2'\) and \(g' \in U_2'\) normalizes \(U_1'\) then on
the actions of g and \(1 \otimes g\) agree. Moreover if U is a neat open compact subgroup of \(G^{(m)}_n({\mathbb {A}}^\infty )\) with image \(U'\) in \(G_n^{(m')}({\mathbb {A}}^\infty )\) then the natural map
is an isomorphism. These isomorphisms are equivariant for the actions of the groups \(G_n^{(m)}({\mathbb {A}}^\infty )\) and \(Q_{m,m'}(F)\).
The natural maps
are \(G_n^{(m')}({\mathbb {A}}^\infty )\) and \(Q_{m,m'}(F)\) equivariant isomorphisms.
Suppose that U is a neat open compact subgroup of \(G_n^{(m)}({\mathbb {A}}^\infty )\) with image \(U'\) in \(G_n^{(m')}({\mathbb {A}}^\infty )\) and \(U''\) in \(G_n({\mathbb {A}}^\infty )\). If U is of the form \(U' \ltimes M\), then the quasi-isogeny \(i_{A^{{\text {univ}}}}: A^{(m)}_{n,U} \rightarrow (A^{{\text {univ}}})^{m-m'}\) over \(A^{(m')}_{n,U'}\) gives rise to an isomorphism
and a canonical embedding
where the first map denotes the diagonal embedding. These maps do not depend on the choice of \(A^{{\text {univ}}}\). They are \(G_n^{(m)}({\mathbb {A}}^\infty )\)-equivariant. The first map is also \(Q_{m,m'}(F)\)-equivariant, where an element \(\gamma \in Q_{m,m'}(F)\) acts on the left hand sides by composition with the inverse of the projection of \(\gamma \) to \(GL_{m-m'}(F)\). This remains true if we do not assume that U has the form \(U' \ltimes M\).
This gives rise to canonical \(G_n^{(m')}({\mathbb {A}}^\infty )\)-equivariant isomorphisms
Moreover the composite maps
are \(G_n^{(m')}({\mathbb {A}}^\infty )\)-equivariant isomorphisms.
4.4.6 Higher direct images from Kuga–Sato varieties to Shimura varieties, mixed characteristic case
If \(m \ge m'\) and if \(U^p\) is a neat open compact subgroup of \(G^{(m)}_n({\mathbb {A}}^{\infty ,p})\) with image \((U^p)'\) in \(G_n^{(m')}({\mathbb {A}}^{\infty ,p})\), and if \(0 \le N_1 \le N_2\) are integers, then the sheaf
depends only on \((U^p)'\) and not on \(U^p\). We will denote it
If \(g \in G_n^{(m)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and \(g^{-1}U^p_1(N_{11},N_{12})g \subset U^p_2(N_{21},N_{22})\), then there is a natural map
where \((U_i^p)'\) denotes the image of \(U^p_i\) in \(G_n^{(m')}({\mathbb {A}}^{\infty ,p})\) and \(g'\) denotes the image of g in \(G_n^{(m')}({\mathbb {A}}^{\infty })^{{\text {ord}}}\). If \(g \in G_n^{(m)}({\mathbb {A}}^{\infty })^{{{\text {ord}}},\times }\) then it is an isomorphism. Moreover this map only depends on \(g'\), \((U_1^p)'(N_{11},N_{12})\) and \((U^p_2)'(N_{21},N_{22})\) and not on g, \(U_1^p(N_{11},N_{12})\) and \(U_2^p(N_{21},N_{22})\). This gives the system of sheaves
a left action of \(G_n^{(m')}({\mathbb {A}}^{\infty })^{{\text {ord}}}\).
If \(\gamma \in Q_{m,m'}({\mathcal {O}}_{F,(p)})\) then \(\gamma : A^{(m)}_{n,U^p(N_1,N_2)} \rightarrow A^{(m)}_{n,\gamma U^p(N_1,N_2)}\) gives a natural isomorphism
which depends only on \((U^p)'(N_1,N_2)\) and not on \(U^p(N_1,N_2)\). This gives the system of sheaves
a right action of \(Q_{m,m'}({\mathcal {O}}_{F,(p)})\). We have \(\gamma \circ g=\gamma (g) \circ \gamma \).
If \((U^p_1)'(N_{11},N_{12}) \supset (U_2^p)'(N_{21},N_{22})\) and \(g \in (U^p_1)' (N_{11},N_{12})\) normalizes the subgroup \((U^p_2)'(N_{21},N_{22})\), then on
the actions of g and \(1 \otimes g\) agree. Moreover if \(U^p\) is a neat open compact subgroup of \(G^{(m)}_n({\mathbb {A}}^{\infty ,p})\) with image \((U^p)'\) in \(G_n^{(m')}({\mathbb {A}}^{\infty ,p})\), and if \(0 \le N_1 \le N_2\), then the natural map
is an isomorphism. These isomorphisms are \(G_n^{(m)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and \(Q_{m,m'}({\mathcal {O}}_{F,(p)})\) equivariant.
The natural maps
are \(G_n^{(m')}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and \(Q_{m,m'}({\mathcal {O}}_{F,(p)})\) equivariant isomorphisms.
Under the identification
the sheaves \(\varOmega ^{{\text {ord}}}_{n, (U^p)'(N_1,N_2)}\) (resp. \(\varXi ^{{\text {ord}}}_{n,(U^p)'(N_1,N_2)}\)) are naturally identified with the sheaves \(\varOmega _{n, (U^p)'(N_1,N_2)}\) (resp. \(\varXi _{n,(U^p)'(N_1,N_2)}\)). Moreover, under the identification
the sheaf \((R^j\pi _* \varOmega ^i_{{\mathcal {A}}^{(m),{{\text {ord}}}}/{\mathcal {A}}_n^{(m'),{{\text {ord}}}}})_{(U^p)'(N_1,N_2)}\) is naturally identified with the sheaf \((R^j\pi _* \varOmega ^i_{A^{(m)}_n/A^{(m')}_n})_{(U^p)'(N_1,N_2)}\). These identifications are equivariant for the actions of \(G_n({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and \(Q_{m,m'}({\mathcal {O}}_{F,(p)})\).
5 Generalized Shimura varieties
We will introduce certain disjoint unions of mixed Shimura varieties, which are associated to \(L_{n,(i),{{\text {lin}}}}\) and \(L_{n,(i)}\) and \(P_{n,(i)}^+/Z(N_{n,(i)})\) and \(P_{n,(i)}^+\); to \(L^{(m)}_{n,(i),{{\text {lin}}}}\) and \(L^{(m)}_{n,(i)}\) and \(P^{(m),+}_{n,(i)}/Z(N^{(m)}_{n,(i)})\) and \(P^{(m),+}_{n,(i)}\); and to \(\widetilde{{P}}^{(m),+}_{n,(i)}\). The differences with the last section are purely book keeping. We then describe certain torus embeddings for these generalized Shimura varieties and discuss their completion along the boundary. These completions will serve as formal local models near the boundary of the toroidal compactifications of the \(X_{n,U}\) and the \(A^{(m)}_{n,\widetilde{{U}}}\) to be discussed in the next section.
We remind the reader of our convention that, if U is a subgroup of G and H is a quotient of G, then we will sometimes use U to denote its image in H. We hope that this causes no confusion as we will only do this when the context makes clear we are referring to a subgroup of H.
5.1 Generalized Shimura varieties
If U is a neat open compact subgroup of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )\) we set
In the case \(m=0\) we will write simply \(Y_{n,(i),U}^+\). Then \(\{ Y^{(m),+}_{n,(i),U} \}\) is a system of schemes (locally of finite type over \({{\text {Spec}}\,}{\mathbb {Q}}\)) with right \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )\)-action. Each \(Y_{n,(i),U}^{(m),+}\) also has a left action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\), and the inverse system has a right action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )\). If \(\delta \in GL_m(F)\) we get a map
which sends \(({{\text {Spec}}\,}{\mathbb {Q}})_{hU} \rightarrow ({{\text {Spec}}\,}{\mathbb {Q}})_{\delta (h) \delta (U)}\) via the identity. This gives a left action of \(GL_m(F)\) on the inverse system of the \(Y_{n,(i),U}^{(m)}\). If \(\delta \in GL_m(F)\) and \(\gamma \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) and \(g \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )\) then \(\delta \circ \gamma =\delta (\gamma ) \circ \delta \) and \(\delta \circ g=\delta (g) \circ \delta \). If \(U'\) denotes the image of U in \(L_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )\) then there is a natural map
These maps are equivariant for
The naive quotient
makes sense. We will denote this space
and drop the (m) if \(m=0\). The inverse system of these spaces has a right action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )\), and a left action of \(GL_m(F)\). The induced map
is an isomorphism, and \(GL_m(F)\) acts trivially on these spaces. (Use the fact that
Similarly if \(U^p\) is a neat open compact subgroup of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^{\infty ,p})\) and if \(N \in {\mathbb {Z}}_{\ge 0}\) we set
In the case \(m=0\) we drop it from the notation. Each \({\mathcal {Y}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+}\) has a left action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) and the inverse system of the \({\mathcal {Y}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+}\) has a commuting right action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^{\infty })^{{\text {ord}}}\). It also has a left action of \(GL_m({\mathcal {O}}_{F,(p)})\). If \(\delta \in GL_m({\mathcal {O}}_{F,(p)})\) and \(\gamma \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) and \(g \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )^{{\text {ord}}}\) then \(\delta \circ \gamma =\delta (\gamma ) \circ \delta \) and \(\delta \circ g=\delta (g) \circ \delta \). There are equivariant maps
We set
There are maps
which are equivariant for the actions of the groups \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) and \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and \(GL_m({\mathcal {O}}_{F,(p)})\). Moreover the maps \({\mathcal {Y}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+} \rightarrow {\mathcal {Y}}_{n,(i),U^p(N)}^{{{\text {ord}}},+}\) and \(Y_{n,(i),U^p(N)}^{(m),+}\rightarrow Y_{n,(i),U^p(N)}^{+}\) are compatible. The induced maps
are isomorphisms.
Suppose now that U is a neat open compact subgroup of \(L^{(m)}_{n,(i)}({\mathbb {A}}^\infty )\). We set
In the case \(m=0\) we will write simply \(X^+_{n,(i),U}\). Then \(\{ X^{(m),+}_{n,(i),U} \}\) is a system of schemes (locally of finite type over \({{\text {Spec}}\,}{\mathbb {Q}}\)) with right \(L^{(m)}_{n,(i)}({\mathbb {A}}^\infty )\)-action via finite etale maps. Each \(X_{n,(i),U}^{(m),+}\) has a left action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\), which commutes with the right \(L^{(m)}_{n,(i)}({\mathbb {A}}^\infty )\)-action. The system also has a left action of \(GL_m(F)\). If \(\delta \in GL_m(F)\) and \(\gamma \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) and \(g \in L^{(m)}_{n,(i)}({\mathbb {A}}^\infty )\) then \(\delta \circ \gamma =\delta (\gamma ) \circ \delta \) and \(\delta \circ g=\delta (g) \circ \delta \). If \(U'\) is an open normal subgroup of U then \(X_{n,(i),U}^{(m),+}\) is identified with \(X_{n,(i),U'}^{(m),+}/U\). Projection to the second factor gives \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}}) \times L^{(m)}_{n,(i)}({\mathbb {A}}^\infty )\) and \(GL_m(F)\) equivariant maps
The fibre over \(g \in L_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )\) is simply \(X_{n-i,U \cap G_{n-i}({\mathbb {A}}^\infty )}\). If \(U'\) denotes the image of U in \(L_{n,(i)}({\mathbb {A}}^\infty )\) then there is a natural, \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}}) \times L^{(m)}_{n,(i)}({\mathbb {A}}^\infty )\)-equivariant, commutative diagram
We have
and
The naive quotient
makes sense and fibres over \(Y^{(m),\natural }_{n,(i),U}\), the fibre over g being \(X_{n-i,U_1}\), where \(U_1\) denotes the projection to \(G_{n-i}({\mathbb {A}}^\infty )\) of the subgroup \(U_2 \subset U\) consisting of elements whose projection to \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )\) lies in \(g^{-1}L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})g\). We sometimes write \(X^{\natural }_{n,(i),U}\) for \(X^{(0),\natural }_{n,(i),U}\). If \(U'\) denotes the projection of U to \(L_{n,(i)}({\mathbb {A}}^\infty )\), then the induced map
is an isomorphism. The action of \(L_{n,(i)}^{(m)}({\mathbb {A}}^\infty )\) is by finite etale maps and if \(U'\) is an open normal subgroup of U then \(X_{n,(i),U}^{(m),\natural }\) is identified with \(X_{n,(i),U'}^{(m),\natural }/U\). We have
We define sheaves \(\varOmega ^{+}_{n,(i),U}\) and \(\varXi ^{+}_{n,(i),U}\) over \(X^+_{n,(i),U}\) as the quotients of
and
by U. Then \(\{ \varOmega ^{+}_{n,(i),U} \}\) and \(\{ \varXi ^{+}_{n,(i),U} \}\) are systems of locally free sheaves on \(X^{+}_{n,(i),U}\) with left \(L_{n,(i)}({\mathbb {A}}^\infty )\)-action and commuting right \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-action.
Let \({\mathcal {E}}_{(i),U}^{+}\) denote the principal \(R_{n,(n),(i)}/N(R_{n,(n),(i)})\)-bundle on \(X^+_{n,(i),U}\) in the Zariski topology defined by setting, for \(W \subset X^+_{n,(i),U}\) a Zariski open, \({\mathcal {E}}_{(i),U}^{+}(W)\) to be the set of triples \((\xi _0,\xi _{11},\xi _{12})\), where
and
and
We define the \(R_{n,(n),(i)}/N(R_{n,(n),(i)})\)-action on \({\mathcal {E}}_{(i),U}^{+}\) by
The inverse system \(\{ {\mathcal {E}}_{(i),U}^{+} \}\) has an action of \(L_{n,(i)}({\mathbb {A}}^\infty )\) and of \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\).
Suppose that \(R_0\) is a \({\mathbb {Q}}\)-algebra and that \(\rho \) is a representation of the algebraic group \(R_{n,(n),(i)}/N(R_{n,(n),(i)})\) on a finite, locally free \(R_0\)-module \(W_\rho \). We define a locally free sheaf \({\mathcal {E}}^{+}_{(i),U,\rho }\) over \(X^{+}_{n,(i),U} \times {{\text {Spec}}\,}R_0\) by setting \({\mathcal {E}}^{+}_{(i),U,\rho }(W)\) to be the set of \((R_{n,(n),(i)}/N(R_{n,(n),(i)}))({\mathcal {O}}_W)\)-equivariant maps of Zariski sheaves of sets
Then \(\{ {\mathcal {E}}^{+}_{(i),U,\rho } \}\) is a system of locally free sheaves with an \(L_{n,(i)}({\mathbb {A}}^\infty )\)-action and an \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-action over the system of schemes \(\{ X^{+}_{n,(i),U} \times {{\text {Spec}}\,}R_0\}\). The restriction of \({\mathcal {E}}^+_{(i),U,\rho }\) to \(X_{n-i,hUh^{-1} \cap G_{n-i}({\mathbb {A}}^\infty )}\) can be identified with \({\mathcal {E}}_{hUh^{-1} \cap G_{n-i}({\mathbb {A}}^\infty ),\rho |_{L_{n-i,(n-i)}}}\). However the description of the actions of \(L_{n,(i)}({\mathbb {A}}^\infty )\) and \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) involve \(\rho \) and not just \(\rho |_{L_{n-i,(n-i)}}\). If \(g \in L_{n,(i)}({\mathbb {A}}^\infty )\) and \(\gamma \in L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\), then the natural maps
and
are isomorphisms.
We will also write
and
locally free sheaves on \(X^{\natural }_{n,(i),U}\). (If \(\rho \) is trivial on \(L_{n,(i),{{\text {lin}}}}\) then one can also form the quotient of \({\mathcal {E}}^+_{(i),U,\rho }\) by \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\), but in general this quotient does not make sense.)
If \(U^p\) is a neat open compact subgroup of \(L^{(m)}_{n,(i)}({\mathbb {A}}^{\infty ,p})\) and \(N_2 \ge N_1 \ge 0\) we set
In the case \(m=0\) we drop it from the notation. Each \({\mathcal {X}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+}\) has a left action of \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\) and the inverse system has a commuting right action of \(L_{n,(i)}^{(m)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\). There is also a left action of \(GL_m({\mathcal {O}}_{F,(p)})\). If \(\delta \in GL_m({\mathcal {O}}_{F,(p)})\) and \(\gamma \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) and \(g \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) then \(\delta \circ \gamma =\delta (\gamma ) \circ \delta \) and \(\delta \circ g=\delta (g) \circ \delta \). If \(g \in L_{n,(i)}^{(m)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and if
then this map is quasi-finite and flat. If \(g \in L_{n,(i)}^{(m)}({\mathbb {A}}^{\infty })^{{{\text {ord}}},\times }\) then it is etale, and, if further \(N_2=N_2'\), then it is finite etale. If \(N_2'>0\) and \(p^{N_2-N_2'}\nu (g_p) \in {\mathbb {Z}}_p^\times \) then the map is finite. On \({\mathbb {F}}_p\)-fibres the map \(\varsigma _p\) is absolute Frobenius composed with the forgetful map. If \((U^p)'\) is an open normal subgroup of \(U^p\) and if \(N_1 \le N_1'\le N_2\) then
There are commutative diagrams
We set
and write \({\mathcal {X}}_{n,(i),U^p(N_1,N_2)}^{{{\text {ord}}},\natural }\) for \({\mathcal {X}}_{n,(i),U^p(N_1,N_2)}^{(0),{{\text {ord}}},\natural }\). The system of these spaces has a right action of \(L_{n,(i)}^{(m)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and a left action of \(GL_m({\mathcal {O}}_{F,(p)})\). If \(\delta \in GL_m({\mathcal {O}}_{F,(p)})\) and \(g \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) then \(\delta \circ g=\delta (g) \circ \delta \). If \(g \in L_{n,(i)}^{(m)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and if
then this map is quasi-finite and flat. If \(g \in L_{n,(i)}^{(m)}({\mathbb {A}}^{\infty })^{{{\text {ord}}},\times }\) then it is etale, and, if further \(N_2=N_2'\), then it is finite etale. If \(N_2'>0\) and \(p^{N_2-N_2'}\nu (g_p) \in {\mathbb {Z}}_p^\times \) then the map is finite. On \({\mathbb {F}}_p\)-fibres the map \(\varsigma _p\) is absolute Frobenius composed with the forgetful map. If \((U^p)'\) is an open normal subgroup of \(U^p\) and if \(N_1 \le N_1'\le N_2\) then
The natural maps
are isomorphisms.
We define sheaves \(\varOmega ^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) and \(\varXi ^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) over \({\mathcal {X}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) as the quotients of
and
by \(U^p\). Then the systems of sheaves \(\varOmega ^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) and \(\varXi ^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) have commuting actions of \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) and \(L_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\).
Let \({\mathcal {E}}_{(i),U^p(N_1,N_2)}^{{{\text {ord}}},+}\) denote the principal \(R_{n,(n),(i)}/N(R_{n,(n),(i)})\)-bundle for the Zariski topology on \({\mathcal {X}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) defined by setting, for \(W \subset {\mathcal {X}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) a Zariski open, \({\mathcal {E}}_{(i),U^p(N_1,N_2)}^{{{\text {ord}}},+}(W)\) to be the set of triples \((\xi _0,\xi _{11},\xi _{12})\), where
and
and
We define the \(R_{n,(n),(i)}/N(R_{n,(n),(i)})\)-action on \({\mathcal {E}}_{(i),U^p(N_1,N_2)}^{{{\text {ord}}},+}\) by
The inverse system \(\{ {\mathcal {E}}_{(i),U^p(N_1,N_2)}^{+} \}\) has an action of \(L_{n,(i)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) and an action of \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\).
Suppose that \(R_0\) is a \({\mathbb {Z}}_{(p)}\)-algebra and that \(\rho \) is a representation of the algebraic group \(R_{n,(n),(i)}/N(R_{n,(n),(i)})\) on a finite, locally free \(R_0\)-module \(W_\rho \). We define a locally free sheaf \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho }\) over \({\mathcal {X}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \times {{\text {Spec}}\,}R_0\) by setting \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho }(W)\) to be the set of \((R_{n,(n),(i)}/N(R_{n,(n),(i)}))({\mathcal {O}}_W)\)-equivariant maps of Zariski sheaves of sets
Then \(\{ {\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho } \}\) is a system of locally free sheaves with \(L_{n,(i)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-action and \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\)-action over the system of schemes \(\{ {\mathcal {X}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \times {{\text {Spec}}\,}R_0\}\). The restriction of \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho }\) to \({\mathcal {X}}^{{{\text {ord}}}}_{n-i,(hU^ph^{-1} \cap G_{n-i}({\mathbb {A}}^{\infty ,p})(N_1,N_2))}\) can be identified with \({\mathcal {E}}^{{{\text {ord}}}}_{(hU^ph^{-1} \cap G_{n-i}({\mathbb {A}}^{\infty ,p}))(N_1,N_2),\rho |_{L_{n-i,(n-i)}}}\). However the description of the actions of the groups \(L_{n,(i)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) and \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) involves \(\rho \) and not just \(\rho |_{L_{n-i,(n-i)}}\). If \(g \in L_{n,(i)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) and \(\gamma \in L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\), then the natural maps
and
are isomorphisms.
We will also write
and
locally free sheaves on \({\mathcal {X}}^{{{\text {ord}}},\natural }_{n,(i),U^p(N_1,N_2)}\).
There are maps
which are equivariant for the actions of the groups \(L_{n,(i)}^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) and \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\) and \(GL_m({\mathcal {O}}_{F,(p)})\). Under these maps the sheaves \(\varOmega ^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) (resp. \(\varXi ^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\), resp. \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho }\)) correspond to \(\varOmega ^+_{n,(i),U^p(N_1,N_2)}\) (resp. to \(\varXi ^+_{n,(i),U^p(N_1,N_2)}\), resp. to \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho \otimes {\mathbb {Q}}}\)). The induced maps
are isomorphisms.
5.2 Generalized Kuga–Sato varieties
Now suppose that U is a neat open compact subgroup of
We set
In the case \(m=0\) we will write simply \(A^+_{n,(i),U}\).
If \(g \in (P^{(m),+}_{n,(i)}/Z(N^{(m)}_{n,(i)}))({\mathbb {A}}^\infty )\) and \(g^{-1} U g \subset U'\), then we define a finite etale map
to be the coproduct of the maps
where \(h,h' \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )\) and \(g' \in G^{(i+m)}_{n-i}({\mathbb {A}}^\infty )\) satisfy \(hg=g' h'\). This makes \(\{ A^{(m),+}_{n,(i),U} \}\) a system of schemes (locally of finite type over \({{\text {Spec}}\,}{\mathbb {Q}}\)) with right action of the group \((P^{(m),+}_{n,(i)}/Z(N^{(m)}_{n,(i)}))({\mathbb {A}}^\infty )\). If \(U'\) is an open normal subgroup of U then \(A_{n,(i),U}^{(m),+}\) is identified with \(A_{n,(i),U'}^{(m),+}/U\).
If \(\gamma \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\), then we define
to be the coproduct of the maps
This gives a left action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) on each \(A_{n,(i),U}^{(m),+}\), which commutes with the action of \((P^{(m),+}_{n,(i)}/Z(N^{(m)}_{n,(i)}))({\mathbb {A}}^\infty )\).
If \(\delta \in GL_m(F)\) define a map
as the coproduct of the maps
This gives a left \(GL_m(F)\)-action on the system of the \(A_{n,(i),U}^{(m),+}\). If \(\delta \in GL_m(F)\) and \(\gamma \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) and \(g \in (P^{(m),+}_{n,(i)}/Z(N^{(m)}_{n,(i)}))({\mathbb {A}}^\infty )\) then \(\delta \circ \gamma =\delta (\gamma ) \circ \delta \) and \(\delta \circ g=\delta (g) \circ \delta \).
There are natural maps
which are equivariant for the actions of \((P^{(m),+}_{n,(i)}/Z(N^{(m)}_{n,(i)}))({\mathbb {A}}^\infty )\) and \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) and \(GL_m(F)\).
If \(U'\) denotes the image of U in \((P_{n,(i)}^+/Z(N_{n,(i)}))({\mathbb {A}}^\infty )\) then there is a natural commutative diagram:
which is \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) and \((P^{(m),+}_{n,(i)}/Z(N^{(m)}_{n,(i)}))({\mathbb {A}}^\infty )\) equivariant.
We have
Note that it does not make sense to divide \(A_{n,(i),U}^{(m),+}\) by \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}})\), so we don’t do so.
We define a semi-abelian scheme \(\widetilde{{G}}^{{\text {univ}}}/A^+_{n,(i),U}\) by requiring that over the open and closed subscheme \(A^{(i)}_{n-i,hUh^{-1} \cap G_{n-i}^{(i)}({\mathbb {A}}^\infty )}\) it restricts to \(G^{{\text {univ}}}\). It is unique up to unique quasi-isogeny. We also define a sheaf \(\widetilde{\varOmega }^{+}_{n,(i),U}\) (resp. \(\widetilde{\varXi }^+_{n,(i),U}\)) over \(A^+_{n,(i),U}\) to be the unique sheaf which, for each h, restricts to \(\varOmega ^{(i)}_{n-i,hUh^{-1} \cap G_{n-i}^{(i)}({\mathbb {A}}^\infty )}\) (resp. \(\varXi _{n-i,hUh^{-1} \cap G_{n-i}^{(i)}({\mathbb {A}}^\infty )}\)) on \(A^{(i)}_{n-i,hUh^{-1} \cap G_{n-i}^{(i)}({\mathbb {A}}^\infty )}\). Thus \(\widetilde{\varOmega }^+_{n,(i),U}\) is the pull-back by the identity section of \(\varOmega ^1_{\widetilde{{G}}^{{\text {univ}}}/A^+_{n,(i),U}}\). Then \(\{ \widetilde{\varOmega }^{+}_{n,(i),U} \}\) (resp. \(\{ \widetilde{\varXi }^{+}_{n,(i),U} \}\)) is a system of locally free sheaves on \(A^{+}_{n,(i),U}\) with a left \((P_{n,(i)}^+/Z(N_{n,(i)}))({\mathbb {A}}^\infty )\)-action and a commuting right \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-action. There are equivariant exact sequences
where \(\pi \) denotes the map \(A^+_{n,(i),U} \rightarrow X^+_{n,(i),U}\).
Let \({\widetilde{{\mathcal {E}}}}_{(i),U}^{+}\) denote the principal \(R_{n,(n),(i)}\)-bundle on \(A^+_{n,(i),U}\) in the Zariski topology defined by setting, for \(W \subset A^+_{n,(i),U}\) a Zariski open, \({\widetilde{{\mathcal {E}}}}_{(i),U}^{+}(W)\) to be the set of pairs \((\xi _0,\xi _1)\), where
and
satisfies
We define the \(R_{n,(n),(i)}\)-action on \({\mathcal {E}}_{(i),U}^{+}\) by
The inverse system \(\{ {\widetilde{{\mathcal {E}}}}_{(i),U}^{+} \}\) has an action of \(P_{n,(i)}^+({\mathbb {A}}^\infty )\) and of \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\).
Suppose that \(R_0\) is a \({\mathbb {Q}}\)-algebra and that \(\rho \) is a representation of \(R_{n,(n),(i)}\) on a finite, locally free \(R_0\)-module \(W_\rho \). We define a locally free sheaf \({\mathcal {E}}^{+}_{(i),U,\rho }\) over \(A^{+}_{n,(i),U} \times {{\text {Spec}}\,}R_0\) by setting \({\mathcal {E}}^{+}_{(i),U,\rho }(W)\) to be the set of \(R_{n,(n),(i)}({\mathcal {O}}_W)\)-equivariant maps of Zariski sheaves of sets
Then \(\{ {\mathcal {E}}^{+}_{(i),U,\rho } \}\) is a system of locally free sheaves with both a \(P_{n,(i)}^+({\mathbb {A}}^\infty )\)-action and an \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-action over the system of schemes \(\{ A^{+}_{n,(i),U} \times {{\text {Spec}}\,}R_0\}\). The restriction of \({\mathcal {E}}^+_{(i),U,\rho }\) to \(A^{(i)}_{n-i,hUh^{-1} \cap G_{n-i}^{(i)}({\mathbb {A}}^\infty )}\) can be identified with \({\mathcal {E}}^{(i)}_{hUh^{-1} \cap G_{n-i}^{(i)}({\mathbb {A}}^\infty ),\rho |_{R_{n-i,(n-i)}^{(i)}}}\). However the description of the actions of \(P_{n,(i)}^+({\mathbb {A}}^\infty )\) and \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) involve \(\rho \) and not just \(\rho |_{R^{(i)}_{n-i,(n-i)}}\). If \(g \in P_{n,(i)}^+({\mathbb {A}}^\infty )\) and \(\gamma \in L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\), then the natural maps
and
are isomorphisms. If \(\rho \) factors through \(R_{n,(n),(i)}/N(R_{n,(n),(i)})\) then \({\mathcal {E}}^{+}_{(i),U,\rho }\) is canonically isomorphic to the pull-back of \({\mathcal {E}}^+_{(i),U,\rho }\) from \(X^+_{n,(i),U}\). In general \(W_\rho \) has a filtration by \(R_{n,(n),(i)}\)-invariant local direct summands such that the action of \(R_{n,(n),(i)}\) on each graded piece factors through \(R_{n,(n),(i)}/N(R_{n,(n),(i)})\). (To see this apply proposition 4.7.3 of exposé I of [53] to the action of \(A_{n,(i),{{\text {lin}}}}\) on \(W_\rho \).) Thus \({\mathcal {E}}^{+}_{(i),U,\rho }\) has a \(P_{n,(i)}^+({\mathbb {A}}^\infty )\) and \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) invariant filtration by local direct summands such that each graded piece is the pull-back of some \({\mathcal {E}}^+_{(i),U,\rho '}\) from \(X^+_{n,(i),U}\).
Similarly if \(U^p\) is a neat open compact subgroup of \((P^{(m),+}_{n,(i)}/Z(N^{(m)}_{n,(i)}))({\mathbb {A}}^{\infty ,p})=(\widetilde{{P}}^{(m),+}_{n,(i)}/Z(\widetilde{{N}}^{(m)}_{n,(i)}))({\mathbb {A}}^{\infty ,p})\) we set
In the case \(m=0\) we will write simply \({\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\). The inverse system of the \({\mathcal {A}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+}\) has a right action of \((P^{(m),+}_{n,(i)}/Z(N^{(m)}_{n,(i)}))({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and a commuting left action of \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\). If \(g \in (P^{(m),+}_{n,(i)}/Z(N^{(m)}_{n,(i)}))({\mathbb {A}}^{\infty })^{{\text {ord}}}\) then the map
is quasi-finite and flat. If \(g \in (P^{(m),+}_{n,(i)}/Z(N^{(m)}_{n,(i)}))({\mathbb {A}}^{\infty })^{{{\text {ord}}},\times }\) then it is etale, and, if further \(N_2=N_2'\), then it is finite etale. If \(N_2'>0\) and \(p^{N_2-N_2'}\nu (g_p) \in {\mathbb {Z}}_p^\times \) then the map is finite. On \({\mathbb {F}}_p\)-fibres the map \(\varsigma _p\) is absolute Frobenius composed with the forgetful map. If \((U^p)'\) is an open normal subgroup of \(U^p\) and if \(N_1 \le N_1' \le N_2\) then \({\mathcal {A}}_{n,(i),(U^p)'(N_1',N_2)}^{(m),{{\text {ord}}},+}/U^p(N_1,N_2)\) is identified with \({\mathcal {A}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+}\). Further there is a left action of \(GL_m({\mathcal {O}}_{F,(p)})\) such that if \(\delta \in GL_m({\mathcal {O}}_{F,(p)})\) and \(\gamma \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) and \(g \in (P^{(m),+}_{n,(i)}/Z(N^{(m)}_{n,(i)}))({\mathbb {A}}^{\infty })^{{\text {ord}}}\), \(\delta \circ \gamma =\delta (\gamma ) \circ \delta \) and \(\delta \circ g=\delta (g) \circ \delta \). There are natural equivariant maps
If \((U^p)'\) denotes the image of \(U^p\) in \((P_{n,(i)}^+/Z(N_{n,(i)}))({\mathbb {A}}^{\infty ,p})\) then there is a natural equivariant, commutative diagram:
There are equivariant embeddings
We define a semi-abelian scheme \({\widetilde{{\mathcal {G}}}}^{{\text {univ}}}/{\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) over \({\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) by requiring that over \({\mathcal {A}}^{(i),{{\text {ord}}}}_{n-i,(hU^ph^{-1} \cap G_{n-i}^{(i)}({\mathbb {A}}^{\infty ,p}))(N_1,N_2)}\) it restricts to \({\mathcal {G}}^{{\text {univ}}}\). It is unique up to unique prime-to-p quasi-isogeny. We define a locally free sheaf \(\widetilde{\varOmega }^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) (resp. \(\widetilde{\varXi }^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\)) over the scheme \({\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) to be the sheaf which, for each h, restricts to the sheaf
(resp. \(\varXi ^{(i),{{\text {ord}}}}_{n-i,(hU^ph^{-1} \cap G_{n-i}^{(i)}({\mathbb {A}}^{\infty ,p}))(N_1,N_2)}\)) on the subscheme \({\mathcal {A}}^{(i),{{\text {ord}}}}_{n-i,(hU^ph^{-1} \cap G_{n-i}^{(i)}({\mathbb {A}}^{\infty ,p}))(N_1,N_2)}\). Then \(\widetilde{\varOmega }^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) is the pull-back by the identity section of \(\varOmega ^1_{{\widetilde{{\mathcal {G}}}}^{{\text {univ}}}/{\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}}\). The collection \(\{\widetilde{\varOmega }^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \}\) (resp. \(\{ \widetilde{\varXi }^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \}\)) is a system of locally free sheaves on \({\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) with a left \((P_{n,(i)}^+/Z(N_{n,(i)}))({\mathbb {A}}^\infty )^{{\text {ord}}}\)-action and a commuting right \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\)-action. Also there are equivariant exact sequences
where \(\pi \) denotes the map \({\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U} \rightarrow {\mathcal {X}}^{{{\text {ord}}},+}_{n,(i),U}\).
Let \({\widetilde{{\mathcal {E}}}}_{(i),U^p(N_1,N_2)}^{{{\text {ord}}},+}\) denote the principal \(R_{n,(n),(i)}\)-bundle on the scheme \({\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) in the Zariski topology defined by setting, for \(W \subset {\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) a Zariski open, \({\widetilde{{\mathcal {E}}}}_{(i),U^p(N_1,N_2)}^{{{\text {ord}}},+}(W)\) to be the set of pairs \((\xi _0,\xi _1)\), where
and
satisfies
We define the \(R_{n,(n),(i)}\)-action on \({\widetilde{{\mathcal {E}}}}_{(i),U^p(N_1,N_2)}^{{{\text {ord}}},+}\) by
The inverse system \(\{ {\widetilde{{\mathcal {E}}}}_{(i),U^p(N_1,N_2)}^{{{\text {ord}}},+} \}\) has an action both of the groups \(P_{n,(i)}^+({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) and of \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\).
Suppose that \(R_0\) is a \({\mathbb {Q}}\)-algebra and that \(\rho \) is a representation of \(R_{n,(n),(i)}\) on a finite, locally free \(R_0\)-module \(W_\rho \). We define a locally free sheaf \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho }\) over \({\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \times {{\text {Spec}}\,}R_0\) by setting \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho }(W)\) to be the set of \(R_{n,(n),(i)}({\mathcal {O}}_W)\)-equivariant maps of Zariski sheaves of sets
Then \(\{ {\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho } \}\) is a system of locally free sheaves with \(P_{n,(i)}^+({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-action and \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\)-action over the system of schemes \(\{ {\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \times {{\text {Spec}}\,}R_0\}\). The restriction of \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho }\) to \({\mathcal {A}}^{(i),{{\text {ord}}}}_{n-i,(hU^ph^{-1} \cap G_{n-i}^{(i)}({\mathbb {A}}^{\infty ,p}))(N_1,N_2)}\) can be identified with
However the description of the actions of the groups \(P_{n,(i)}^+({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) and \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) involves \(\rho \) and not just \(\rho |_{R^{(i)}_{n-i,(n-i)}}\). If \(g \in P_{n,(i)}^+({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) and \(\gamma \in L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\), then the natural maps
and
are isomorphisms. If \(\rho \) factors through \(R_{n,(n),(i)}/N(R_{n,(n),(i)})\) then \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho }\) is canonically isomorphic to the pull-back of \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho }\) from \({\mathcal {X}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\). In general \(W_\rho \) has a filtration by \(R_{n,(n),(i)}\)-invariant local direct summands such that the action of \(R_{n,(n),(i)}\) on each graded piece factors through \(R_{n,(n),(i)}/N(R_{n,(n),(i)})\). (To see this apply proposition 4.7.3 of exposé I of [53] to the action of \(A_{n,(i),{{\text {lin}}}}\) on \(W_\rho \).) Thus \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho }\) has a \(P_{n,(i)}^+({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) and \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) invariant filtration by local direct summands such that each graded piece is the pull-back of some \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho '}\) from \({\mathcal {X}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\).
The next lemma follows from the discussion in Sect. 3.4.
Lemma 4.1
If \(U'\) is the image of U (resp. \(U^p\)) and if \(\pi \) denotes the map \(A_{n,(i),U}^{(m),+} \rightarrow A_{n,(i),U'}^+\) then there are \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-equivariant, \((P^{+}_{n,(i)}/Z(N_{n,(i)}))({\mathbb {A}}^\infty )\)-equivariant and \(GL_m(F)\)-equivariant isomorphisms
5.3 Generalized mixed Shimura varieties
Next suppose \(\widetilde{{U}}\) is an open compact subgroup of \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\). We define a split torus \(\widetilde{{S}}^{(m),+}_{n,(i),\widetilde{{U}}}/Y_{n,(i),\widetilde{{U}}}^{(m),+}\) as
Thus \(X_*(\widetilde{{S}}^{(m),+}_{n,(i),\widetilde{{U}}})_{\mathbb {Q}}\) is a constant sheaf:
If \(\widetilde{{g}}\in \widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) and \(\widetilde{{g}}^{-1} \widetilde{{U}}\widetilde{{g}}\subset \widetilde{{U}}'\), then we define
to be the coproduct of the maps
where \(h,h' \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )\) and \(\widetilde{{g}}' \in \widetilde{{G}}^{(m+i)}_{n-i}({\mathbb {A}}^\infty )\) satisfy \(h\widetilde{{g}}=\widetilde{{g}}' h'\). This makes \(\{ \widetilde{{S}}^{(m),+}_{n,(i),\widetilde{{U}}} \}\) a system of relative tori with right \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\)-action. If \(\gamma \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\), then we define
to be the coproduct of the maps
This gives a left action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) on each \(\widetilde{{S}}_{n,(i),\widetilde{{U}}}^{(m),+}\), which commutes with the action of \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\).
Similarly suppose \(\widetilde{{U}}^p\) is an open compact subgroup of \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty ,p})\) and that N is a non-negative integer. We define a split torus \(\widetilde{{{\mathcal {S}}}}^{(m),{{\text {ord}}},+}_{n,(i),\widetilde{{U}}^p(N)}/{\mathcal {Y}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),{{\text {ord}}},+}\) as
for any \(N'\ge N\). Thus \(X_*(\widetilde{{S}}^{(m),{{\text {ord}}},+}_{n,(i),\widetilde{{U}}^p(N)})_{{\mathbb {Z}}_{(p)}}\) is a constant sheaf:
If \(\widetilde{{g}}\in \widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and \(\widetilde{{g}}^{-1} \widetilde{{U}}^p(N) \widetilde{{g}}\subset (\widetilde{{U}}^p)'(N')\), then we define
to be the coproduct of the maps
where \(h,h' \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and \(\widetilde{{g}}' \in \widetilde{{G}}^{(m+i)}_{n-i}({\mathbb {A}}^\infty )^{{\text {ord}}}\) satisfy \(h\widetilde{{g}}=\widetilde{{g}}' h'\). This makes \(\{ \widetilde{{{\mathcal {S}}}}^{(m),+}_{n,(i),\widetilde{{U}}^p(N)} \}\) a system of relative tori with right \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-action. If \(\gamma \) is an element of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\), then we define
to be the coproduct of the maps
This gives a left action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) on each \(\widetilde{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),+}\), which commutes with the action of \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\).
The sheaves \(X^*(\widetilde{{S}}_{n,(i),\widetilde{{U}}}^{(m),+})\) and \(X_*(\widetilde{{S}}_{n,(i),\widetilde{{U}}}^{(m),+})\) have actions of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\). The sheaves \(X^*(\widetilde{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),{{\text {ord}}},+})\) and \(X_*(\widetilde{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),{{\text {ord}}},+})\) have actions of the group \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\). The systems of sheaves \(\{ X^*(\widetilde{{S}}_{n,(i),\widetilde{{U}}}^{(m),+}) \}\) and \(\{ X_*(\widetilde{{S}}_{n,(i),\widetilde{{U}}}^{(m),+}) \}\) (resp. \(\{ X^*(\widetilde{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),{{\text {ord}}},+})\}\) and \(\{X_*(\widetilde{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),{{\text {ord}}},+})\}\)) have actions of \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) (resp. \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )^{{\text {ord}}}\)).
The sheaf
is a subsheaf of \(X_*(\widetilde{{S}}_{n,(i),\widetilde{{U}}}^{(m),+})\). (Recall the embedding
It is invariant by the actions of the groups \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) and \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\). We define a split torus
by
If U denote the image of \(\widetilde{{U}}\) in \(P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )\), then we will write
It depends only on U and not on the choice of \(\widetilde{{U}}\) mapping onto U. The sheaf \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {Q}}\) is constant:
In the case \(m=0\) we will write simply \(S^+_{n,(i),U}\). The tori \(\widetilde{{S}}_{n,(i),\widetilde{{U}}}^{(m),+}\) and \(S_{n,(i),\widetilde{{U}}}^{(m),+}\) inherit a left action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) and a right action of \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\). In the case of \(S_{n,(i),\widetilde{{U}}}^{(m),+}\) the latter factors through \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\). If \(\widetilde{{U}}\) is a neat open compact subgroup of \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) with image U in \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) and image \(U'\) in \(P^{+}_{n,(i)}({\mathbb {A}}^\infty )\), then there is a natural, \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-equivariant and \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\)-equivariant, commutative diagram:
Similarly the sheaf
is a subsheaf of \(X_*(\widetilde{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),{{\text {ord}}},+})\). It is invariant by the actions of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) and \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )^{{\text {ord}}}\). We define a split torus
by
If \(U^p\) denotes the image of \(\widetilde{{U}}^p\) in \(P_{n,(i)}^{(m)}({\mathbb {A}}^{\infty ,p})\), then we will write
It depends only on \(U^p\) and not the \(\widetilde{{U}}^p\) mapping to \(U^p\). The sheaf \(X_*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{{\mathbb {Z}}_{(p)}}\) is constant:
In the case \(m=0\) we will write simply \({\mathcal {S}}^+_{n,(i),U^p(N)}\). The tori \(\widehat{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),{{\text {ord}}},+}\) and \({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+}\) inherit a left action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) and a right action of \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )^{{\text {ord}}}\). In the case of \({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+}\) the latter factors through \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )^{{\text {ord}}}\). If \(\widetilde{{U}}^p\) is a neat open compact subgroup of \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty ,p})\) with image \(U^p\) in \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty ,p})\) and image \((U^p)'\) in \(P^{+}_{n,(i)}({\mathbb {A}}^{\infty ,p})\), then there is a natural, \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\)-equivariant and \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )^{{\text {ord}}}\)-equivariant, commutative diagram:
There are natural equivariant embeddings
and
and
We write \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\succ 0}\) (resp. \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{>0}\), resp. \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\ge 0}\)) for the subsheaves (of monoids) of \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}\) corresponding to the subset \({\mathfrak {C}}^{(m),\succ 0}(V_{n,(i)}) \subset Z(N^{(m)}_{n,(i)})({\mathbb {R}})\) (resp. to the subset \({\mathfrak {C}}^{(m),> 0}(V_{n,(i)}) \subset Z(N^{(m)}_{n,(i)})({\mathbb {R}})\), resp. to the subset \({\mathfrak {C}}^{(m),\ge 0}(V_{n,(i)}) \subset Z(N^{(m)}_{n,(i)})({\mathbb {R}})\)).
We will also write \(X^*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\ge 0}\) (resp. \(X^*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{>0}\), resp. \(X^*(S_{n,(i),U}^{(m),+})^{\ge 0}\), resp. \(X^*(S_{n,(i),U}^{(m),+})^{>0}\)) for the subsheaves (of monoids) of \(X^*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}\) (resp. \(X^*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}\), resp. \(X^*(S_{n,(i),U}^{(m),+})\), resp. \(X^*(S_{n,(i),U}^{(m),+})\)) consisting of sections that have non-negative (resp. strictly positive, resp. non-negative, resp. strictly positive) pairing with each nonzero section of \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\ge 0}\). All these sheaves have (compatible) actions of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\). The system of sheaves \(\{ X_*(S_{n,(i),U}^{(m),+}) \}\) has an action of \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\), and the same is true for all the other systems of sheaves we are considering in this paragraph.
We may take the quotients of the sheaves \(X^*(S_{n,(i),U}^{(m),+})\) (resp. \(X^*(S_{n,(i),U}^{(m),+})^{>0}\), resp. \(X^*(S_{n,(i),U}^{(m),+})^{\ge 0}\)) by \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) to give sheaves of sets on \(Y_{n,(i),U}^{(m),\natural }\), which we will denote \(X^*(S_{n,(i),U}^{(m),+})^\natural \) (resp. \(X^*(S_{n,(i),U}^{(m),+})^{>0,\natural }\), resp. \(X^*(S_{n,(i),U}^{(m),+})^{\ge 0,\natural }\)). If \(y=h U\) lies in \(Y^{(m),+}_{n,(i),U}\) above \(y^\natural \in Y^{(m),\natural }_{n,(i),U}\) then the stalk of \(X^*(S_{n,(i),U}^{(m),+})^\natural \) at \(y^\natural \) equals
We will write \(X_*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\ge 0}\) (resp. \(X_*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{>0}\), resp. \(X_*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\ge 0}\)) for the subsheaves (of monoids) of \(X_*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}\) corresponding to
(resp.
resp.
Again we will write \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\ge 0}\) (resp. \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})^{\ge 0}\)) for the subsheaves (of monoids) of \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}\) (resp. \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})\)) consisting of sections that have non-negative pairing with each section of \(X_*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\ge 0}\). We will also write \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{>0}\) (resp. \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})^{>0}\)) for the subsheaves (of monoids) of the sheaves \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}\) (resp. \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})\)) consisting of sections that have strictly positive pairing with each nonzero section of \(X_*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\ge 0}\). All these sheaves have actions of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\). The system of sheaves \(\{ X_*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+}) \}\) has an action of \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )^{{\text {ord}}}\), and the same is true for all the other systems of sheaves we are considering in this paragraph.
We may take the quotients of the sheaves \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})\) and \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})^{>0}\) and \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})^{\ge 0}\) by \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) to give sheaves of sets on \({\mathcal {Y}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},\natural }\), which we will denote \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})^\natural \) and \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})^{>0,\natural }\) and \(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})^{\ge 0,\natural }\).
Suppose again that \(\widetilde{{U}}\) is a neat open compact subgroup of \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) and set
It is an \(\widetilde{{S}}_{n,(i),\widetilde{{U}}}^{(m),+}\)-torsor over \(A_{n,(i),\widetilde{{U}}}^{(m),+}\). If U denotes the image of \(\widetilde{{U}}\) in \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) then the push-out of \(\widetilde{{T}}_{n,(i),\widetilde{{U}}}^{(m),+}\) under \(\widetilde{{S}}_{n,(i),\widetilde{{U}}}^{(m),+} \twoheadrightarrow S_{n,(i),U}^{(m),+}\) is an \(S_{n,(i),U}^{(m),+}\)-torsor over \(A_{n,(i),U}^{(m),+}=A_{n,(i),\widetilde{{U}}}^{(m),+}\), which only depends on U (and not \(\widetilde{{U}}\)), and which we will denote \(T_{n,(i),U}^{(m),+}\). In the case \(m=0\) we will write simply \(T^+_{n,(i),U}\). Note that \(\widetilde{{T}}_{n,(i),\widetilde{{U}}}^{(m),+}\) is an \(\widehat{{S}}^{(m),+}_{n,(i),\widetilde{{U}}}\)-torsor over \(T_{n,(i),\widetilde{{U}}}^{(m),+}\).
If \(\widetilde{{g}}\in \widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) and \(\widetilde{{g}}^{-1} \widetilde{{U}}\widetilde{{g}}\subset \widetilde{{U}}'\), then we define
to be the coproduct of the maps
where \(h,h' \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )\) and \(\widetilde{{g}}' \in \widetilde{{G}}^{(m+i)}_{n-i}({\mathbb {A}}^\infty )\) satisfy \(h\widetilde{{g}}=\widetilde{{g}}' h'\). This makes \(\{ \widetilde{{T}}^{(m),+}_{n,(i),\widetilde{{U}}} \}\) a system of \(\{\widetilde{{S}}_{n,(i),\widetilde{{U}}}^{(m),+} \} \)-torsors over \(\{ A_{n,(i),\widetilde{{U}}}^{(m),+}\} \) with right \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\)-action. It also induces an action of \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) on \(\{ T^{(m),+}_{n,(i),U} \}\), which makes \(\{ T^{(m),+}_{n,(i),U} \}\) a system of \(\{S_{n,(i),U}^{(m),+} \} \)-torsors over \(\{ A_{n,(i),U}^{(m),+}\} \) with right \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\)-action. If \(\gamma \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\), then we define
to be the coproduct of the maps
This gives a left action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) on each \(\widetilde{{T}}_{n,(i),\widetilde{{U}}}^{(m),+}\), which commutes with the action of \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\). It induces a left action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) on each \(T_{n,(i),U}^{(m),+}\), which commutes with the action of \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\). Suppose that \(\widetilde{{U}}\) is a neat open compact subgroup of \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) with image U in \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) and image \(U'\) in \(P^{+}_{n,(i)}({\mathbb {A}}^\infty )\). Then there is a commutative diagram
This diagram is \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-equivariant and \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\)-equivariant. We have
Similarly if \(\widetilde{{U}}^p\) is a neat open compact subgroup of \(\widetilde{{P}}^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty ,p})\) and \(0 \le N_1 \le N_2\) we set
It is an \(\widetilde{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N_1)}^{(m),{{\text {ord}}},+}\)-torsor over \({\mathcal {A}}_{n,(i),\widetilde{{U}}^p(N_1,N_2)}^{(m),{{\text {ord}}},+}\). If \(U^p\) denotes the image of \(\widetilde{{U}}^p\) in \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty ,p})\) then the push-out of \({\widetilde{{\mathcal {T}}}}_{n,(i),\widetilde{{U}}^p(N_1,N_2)}^{(m),{{\text {ord}}},+}\) under \(\widetilde{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N_1)}^{(m),{{\text {ord}}},+} \twoheadrightarrow {\mathcal {S}}_{n,(i),U^p(N_1)}^{(m),{{\text {ord}}},+}\) is an \({\mathcal {S}}_{n,(i),U^p(N_1)}^{(m),{{\text {ord}}},+}\)-torsor over \({\mathcal {A}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+}\), which only depends on \(U^p\) (and not \(\widetilde{{U}}^p\)) and \(N_1,N_2\), and which we will denote \({\mathcal {T}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+}\). In the case \(m=0\) we will write simply \({\mathcal {T}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\). Note that \({\widetilde{{\mathcal {T}}}}_{n,(i),\widetilde{{U}}^p(N_1,N_2)}^{(m),{{\text {ord}}},+}\) is a \(\widehat{{{\mathcal {S}}}}^{(m),{{\text {ord}}},+}_{n,(i),\widetilde{{U}}^p(N_1)}\)-torsor over \({\mathcal {T}}_{n,(i),\widetilde{{U}}^p(N_1,N_2)}^{(m),{{\text {ord}}},+}\).
As above the system \(\{ {\widetilde{{\mathcal {T}}}}_{n,(i),\widetilde{{U}}^p(N_1,N_2)}^{(m),{{\text {ord}}},+} \}\) has a right action of \(\widetilde{{P}}_{n,(i)}^{(m),+}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) and a commuting left action of \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\). If \(g \in \widetilde{{P}}_{n,(i)}^{(m),+}({\mathbb {A}}^{\infty })^{{{\text {ord}}},\times }\) then the map g is finite etale. The map
equals absolute Frobenius composed with the forgetful map. If \(N_2>1\) then the map
is finite flat. Further there is a left action of \(GL_m({\mathcal {O}}_{F,(p)})\) such that if \(\delta \in GL_m({\mathcal {O}}_{F,(p)})\) and \(\gamma \in L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) and \(g \in \widetilde{{P}}_{n,(i)}^{(m),+}({\mathbb {A}}^{\infty })^{{\text {ord}}}\), then \(\gamma \) followed by \(\delta \) equals \(\delta \) followed by \(\delta \gamma \delta ^{-1}\), and g followed by \(\delta \) equals \(\delta \) followed by \(\delta g \delta ^{-1}\). These actions are also all compatible with the actions on \(\{ \widetilde{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N_1)}^{(m),{{\text {ord}}},+} \}\). There are induced actions of the groups \(GL_m({\mathcal {O}}_{F,(p)})\) and \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) and \(P_{n,(i)}^{(m),+}({\mathbb {A}}^{\infty })^{{\text {ord}}}\) on \(\{ {\mathcal {T}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+} \}\), which are compatible with the actions on \(\{ {\mathcal {S}}_{n,(i),U^p(N_1)}^{(m),{{\text {ord}}},+} \}\). There is an equivariant commutative diagram
There are natural equivariant embeddings
and
If a is a section of \(X^*(S^{(m),+}_{n,(i),U})\) over \(W\subset Y^{(m)}_{n,(i),U}\) then we can associate to it a line bundle
over \(A^{(m),+}_{n,(i),U}|_W\) as in Sect. 2.1. There are natural isomorphisms
Suppose that \(R_0\) is a noetherian \({\mathbb {Q}}\)-algebra. Suppose also that U is a neat open compact subgroup of \(P_{n,(i)}^+({\mathbb {A}}^\infty )\). If a is a section in \(X^*(S^{+}_{n,(i),U})^{>0}(W)\) then \({\mathcal {L}}_{U}^+(a)\) is relatively ample for \(A^{+}_{n,(i),U}|_W/X^{+}_{n,(i),U}|_W\). If \(\pi ^+\) denotes the map
then we see that
for \(i>0\). (Because \(A_{n,(i),U}^+|_W/X_{n,(i),U}^+|_W\) is a torsor for an abelian scheme and \({\mathcal {L}}_U^+(a)\) is relatively ample for this morphism.) We will denote by \((\pi _{A^+/X^+,*} {\mathcal {L}})^+_U(a)\) the image \(\pi ^+_* {\mathcal {L}}_U^+(a)\). Suppose further that \({\mathcal {F}}\) is a locally free sheaf on \(X^+_{n,(i),U} \times {{\text {Spec}}\,}R_0\) with \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-action. If \(a^\natural \) is a section of \(X^*(S^{+}_{n,(i),U})^{>0,\natural }\) we will define
as follows: Over a point \(y^\natural \) of \(Y_{n,(i),U}^\natural \) we take the sheaf
over \(X^\natural _{n,(i),U,y^\natural } \times {{\text {Spec}}\,}R_0\), where y runs over points of \(Y_{n,(i),U}^+ \) above \(y^\natural \) and a runs over sections of \(X^*(S_{n,(i),U}^+)_y\) above \(a^\natural \). It is a sheaf with an action of \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\).
Lemma 4.2
Keep the notation and assumptions of the previous paragraph.
-
(1)
$$\left( \pi _{A^+/X^\natural ,*}{\mathcal {L}}\otimes {\mathcal {F}}\right) _{U}^+(a^\natural ) \cong {{\text {Ind}}}_{\{1\}}^{L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})} \left( \left( \pi _{A^+/X^\natural ,*}{\mathcal {L}}\otimes {\mathcal {F}}\right) _{U}^+(a^\natural )^{L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})} \right) $$
as a sheaf on \(X^{\natural }_{n,(i),U} \times {{\text {Spec}}\,}R_0\) with \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-action.
-
(2)
If
$$\pi : A^{+}_{n,(i),U} \times {{\text {Spec}}\,}R_0 \longrightarrow X^{\natural }_{n,(i),U} \times {{\text {Spec}}\,}R_0$$then
$$\begin{array}{l} R^i\pi _* \prod _{a \in X^*\left( S^{+}_{n,(i),U}\right) ^{>0}} \left( {\mathcal {L}}^+_{U}(a)\otimes {\mathcal {F}}\right) \\ \quad \cong \left\{ \begin{array}{ll} \prod _{a^\natural \in X^*(S^{+}_{n,(i),U})^{>0,\natural }} \left( \pi _{A^+/X^\natural ,*}{\mathcal {L}}\otimes {\mathcal {F}}\right) _{U}^+(a^\natural )&{} \quad \mathrm{if}\,\,\, i=0 \\ (0) &{} \quad \mathrm{otherwise}. \end{array} \right. \end{array}$$
Proof
For the first part note that if y in \(Y^+_{n,(i),U}\) and if \(a \in X^*(S^+_{n,(i),U})^{>0}_y\) then the stabilizer of a in \(\{ \gamma \in L_{n,(i)}({\mathbb {Q}}): \gamma y=y\}\) is finite, and that if U is neat then it is trivial. The second part follows from the observations of the previous paragraph together with proposition 0.13.3.1 of [23]. \(\square \)
Similarly if a is a section of \(X^*({\mathcal {S}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1)})\) over \(W \subset {\mathcal {Y}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1)}\) then we can associate to it a line bundle
over \({\mathcal {A}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}|_W\). There are natural isomorphisms
Suppose that \(R_0\) is a noetherian \({\mathbb {Z}}_{(p)}\)-algebra. Suppose that \(U^p\) is a neat open compact subgroup of \(P^+_{n,(i)}({\mathbb {A}}^{\infty ,p})\) and that \(0 \le N_1 \le N_2\). If a is a section in \(X^*({\mathcal {S}}^{+}_{n,(i),U^p(N_1)})^{>0}(W)\) then \({\mathcal {L}}_{U^p(N_1,N_2)}^+(a)\) is relatively ample for \({\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}|_W\) over \({\mathcal {X}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}|_W\). If \(\pi ^+\) denotes the map
then we see that
for \(i>0\). (Again because \({\mathcal {A}}_{n,(i),U^p(N_1,N_2)}^{{{\text {ord}}},+}|_W/{\mathcal {X}}_{n,(i),U^p(N_1,N_2)}^{{{\text {ord}}},+}|_W\) is a torsor for an abelian scheme and \({\mathcal {L}}_{U^p(N_1,N_2)}^+(a)\) is relatively ample for this morphism.) We will denote by \((\pi _{{\mathcal {A}}^{{{\text {ord}}},+}/{\mathcal {X}}^{{{\text {ord}}},+},*} {\mathcal {L}})^+_{U^p(N_1,N_2)}(a)\) the image \(\pi ^+_* {\mathcal {L}}^+(a)\). Suppose further that \({\mathcal {F}}\) is a locally free sheaf on \({\mathcal {X}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \times {{\text {Spec}}\,}R_0\) with \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\)-action. If \(a^\natural \) is a section of \(X^*({\mathcal {S}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1)})^{>0,\natural }\) we define a sheaf
as follows: Over a point \(y^\natural \) of \({\mathcal {Y}}_{n,(i),U^p(N_1,N_2)}^{{{\text {ord}}},\natural }\) we take the sheaf
over \({\mathcal {X}}^{{{\text {ord}}},\natural }_{n,(i),U^p(N_1,N_2),y^\natural } \times {{\text {Spec}}\,}R_0\), where y runs over points of \({\mathcal {Y}}^{{{\text {ord}}},\natural }_{n,(i),U^p(N_1,N_2)}\) above \(y^\natural \) and a runs over sections of \(X^*({\mathcal {S}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1)})_y\) above \(a^\natural \). It is a sheaf with an action of \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\). As above we have the following lemma.
Lemma 4.3
Keep the notation and assumptions of the previous paragraph.
-
(1)
$$\begin{array}{l} \left( \pi _{{\mathcal {A}}^{{{\text {ord}}},+}/{\mathcal {X}}^{{{\text {ord}}},\natural },*}{\mathcal {L}}\otimes {\mathcal {F}}\right) _{U^p(N_1,N_2)}^+(a^\natural ) \\ \quad \cong {{\text {Ind}}}_{\{1\}}^{L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})} \left( \left( \pi _{{\mathcal {A}}^{{{\text {ord}}},+}/{\mathcal {X}}^{{{\text {ord}}},+},*}{\mathcal {L}}\otimes {\mathcal {F}}\right) _{U^p(N_1,N_2)}^+(a^\natural )^{L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})} \right) \end{array}$$
as a sheaf on \({\mathcal {X}}^{{{\text {ord}}},\natural }_{n,(i),U^p(N_1,N_2)} \times {{\text {Spec}}\,}R_0\) with \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\)-action.
-
(2)
If
$$\begin{aligned} \pi : {\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \times {{\text {Spec}}\,}R_0\longrightarrow {\mathcal {X}}^{{{\text {ord}}},\natural }_{n,(i),U^p(N_1,N_2)} \times {{\text {Spec}}\,}R_0 \end{aligned}$$then
$$R^i\pi _* \prod _{a \in X^*\left( {\mathcal {S}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1)}\right) ^{>0}} \left( {\mathcal {L}}^+_{U^p(N_1,N_2)}(a) \otimes {\mathcal {F}}\right) $$is isomorphic to
$$\prod _{a^\natural \in X^*\left( {\mathcal {S}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1)}\right) ^{>0,\natural }} \left( \pi _{{\mathcal {A}}^{{{\text {ord}}},+}/{\mathcal {X}}^{{{\text {ord}}},\natural },*}{\mathcal {L}}\otimes {\mathcal {F}}\right) _{U^p(N_1,N_2)}^+(a^\natural )$$if \(i=0\), and otherwise is (0).
5.4 Partial compactifications
We will now turn to the partial compactification of the generalized Shimura varieties, \(T^{(m)}_{n,(i),U}\), we discussed in the last section. These will serve as models for the full compactification of the \(A^{(m)}_{n,U}\), which near the boundary can be formally modelled on the partial compactifications of the \(T^{(m)}_{n,(i),U}\).
Suppose that U (resp. \(U^p\)) is a neat open compact subgroup of \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^\infty )\) (resp. \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^{\infty ,p})\)) and that N is a non-negative integer. By an admissible cone decomposition \(\varSigma _0\) for \(X_*(S^{(m),+}_{n,(i),U})_{\mathbb {R}}^{\succ 0}\) (resp. \(X_*({\mathcal {S}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N)})_{\mathbb {R}}^{\succ 0}\)) we shall mean a partial fan \(\varSigma _0\) in \(X_*(S^{(m),+}_{n,(i),U})_{\mathbb {R}}\) (resp. \(X_*({\mathcal {S}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N)})_{\mathbb {R}}\)) such that
-
\(|\varSigma _0|=X_*(S^{(m),+}_{n,(i),U})_{\mathbb {R}}^{\succ 0}\) (resp. \(X_*({\mathcal {S}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N)})_{\mathbb {R}}^{\succ 0}\));
-
\(|\varSigma _0|^0=X_*(S^{(m),+}_{n,(i),U})_{\mathbb {R}}^{> 0}\) (resp. \(X_*({\mathcal {S}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N)})_{\mathbb {R}}^{> 0}\));
-
\(\varSigma _0\) is invariant under the left action of \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}})\) (resp. \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\));
-
\(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}}) \backslash \varSigma _0\) (resp. \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)}) \backslash \varSigma _0\)) is a finite set;
-
if \(\sigma \in \varSigma _0\) and \(1 \ne \gamma \in L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}})\) (resp. \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\)) then
$$\begin{aligned} \sigma \cap \gamma \sigma \not \in \varSigma _0. \end{aligned}$$
(Many authors would not include the last condition in the definition of an ‘admissible cone decomposition’.) In concrete terms \(\varSigma _0\) consists of a partial fan \(\varSigma _{g,0}\) in \(Z(N_{n,(i)}^{(m)})({\mathbb {R}})\) for each \(g \in L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^\infty )\) (resp. \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)), such that
-
\(\varSigma _{\gamma g u,0}=\gamma \varSigma _{g,0}\) for all \(\gamma \in L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}})\) (resp. \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\)) and \(u \in U\) (resp. \(U^p(N)\));
-
\(|\varSigma _{g,0}|={\mathfrak {C}}^{(m),\succ 0}(V_{n,(i)})\) and \(|\varSigma _{g,0}|^0={\mathfrak {C}}^{(m),>0}(V_{n,(i)})\) for each g;
-
\((L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}}) \cap gUg^{-1})\backslash \varSigma _{g,0}\) (resp. \((L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)}) \cap gU^p(N)g^{-1})\backslash \varSigma _g\)) is finite for all g;
-
for each g and each \(\sigma \in \varSigma _{g,0}\), if \(1 \ne \gamma \in (L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}}) \cap gUg^{-1})\) (resp. \((L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)}) \cap gU^p(N)g^{-1})\)) then
$$\begin{aligned} \sigma \cap \gamma \sigma \not \in \varSigma _{g,0}. \end{aligned}$$
Note that an admissible cone decomposition \(\varSigma _0\) for \(X_*(S^{(m),+}_{n,(i),U^p(N)})_{\mathbb {R}}^{\succ 0}\) induces (by restriction) one, which we will denote \(\varSigma _0^{{\text {ord}}}\), for \(X_*({\mathcal {S}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N)})_{\mathbb {R}}^{\succ 0}\). This sets up a bijection between admissible cone decompositions for \(X_*(S^{(m),+}_{n,(i),U^p(N)})_{\mathbb {R}}^{\succ 0}\) and for \(X_*({\mathcal {S}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N)})_{\mathbb {R}}^{\succ 0}\).
Lemma 4.4
Suppose that U (resp. \(U^p\)) is a neat open compact subgroup of the group \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^\infty )\) (resp. \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^\infty )\)) and that N is a non-negative integer. Suppose also that \(\varSigma _0\) is an admissible cone decomposition for \(X_*(S^{(m),+}_{n,(i),U})_{\mathbb {R}}^{\succ 0}\) (resp. for \(X_*({\mathcal {S}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N)})_{\mathbb {R}}^{\succ 0}\)). Also suppose that \(\tau \subset |\varSigma _0|\) is a rational polyhedral cone. Then the set
(resp.
is finite.
Proof
We treat the case of \(X_*(S^{(m),+}_{n,(i),U})_{\mathbb {R}}^{\succ 0}\), the other being exactly similar. Suppose that \(\tau \) has support \(y=hU\) and set \(\varGamma =L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}}) \cap hUh^{-1}\) a discrete subgroup of \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}})\). We certainly have
That this set is finite follows from theorem II.4.6 and the remark (ii) at the end of section II.4.1 of [3]. \(\square \)
Corollary 4.5
If \(\varSigma _0\) is an admissible cone decomposition for \(X_*(S^{(m),+}_{n,(i),U})_{\mathbb {R}}^{\succ 0}\) or for \(X_*({\mathcal {S}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N)})_{\mathbb {R}}^{\succ 0}\), then \(\varSigma _0\) is locally finite.
Proof
Let \(\tau \subset |\varSigma _0|\) be a rational polyhedral cone. Let \(\sigma _1,\ldots ,\sigma _r\) be representatives for \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^\infty )\backslash \varSigma _0\) (resp. \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)}) \backslash \varSigma _0\)); and suppose they are chosen with the same support as \(\tau \) whenever possible. Let \(\tau '\) be the rational polyhedral cone spanned by \(\tau \) and those \(\sigma _i\) with the same support as \(\tau \). If \(\gamma \in L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^\infty )\) (resp. \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\)) and
then
and so by the previous lemma \(\gamma \) lies in a finite set. The corollary follows. \(\square \)
If \(g \in P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\), if \(U' \supset g^{-1} U g\) are neat open compact subgroups of the group \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) and if \(\varSigma '_0\) is a \(U'\)-admissible cone decomposition for \(X_*(S_{n,(i),U'}^{(m),+})_{\mathbb {R}}^{\succ 0}\), then \(\varSigma '_0 g^{-1}\) is a U-admissible cone decomposition for \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\succ 0}\). We will call a U-admissible cone decomposition \(\varSigma _0\) for \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\succ 0}\) compatible with \(\varSigma _0'\) with respect to g if \(\varSigma _0\) refines \(\varSigma '_0 g^{-1}\). Similarly if \(g \in P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\), if \((U^p)' (N')\supset (g^{-1} U^p g)(N)\) and if \(\varSigma '_0\) is a \((U^p)'(N')\)-admissible cone decomposition for \(X_*({\mathcal {S}}_{n,(i),(U^p)'(N')}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\), then \((\varSigma ' g^{-1},\varSigma '_0 g^{-1})\) is an admissible cone decomposition for \(X_*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\). We will call a \(U^p(N)\)-admissible cone decomposition \(\varSigma _0\) for \(X_*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\) compatible with \(\varSigma _0'\) with respect to g if \(\varSigma _0\) refines \(\varSigma '_0 g^{-1}\).
If \(U'\) is a neat open compact subgroup of \(P_{n,(i)}^+({\mathbb {A}}^\infty )\) which contains the image of U, we will call an admissible cone decomposition \(\varSigma _0\) of \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\succ 0}\) and an admissible cone decomposition \(\varDelta _0\) of \(X_*(S_{n,(i),U'}^{+})_{\mathbb {R}}^{\succ 0}\) compatible if, under the natural map
the image of each \(\sigma \in \varSigma _0\) is contained in some element of \(\varDelta _0\). Similarly if \((U^p)'\) is a neat open compact subgroup of \(P_{n,(i)}^+({\mathbb {A}}^{\infty ,p})\) which contains the image of \(U^p\) and if \(N' \ge N\), we will call an admissible cone decomposition \(\varSigma _0\) of \(X_*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\) and an admissible cone decomposition \(\varDelta _0\) of \(X_*({\mathcal {S}}_{n,(i),(U^p)'(N')}^{{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\) compatible if, under the natural map
the image of each \(\sigma \in \varSigma _0\) is contained in some element of \(\varDelta _0\).
If \(\varSigma _0\) is a smooth admissible cone decomposition for \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\succ 0}\) (resp. for \(X_*({\mathcal {S}}_{n,(i),U^p(N_1,N_2))}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\)), then the log smooth, log scheme
(resp. the log smooth, log scheme
has a left action of \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}})\) (resp. \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\)) extending that on \(T^{(m),+}_{n,(i),U}\) (resp. \({\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\)). (Recall the definition of \(\widetilde{\varSigma }_0\) from Sect. 2.5.) If \(g \in P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) (resp. \(g \in P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)) and if \(\varSigma _0\) is compatible with \(\varSigma '_0\) with respect to g then the map
(resp.
uniquely extends to an \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-equivariant (resp. \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\)-equivariant) log etale map
(resp.
This makes \(\{ (T^{(m),+}_{n,(i),U,\widetilde{\varSigma }_0},{\mathcal {M}}_{\widetilde{\varSigma }_0}) \}\) (resp. \(\{ ( {\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2),\widetilde{\varSigma }_0},{\mathcal {M}}_{\widetilde{\varSigma }_0})\}\)) a system of log schemes with \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\)-action (resp. \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-action). There are equivariant embeddings
We have
If \(U'\) (resp. \((U')^p\)) is a neat open compact subgroup of the group \(P_{n,(i)}^+({\mathbb {A}}^\infty )\) (resp. \(P_{n,(i)}^+({\mathbb {A}}^{\infty ,p})\)) which contains the image of U (resp. \(U^p\)), if \(\varDelta _0\) is a smooth admissible cone decomposition of \(X_*(S_{n,(i),U'}^{+})_{\mathbb {R}}^{\succ 0}\) (resp. \(X_*({\mathcal {S}}_{n,(i),(U')^p(N_1)}^{{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\)) and if \(\varSigma _0\) is a compatible smooth admissible cone decomposition of \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\succ 0}\) (resp. \(X_*({\mathcal {S}}_{n,(i),U^p(N_1)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\)), then the map
(resp.
extends to an \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-equivariant (resp. \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\)-equivariant) log smooth map
(resp.
This gives rise to a \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\)-equivariant (resp. \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-equivariant) map of systems of log schemes
(resp.
These maps are compatible with the embeddings
and
5.5 Completions
If \(\varSigma _0\) denotes a smooth admissible cone decomposition of \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\succ 0}\) (resp. of \(X_*({\mathcal {S}}_{n,(i),U^p(N_1,N_2))}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\)), then the associated log formal scheme \((T^{(m),+,\wedge }_{n,(i),U,\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge )\) (resp. \(({\mathcal {T}}^{(m),{{\text {ord}}},+,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge )\)) inherits a left action of \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}})\) (resp. \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\)). If \(g \in P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) (resp. \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)) and if \(\varSigma _0\) is compatible with \(\varSigma '_0\) with respect to g, then there is an induced \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-equivariant (resp. \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\)-equivariant) map
(resp.
This makes \(\{ (T^{(m),+,\wedge }_{n,(i),U,\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge ) \}\) (resp. \(\{ ({\mathcal {T}}^{(m),{{\text {ord}}},+,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge ) \}\)) a system of log formal schemes with \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\)-action (resp. \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-action).
Similarly the schemes \(\partial _{\varSigma _0} T^{(m),+}_{n,(i),U}\) (resp. \(\partial _{\varSigma _0} {\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\)) inherit a left action of the group \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}})\) (resp. \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\)). If \(g \in P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) (resp. \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)) and if \(\varSigma _0\) is compatible with \(\varSigma '_0\) with respect to g, then there is an induced \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-equivariant (resp. \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\)-equivariant) map
(resp.
This makes \(\{ \partial _{\varSigma _0} T^{(m),+}_{n,(i),U} \}\) (resp. \(\{ \partial _{\varSigma _0} {\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \}\)) a system of log formal schemes with \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\)-action (resp. \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-action).
If \(U'\) (resp. \((U^p)'\)) is a neat open compact subgroup of the group \(P_{n,(i)}^+({\mathbb {A}}^\infty )\) (resp. \(P_{n,(i)}^+({\mathbb {A}}^{\infty ,p})\)) which contains the image of U (resp. \(U^p\)), if \(\varDelta _0\) is a smooth admissible cone decomposition of \(X_*(S_{n,(i),U'}^{+})_{\mathbb {R}}^{\succ 0}\) (resp. \(X_*({\mathcal {S}}_{n,(i),(U^p)'(N_1)}^{{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\)) and if \(\varSigma _0\) is a compatible smooth admissible cone decomposition of \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\succ 0}\) (resp. \(X_*({\mathcal {S}}_{n,(i),U^p(N_1)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\)), then there are induced maps
and
(resp.
and
which are \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-equivariant (resp. \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\)-equivariant). This gives rise to \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\)-equivariant (resp. \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-equivariant) maps of systems of log formal schemes
and of systems of schemes
(resp.
and
If \(\sigma \in \varSigma _0\) and if \(1 \ne \gamma \in L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}})\) (resp. \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\)) then \(\sigma \cap \gamma \sigma \not \in \varSigma _0\). Thus
and
(resp.
and
It follows we can form log formal schemes
(resp.
and
(resp.
We can also form schemes
(resp.
The quotient maps
and
(resp.
and
are Zariski locally isomorphisms. The log formal scheme \((T^{(m),\natural +,\wedge }_{n,(i),U,\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge )\) (resp. \(({\mathcal {T}}^{(m),{{\text {ord}}},\natural +,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge )\)) inherits an action of \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) (resp. \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\)).
If \(g \in P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) (resp. \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)) and if \(\varSigma _0\) is compatible with \(\varSigma '_0\) with respect to g then there are induced maps
(resp.
and
(resp.
and
(resp.
This makes the collections \(\{ (T^{(m),\natural ,\wedge }_{n,(i),U,\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge ) \}\) (resp. \(\{ ({\mathcal {T}}^{(m),{{\text {ord}}},\natural ,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge )\}\)) and \(\{ (T^{(m),\natural +,\wedge }_{n,(i),U,\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge ) \}\) (resp. \(\{ ({\mathcal {T}}^{(m),{{\text {ord}}},\natural +,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0},{\mathcal {M}}_{\varSigma _0}^\wedge ) \}\)) systems of log formal schemes with \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\)-action (resp. \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-action). It also makes the collections \(\{ \partial _{\varSigma _0} T^{(m),\natural }_{n,(i),U} \}\) (resp. \(\{ \partial _{\varSigma _0} {\mathcal {T}}^{(m),{{\text {ord}}},\natural }_{n,(i),U^p(N_1,N_2)} \}\)) systems of schemes with \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\)-action (resp. \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-action).
If \(U'\) (resp. \((U^p)'\)) is a neat open compact subgroup of the group \(P_{n,(i)}^+({\mathbb {A}}^\infty )\) (resp. \(P_{n,(i)}^+({\mathbb {A}}^{\infty ,p})\)) which contains the image of U (resp. \(U^p\)), if \(\varDelta _0\) is a smooth admissible cone decomposition of \(X_*(S_{n,(i),U'}^{+})_{\mathbb {R}}^{\succ 0}\) (resp. \(X_*({\mathcal {S}}_{n,(i),(U^p)'(N_1)}^{{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\)) and if \(\varSigma _0\) is a compatible smooth admissible cone decomposition of \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\succ 0}\) (resp. \(X_*({\mathcal {S}}_{n,(i),U^p(N_1)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\)), then there are induced maps
(resp.
and
(resp.
and
(resp.
These give rise to \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\)-equivariant (resp. \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-equivariant) maps of systems of log formal schemes
(resp.
and
(resp.
They also give rise to a \(P^{(m)}_{n,(i)}({\mathbb {A}}^\infty )\)-equivariant (resp. \(P^{(m)}_{n,(i)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-equivariant) map of systems of schemes
(resp.
We will write
It is independent of \(N_2\).
We also get a commutative diagram
(resp.
We will let \({\mathcal {I}}^{(m),\natural +,\wedge }_{\partial , n,(i),U,\varSigma _0}\) denote the formal completion of the ideal sheaf defining
We will let \({\mathcal {I}}^{(m),\natural +,\wedge }_{\partial , n,(i),U,\varSigma _0}\) denote its quotient by \({{\text {Hom}}}_F(F^m,F^i)\) and \({\mathcal {I}}^{(m),\natural ,\wedge }_{\partial , n,(i),U,\varSigma _0}\) denote its quotient by \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}})\). Similarly we will let \({\mathcal {I}}^{(m),{{\text {ord}}},+,\wedge }_{\partial , n,(i),U^p(N_1,N_2),\varSigma _0}\) denote the formal completion of the ideal sheaf defining
We will let \({\mathcal {I}}^{(m),{{\text {ord}}},\natural +,\wedge }_{\partial , n,(i),U^p(N_1,N_2),\varSigma _0}\) denote its quotient by \({{\text {Hom}}}_{{\mathcal {O}}_{F,(p)}}({\mathcal {O}}_{F,(p)}^m,{\mathcal {O}}_{F,(p)}^i)\) and \({\mathcal {I}}^{(m),{{\text {ord}}},\natural ,\wedge }_{\partial , n,(i),U^p(N_1,N_2),\varSigma _0}\) denote its quotient by \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\).
There are \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) and \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) equivariant maps
if \(\varSigma _0^{{\text {ord}}}\) and \(\varSigma _0\) correspond under the bijection of Sect. 4.4. These embeddings are compatible with the maps
and
Moreover they are also compatible with the log structures and with the sheaves \({\mathcal {I}}^{(m),{{\text {ord}}},+,\wedge }_{\partial , n,(i),U^p(N_1,N_2),\varSigma _0^{{\text {ord}}}}\) and \({\mathcal {I}}^{(m),+,\wedge }_{\partial , n,(i),U^p(N_1,N_2),\varSigma _0}\). They induce isomorphisms
Lemma 4.6
Suppose that \(R_0\) is an irreducible noetherian \({\mathbb {Q}}\)-algebra (resp. \({\mathbb {Z}}_{(p)}\)-algebra) with the discrete topology. Suppose also that \(U \supset U'\) (resp. \(U^p \supset (U^p)'\)) are neat open compact subgroups of \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) (resp. \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty ,p})\)), that \(N_2' \ge N_1' \ge 0\) and \(N_2 \ge N_1 \ge 0\) are integers with \(N_2' \ge N_2\) and \(N_1' \ge N_1\) and that \(\varSigma _0\) and \(\varSigma '_0\) are compatible smooth admissible cone decompositions for \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\succ 0}\) and \(X_*(S_{n,(i),U'}^{(m),+})_{\mathbb {R}}^{\succ 0}\) (resp. for \(X_*({\mathcal {S}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\) and \(X_*({\mathcal {S}}_{n,(i),(U^p)'(N_1',N_2')}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\)). Let \(\pi _{(U',\varSigma _0'),(U,\varSigma _0)}\) (resp. \(\pi _{((U^p)'(N_1',N_2'),\varSigma _0'),(U^p(N_1,N_2),\varSigma _0)}\)) denote the map
(resp.
-
(1)
If \(i>0\) then
$$R^i\pi _{(U',\varSigma _0'),(U,\varSigma _0),*} \left( {\mathcal {I}}^{(m),\natural ,\wedge }_{\partial , n,(i),U',\varSigma _0'} {\widehat{\otimes }}R_0\right) = R^i\pi _{(U',\varSigma _0'),(U,\varSigma _0),*} {\mathcal {O}}_{ T^{(m),\natural ,\wedge }_{n,(i),U',\varSigma _0'} \times {{\text {Spf}}\,}R_0}=(0)$$(resp.
$$R^i\pi _{((U^p)'(N_1',N_2'),\varSigma _0'),(U^p(N_1,N_2),\varSigma _0),*} \left( {\mathcal {I}}^{(m),{{\text {ord}}},\natural ,\wedge }_{\partial , n,(i),(U^p)'(N_1',N_2') ,\varSigma _0'} {\widehat{\otimes }}R_0\right) =(0)$$and
$$\begin{aligned} R^i\pi _{((U^p)'(N_1',N_2'),\varSigma _0'),(U^p(N_1,N_2),\varSigma _0),*} {\mathcal {O}}_{ {\mathcal {T}}^{(m),{{\text {ord}}},\natural ,\wedge }_{n,(i),(U^p)'(N_1',N_2'),\varSigma _0'} \times {{\text {Spf}}\,}R_0}=(0) \big ). \end{aligned}$$ -
(2)
Suppose further that \(U'\) (resp. \((U^p)'\)) is a normal subgroup of U (resp. \(U^p\)) and that \(\varSigma _0'\) is U-invariant (resp. \(U^p(N_1,N_2)\)-invariant). Then the natural maps
$${\mathcal {O}}_{T^{(m),\natural ,\wedge }_{n,(i),U,\varSigma _0} \times {{\text {Spf}}\,}R_0} \longrightarrow \left( \pi _{(U',\varSigma '_0),(U,\varSigma _0),*} {\mathcal {O}}_{T^{(m),\natural ,\wedge }_{n,(i),U',\varSigma _0'} \times {{\text {Spf}}\,}R_0}\right) ^{U}$$(resp.
$$\begin{array}{l} {\mathcal {O}}_{{\mathcal {T}}^{(m),{{\text {ord}}},\natural ,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0} \times {{\text {Spf}}\,}R_0} \\ \quad \left. \longrightarrow \left( \pi _{(((U^p)'(N_2,N_2),\varSigma _0'),(U^p(N_1,N_2),\varSigma _0),*} {\mathcal {O}}_{{\mathcal {T}}^{(m),{{\text {ord}}},\natural ,\wedge }_{n,(i),(U^p)'(N_2,N_2),\varSigma _0'} \times {{\text {Spf}}\,}R_0}\right) ^{U^p(N_1,N_2)} \right) \end{array}$$and
$${\mathcal {I}}^{(m),\natural ,\wedge }_{\partial ,n,(i),U,\varSigma _0}{\widehat{\otimes }}R_0 \longrightarrow \left( \pi _{(U',\varSigma _0'),(U,\varSigma _0),*} \left( {\mathcal {I}}^{(m),\natural ,\wedge }_{\partial ,n,(i),U',\varSigma _0'} {\widehat{\otimes }}R_0\right) \right) ^{U}$$(resp.
$$\begin{array}{l} {\mathcal {I}}^{(m),{{\text {ord}}},\natural ,\wedge }_{\partial ,n,(i),U^p(N_1,N_2), \varSigma _0} {\widehat{\otimes }}R_0 \\ \left. \quad \longrightarrow \left( \pi _{((U^p)'(N_2,N_2),\varSigma _0'),(U^p(N_1,N_2),\varSigma _0),*} \left( {\mathcal {I}}^{(m),{{\text {ord}}},\natural ,\wedge }_{\partial ,n,(i),(U^p)'(N_2,N_2),\varSigma _0'}{\widehat{\otimes }}R_0\right) \right) ^{U^p(N_1,N_2)} \right) \end{array}$$are isomorphisms.
The same statements are true with \(\natural \) replaced by \(+\) or by \(\natural +\).
Proof
It suffices to treat the case of \(+\). We treat the case of \(T^{(m),+,\wedge }_{n,(i),U',\varSigma _0'} \times {{\text {Spf}}\,}R_0\), the case of \({\mathcal {T}}^{(m),{{\text {ord}}},+,\wedge }_{n,(i),(U^p)'(N_1,N_2),\varSigma _0'} \times {{\text {Spf}}\,}R_0\) being exactly similar.
Let \(U''\) denote the open compact subgroup of \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) generated by \(U'\) and \(U \cap Z(N_{n,(i)}^{(m)})({\mathbb {A}}^\infty )\). Then \(\varSigma _0\) is a \(U''\) admissible smooth cone decomposition of \(X_*(S^{(m),+}_{n,(i),U''})_{\mathbb {R}}^{\succ 0}\). Moreover
is finite etale, and if \(U'\) is normal in U then it is Galois with group \(U/U''\). Thus we may replace U by \(U''\) and reduce to the case that U and \(U'\) have the same projection to \((P^{(m),+}_{n,(i)}/Z (N_{n,(i)}^{(m)}))({\mathbb {A}}^\infty )\). In this case the result follows from Lemma 2.15. \(\square \)
Define \(\widetilde{\varOmega }_{n,(i),U,\varDelta _0}^\natural \) on \(T^{\natural ,\wedge }_{n,(i),U,\varDelta _0}\) as the quotient by \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) of the pull-back of \(\widetilde{\varOmega }_{n,(i),U}^+\) from \(A^+_{n,(i),U}\) to \(T^{+,\wedge }_{n,(i),U,\varDelta _0}\). Also define \(\widetilde{\varOmega }_{n,(i),U^p(N_1,N_2),\varDelta _0}^{{{\text {ord}}},\natural }\) on \({\mathcal {T}}^{{{\text {ord}}},\natural ,\wedge }_{n,(i),U^p(N_1,N_2),\varDelta _0}\) as the quotient by \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) of the pull-back of the sheaf \(\widetilde{\varOmega }_{n,(i),U^p(N_1,N_2)}^{{{\text {ord}}},+}\) from \({\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) to \({\mathcal {T}}^{{{\text {ord}}},+,\wedge }_{n,(i),U^p(N_1,N_2),\varDelta _0}\).
Suppose that \(R_0\) is a \({\mathbb {Q}}\)-algebra and that \(\rho \) is a representation of \(R_{n,(n),(i)}\) on a finite, locally free \(R_0\)-module \(W_\rho \). Then we define a locally free sheaf \({\mathcal {E}}_{(i),U,\varDelta _0,\rho }^\natural \) on \(T^{\natural ,\wedge }_{n,(i),U,\varDelta _0}\) as the quotient by \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) of the pull-back of \({\mathcal {E}}^+_{(i),U,\rho }\) from \(A^+_{n,(i),U}\) to \(T^{+,\wedge }_{n,(i),U,\varDelta _0}\). Then the system of sheaves \(\{ {\mathcal {E}}_{(i),U,\varDelta _0,\rho }^\natural \}\) over \(\{ T^{\natural ,\wedge }_{n,(i),U,\varDelta _0} \}\) has an action of \(P_{n,(i)}^+({\mathbb {A}}^\infty )\). If \(g \in P_{n,(i)}^+({\mathbb {A}}^\infty )\), then the natural map
is an isomorphism. The sheaves \({\mathcal {E}}_{(i),U,\varDelta _0,\rho }^\natural \) have \(P_{n,(i)}^+({\mathbb {A}}^\infty )\)-invariant filtrations by local direct summands whose graded pieces pull-backed to \(T^{+,\wedge }_{n,(i),U,\varDelta _0}\) are equivariantly isomorphic to the pull-backs of sheaves of the form \({\mathcal {E}}_{(i),U,\rho '}^+\) on \(X^+_{n,(i),U}\).
Similarly in the case of mixed characteristic suppose that \(R_0\) is a \({\mathbb {Z}}_{(p)}\)-algebra and that \(\rho \) is a representation of \(R_{n,(n),(i)}\) on a finite, locally free \(R_0\)-module \(W_\rho \). Then we define a locally free sheaf \({\mathcal {E}}_{(i),U^p(N_1,N_2),\varDelta _0,\rho }^{{{\text {ord}}},\natural }\) on \({\mathcal {T}}^{{{\text {ord}}},\natural ,\wedge }_{n,(i),U^p(N_1,N_2),\varDelta _0}\) as the quotient by \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) of the pull-back of \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho }\) from \({\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\) to \({\mathcal {T}}^{{{\text {ord}}},+,\wedge }_{n,(i),U^p(N_1,N_2),\varDelta _0}\). Then the collection \(\{ {\mathcal {E}}_{(i),U^p(N_1,N_2),\varDelta _0,\rho }^{{{\text {ord}}},\natural }\}\) is a system of sheaves over \(\{ {\mathcal {T}}^{{{\text {ord}}},\natural ,\wedge }_{n,(i),U^p(N_1,N_2),\varDelta _0} \}\) with an action of \(P_{n,(i)}^+({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\). If \(g \in P_{n,(i)}^+({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\), then the natural map
is an isomorphism. The sheaves \({\mathcal {E}}_{(i),U^p(N_1,N_2),\varDelta _0,\rho }^{{{\text {ord}}},\natural }\) have \(P_{n,(i)}^+({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-invariant filtrations by local direct summands whose graded pieces pull-backed to the formal scheme \({\mathcal {T}}^{{{\text {ord}}},+,\wedge }_{n,(i),U^p(N_1,N_2),\varDelta _0}\) are equivariantly isomorphic to the pull-backs of sheaves of the form \({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho '}\) on \({\mathcal {X}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\).
Corollary 4.7
Suppose that \(R_0\) is an irreducible noetherian \({\mathbb {Q}}\)-algebra (resp. \({\mathbb {Z}}_{(p)}\)-algebra) with the discrete topology. Let \(\rho \) be a representation of \(R_{n,(n),(i)}\) on a finite, locally free \(R_0\)-module \(W_\rho \). Suppose also that \(U \supset U'\) (resp. \(U^p \supset (U^p)'\)) are neat open compact subgroups of \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) (resp. \(P^{(m),+}_{n,(i)}({\mathbb {A}}^{\infty ,p})\)), that \(N_2' \ge N_1' \ge 0\) and \(N_2 \ge N_1 \ge 0\) are integers with \(N_2' \ge N_2\) and \(N_1' \ge N_1\) and that \(\varSigma _0\) and \(\varSigma '_0\) are compatible smooth admissible cone decompositions for \(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\succ 0}\) and \(X_*(S_{n,(i),U'}^{(m),+})_{\mathbb {R}}^{\succ 0}\) (resp. \(X_*({\mathcal {S}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\) and \(X_*({\mathcal {S}}_{n,(i),(U^p)'(N_1',N_2')}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\)). Let \(\pi _{(U',\varSigma _0'),(U,\varSigma _0)}\) (resp. \(\pi _{((U^p)'(N_1',N_2'),\varSigma _0'),(U^p(N_1,N_2),\varSigma _0)}\)) denote the map
(resp.
-
(1)
If \(i>0\) then
$$R^i\pi _{(U',\varSigma _0'),(U,\varSigma _0),*} \left( {\mathcal {I}}^{(m),\natural ,\wedge }_{\partial , n,(i),U',\varSigma _0'} {\widehat{\otimes }}{\mathcal {E}}_{(i),U',\varSigma _0',\rho }^\natural \right) = R^i\pi _{(U',\varSigma _0'),(U,\varSigma _0),*} {\mathcal {E}}_{(i),U',\varSigma _0',\rho }^\natural =(0)$$(resp.
$$R^i\pi _{((U^p)'(N_1',N_2'),\varSigma _0'),(U^p(N_1,N_2),\varSigma _0),*} \left( {\mathcal {I}}^{(m),{{\text {ord}}},\natural ,\wedge }_{\partial , n,(i),(U^p)'(N_1',N_2') ,\varSigma _0'} {\widehat{\otimes }}{\mathcal {E}}_{(i),(U^p)'(N_1',N_2'),\varSigma _0',\rho }^{{{\text {ord}}},\natural }\right) =(0)$$and
$$\begin{aligned} R^i\pi _{((U^p)'(N_1',N_2'),\varSigma _0'),(U^p(N_1,N_2),\varSigma _0),*} {\mathcal {E}}_{(i),(U^p)'(N_1',N_2'),\varSigma _0',\rho }^{{{\text {ord}}},\natural }=(0) \big ). \end{aligned}$$ -
(2)
Suppose further that \(U'\) (resp. \((U^p)'\)) is a normal subgroup of U (resp. \(U^p\)) and that \(\varSigma _0'\) is U-invariant (resp. \(U^p(N_1,N_2)\)-invariant). Then the natural maps
$${\mathcal {E}}_{(i),U,\varSigma _0,\rho }^\natural \longrightarrow \left( \pi _{(U',\varSigma _0'),(U,\varSigma _0),*} {\mathcal {E}}_{(i),U',\varSigma _0',\rho }^\natural \right) ^{U}$$(resp.
$$\begin{array}{l} {\mathcal {E}}_{(i),U^p(N_1,N_2),\varSigma _0,\rho }^{{{\text {ord}}},\natural } \\ \quad \longrightarrow \left. \left( \pi _{(((U^p)'(N_2,N_2),\varSigma _0'),(U^p(N_1,N_2),\varSigma _0),*} {\mathcal {E}}_{(i),(U^p)'(N_2,N_2),\varSigma _0',\rho }^{{{\text {ord}}},\natural }\right) ^{U^p(N_1,N_2)} \right) \end{array}$$and
$${\mathcal {I}}^{(m),\natural ,\wedge }_{\partial ,n,(i),U,\varSigma _0}{\widehat{\otimes }}{\mathcal {E}}_{(i),U,\varSigma _0,\rho }^\natural \longrightarrow \left( \pi _{(U',\varSigma _0'),(U,\varSigma _0),*} ({\mathcal {I}}^{(m),\natural ,\wedge }_{\partial ,n,(i),U',\varSigma _0'} {\widehat{\otimes }}{\mathcal {E}}_{(i),U',\varSigma '_0,\rho }^\natural ) \right) ^{U}$$(resp.
$$\begin{array}{l} {\mathcal {I}}^{(m),{{\text {ord}}},\natural ,\wedge }_{\partial ,n,(i),U^p(N_1,N_2), \varSigma _0} {\widehat{\otimes }}{\mathcal {E}}_{(i),U^p(N_1,N_2),\varSigma _0,\rho }^{{{\text {ord}}},\natural } \longrightarrow \\ \quad \left. \left( \pi _{((U^p)'(N_2,N_2),\varSigma '),(U^p(N_1,N_2),\varSigma ),*} \left( {\mathcal {I}}^{(m),{{\text {ord}}},\natural ,\wedge }_{\partial ,n,(i),(U^p)'(N_2,N_2),\varSigma _0'}{\widehat{\otimes }}{\mathcal {E}}_{(i),(U^p)'(N_2,N_2),\varSigma _0',\rho }^{{{\text {ord}}},\natural }\right) \right) ^{U^p(N_1,N_2)} \right) \end{array}$$are isomorphisms.
Lemma 4.8
Suppose that U is a neat open compact subgroup of \(P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )\) and let \(U'\) denote the image of U in \(P_{n,(i)}^+({\mathbb {A}}^\infty )\). Let \(\varDelta _0\) be a smooth admissible cone decomposition for \(X_*(S^+_{n,(i),U'})\) and let \(\varSigma _0\) be a compatible smooth admissible cone decomposition for \(X_*(S^{(m),+}_{n,(i),U})\). Let \(\pi ^+=\pi _{(U,\varSigma _0),(U',\varDelta _0)}^+\) denote the map
and let \(\pi ^\natural =\pi _{(U,\varSigma _0),(U',\varDelta _0)}^\natural \) denote the map
-
(1)
The maps \(\pi _{(U,\varSigma _0),(U',\varDelta _0)}^+\) and \(\pi _{(U,\varSigma _0),(U',\varDelta _0)}^\natural \) are proper.
-
(2)
The natural maps
$$\begin{aligned} {\mathcal {O}}_{T^{+,\wedge }_{n,(i),U',\varDelta _0}} \longrightarrow \pi _{(U,\varSigma _0),(U',\varDelta _0),*}^+ {\mathcal {O}}_{T^{(m),\natural +,\wedge }_{n,(i),U,\varSigma _0}} \end{aligned}$$and
$$\begin{aligned} {\mathcal {I}}^{+,\wedge }_{\partial ,n,(i),U',\varDelta _0} \longrightarrow \pi _{(U,\varSigma _0),(U',\varDelta _0),*}^+ {\mathcal {I}}^{(m),\natural +,\wedge }_{\partial ,n,(i),U,\varSigma _0} \end{aligned}$$and
$$\begin{aligned} {\mathcal {O}}_{T^{\natural ,\wedge }_{n,(i),U',\varDelta _0}} \longrightarrow \pi _{(U,\varSigma _0),(U',\varDelta _0),*}^\natural {\mathcal {O}}_{T^{(m),\natural ,\wedge }_{n,(i),U,\varSigma _0}} \end{aligned}$$and
$$\begin{aligned} {\mathcal {I}}^{\natural ,\wedge }_{\partial ,n,(i),U',\varDelta _0} \longrightarrow \pi _{(U,\varSigma _0),(U',\varDelta _0),*}^\natural {\mathcal {I}}^{(m),\natural ,\wedge }_{\partial ,n,(i),U,\varSigma _0} \end{aligned}$$are isomorphisms.
-
(3)
The natural maps
$$\begin{aligned} {\mathcal {I}}^{+,\wedge }_{\partial ,n,(i),U',\varDelta _0} \otimes R^j\pi _{(U,\varSigma _0),(U',\varDelta _0),*}^+ {\mathcal {O}}_{T^{(m),\natural +,\wedge }_{n,(i),U,\varSigma _0}} \longrightarrow R^j\pi _{(U,\varSigma _0),(U',\varDelta _0),*}^+ {\mathcal {I}}^{(m),\natural +,\wedge }_{\partial ,n,(i),U,\varSigma _0} \end{aligned}$$and
$$\begin{aligned} {\mathcal {I}}^{\natural ,\wedge }_{\partial ,n,(i),U',\varDelta _0} \otimes R^j\pi _{(U,\varSigma _0),(U',\varDelta _0),*}^\natural {\mathcal {O}}_{T^{(m),\natural ,\wedge }_{n,(i),U,\varSigma _0}} \longrightarrow R^j\pi _{(U,\varSigma _0),(U',\varDelta _0),*}^\natural {\mathcal {I}}^{(m),\natural ,\wedge }_{\partial ,n,(i),U,\varSigma _0} \end{aligned}$$are isomorphisms.
Proof
It suffices to treat the \(+\) case.
The first part follows from Lemma 2.19. We deduce that all the sheaves mentioned in the remaining parts are coherent.
Thus, by theorem 4.1.5 of [23] (‘the theorem on formal functions’), it suffices to prove the remaining assertions after completing at a point of \(T^{+,\wedge }_{n,(i),U',\varDelta _0}\). The set points where the assertions are true after completing at that point is open. (Again because the sheaves involved are all coherent.) This open set is \(S^{+}_{n,(i),U'}\)-invariant. (The sheaves in question do not all have \(S^{+}_{n,(i),U'}\)-actions. However locally on \(T^{+,\wedge }_{n,(i),U',\varDelta _0}\) they do.) Thus it will do to prove the lemma after completion at \(\partial _\sigma T^{+}_{n,(i),U',\widetilde{\varDelta }_0}\), for \(\sigma \in \varDelta _0\) maximal. We will add a subscript \(\sigma \) to denote completion along \(\partial _\sigma T^{+}_{n,(i),U',\widetilde{\varDelta }_0}\).
We write \({\widetilde{\pi }} \) for the map
and factor \({\widetilde{\pi }} = \pi _2 \circ \pi _1\), where
and
Also write \(\pi _3\) for the other projection
We will first show that
and
As \(T^{+,\wedge }_{n,(i),U',\varDelta _0,\sigma } \times _{A^+_{n,(i),U'}} A^{(m),+}_{n,(i),U}\) has the same underlying topological space as \(A^{(m),+}_{n,(i),U}\), i.e. \(\pi _{3,\sigma }\) is a homeomorphism on the underlying topological space, it suffices to show that
and
This would follow from Lemma 2.22 as long as we can show that, for all \(y \in Y^{(m),+}_{n,(i),U}\) with image \(y'\) in \(Y^{+}_{n,(i),U'}\), we have \(|\varSigma _0|^\vee (y)= |\varDelta _0|^\vee (y')\) and \(|\varSigma _0|^{\vee ,0}(y)=|\varDelta _0|^{\vee ,0} (y')\). Concretely these required equalities are
and
If \(\chi \) lies in one of the right hand sides then \(\chi (z,f) \ge 0\) for all \(z \in {\mathfrak {C}}^{ > 0}(V_{n,(i)})\) and all \(f \in {{\text {Hom}}}(F^m, V_{n,(i)}) \otimes _{\mathbb {Q}}{\mathbb {R}}\). Taking the limit as \(z \rightarrow 0\) we see that \(\chi \ge 0\) on the vector space \({{\text {Hom}}}(F^m, V_{n,(i)}) \otimes _{\mathbb {Q}}{\mathbb {R}}\) and so \(\chi =0\) on this space. Thus the right hand sides are the set of \(\chi \in {\mathfrak {C}}(V_{n,(i)})^\vee \) such that \(\chi \ge 0\) (resp. \(\chi >0\)) on the images of
(resp.
But these images are \({\mathfrak {C}}^{ \succ 0}(V_{n,(i)})\) (resp. \({\mathfrak {C}}^{\succ 0}(V_{n,(i)})-\{0\}\)) and so the required equalities hold.
We deduce that
and
As \(T^{(m),\natural +,\wedge }_{n,(i),U,\varSigma _0,\sigma }\) is the quotient of \(T^{(m),+,\wedge }_{n,(i),U,\varSigma _0,\sigma }\) by \({{\text {Hom}}}_F(F^m,F^i)\), we obtain spectral sequences
and
These can also be written
and
The lemma follows (as \({\mathcal {I}}^{+,\wedge }_{\partial ,n,(i),U',\varDelta _0,\sigma }\) is flat over \({\mathcal {O}}_{T^{+,\wedge }_{n,(i),U',\varDelta _0,\sigma }}\)). \(\square \)
The following lemma is equation (1.3.2.86) in lemma 1.3.2.79 of [44].
Lemma 4.9
Suppose that U is a neat open compact subgroup of \(P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )\) and let \(U'\) denote the image of U in \(P_{n,(i)}^+({\mathbb {A}}^\infty )\). Let \(\varDelta _0\) be a smooth admissible cone decomposition for \(X_*(S^+_{n,(i),U'})\) and let \(\varSigma _0\) be a compatible smooth admissible cone decomposition for \(X_*(S^{(m),+}_{n,(i),U})\). There are canonical equivariant isomorphisms
We deduce the following lemmas.
Lemma 4.10
Suppose that U is a neat open compact subgroup of \(P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )\) and let \(U'\) denote the image of U in \(P_{n,(i)}^+({\mathbb {A}}^\infty )\). Let \(\varDelta _0\) be a smooth admissible cone decomposition for \(X_*(S^+_{n,(i),U'})\) and let \(\varSigma _0\) be a compatible smooth admissible cone decomposition for \(X_*(S^{(m),+}_{n,(i),U})\). Let \(\pi ^+=\pi _{(U,\varSigma _0),(U',\varDelta _0)}^+\) denote the map
and let \(\pi ^\natural =\pi _{(U,\varSigma _0),(U',\varDelta _0)}^\natural \) denote the map
-
(1)
\(\pi ^\natural _*\varOmega ^1_{T^{(m),\natural ,\wedge }_{n,(i),U,\varSigma _0}/T^{\natural ,\wedge }_{n,(i),U',\varDelta _0}} (\log \infty ) \cong {{\text {Hom}}}_F(F^m,\widetilde{\varOmega }_{n,(i),U'}^\natural ) \) is locally free of finite rank.
-
(2)
The natural map
$$\begin{array}{l} \pi _{(U,\varSigma _0),(U',\varDelta _0)}^{\natural ,*}\pi _{(U,\varSigma _0),(U',\varDelta _0),*}^\natural \varOmega ^1_{T^{(m),\natural ,\wedge }_{n,(i),U,\varSigma _0}/T^{\natural ,\wedge }_{n,(i),U',\varDelta _0}} (\log \infty ) \\ \quad \longrightarrow \varOmega ^1_{T^{(m),\natural ,\wedge }_{n,(i),U,\varSigma _0}/T^{\natural ,\wedge }_{n,(i),U',\varDelta _0}} (\log \infty ) \end{array}$$is an isomorphism.
-
(3)
The natural maps
$$\begin{array}{l} \left( R^{j_1}\pi ^\natural _* {\mathcal {O}}_{T^{(m),\natural ,\wedge }_{n,(i),U,\varSigma _0}}\right) \otimes \left( \wedge ^{j_2} \pi ^\natural _* \varOmega ^1_{T^{(m),\natural ,\wedge }_{n,(i),U,\varSigma _0}/T^{\natural ,\wedge }_{n,(i),U',\varDelta _0}} (\log \infty )\right) \\ \quad \longrightarrow R^{j_1}\pi ^\natural _* \varOmega ^{j_2}_{T^{(m),\natural ,\wedge }_{n,(i),U,\varSigma _0}/T^{\natural ,\wedge }_{n,(i),U',\varDelta _0}} (\log \infty ) \end{array}$$and
$$\begin{array}{l} \left( R^{j_1}\pi ^\natural _* {\mathcal {O}}_{T^{(m),\natural ,\wedge }_{n,(i),U,\varSigma _0}}\right) \otimes \left( \wedge ^{j_2} \pi ^\natural _* \varOmega ^1_{T^{(m),\natural ,\wedge }_{n,(i),U,\varSigma _0}/T^{\natural ,\wedge }_{n,(i),U',\varDelta _0}} (\log \infty )\right) \otimes {\mathcal {I}}^{\natural ,\wedge }_{\partial ,n,(i),U',\varDelta _0} \\ \quad \longrightarrow R^{j_1}\pi ^\natural _* (\varOmega ^{j_2}_{T^{(m),\natural ,\wedge }_{n,(i),U,\varSigma _0}/T^{\natural ,\wedge }_{n,(i),U',\varDelta _0}} (\log \infty ) \otimes {\mathcal {I}}^{(m),\natural ,\wedge }_{\partial ,n,(i),U,\sigma _0}) \end{array}$$are isomorphisms.
Lemma 4.11
Suppose that \(U \supset U'\) are neat open compact subgroups of the group \(P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )\) and let V and \(V'\) denote the images of U and \(U'\) in \(P_{n,(i)}^+({\mathbb {A}}^\infty )\). Let \(\varDelta _0\) (resp. \(\varDelta _0'\)) be a smooth admissible cone decomposition for \(X_*(S^+_{n,(i),V})\) (resp. \(X_*(S^+_{n,(i),V'})\)) and let \(\varSigma _0\) (resp. \(\varSigma _0'\)) be a compatible smooth admissible cone decomposition for \(X_*(S^{(m),+}_{n,(i),U})\) (resp. \(X_*(S^{(m),+}_{n,(i),U'})\)). Further suppose that \(\varSigma _0\) and \(\varSigma _0'\) are compatible and that \(\varDelta _0\) and \(\varDelta _0'\) are compatible.
-
(1)
The natural map
$$\begin{aligned} \pi _{(U',\varSigma _0'),(U,\varSigma _0)}^* \varOmega ^1_{T^{(m),\natural ,\wedge }_{n,(i),U,\varSigma _0}/T^{\natural ,\wedge }_{n,(i),V,\varDelta _0}}(\log \infty ) \longrightarrow \varOmega ^1_{T^{(m),\natural ,\wedge }_{n,(i),U',\varSigma _0'}/T^{\natural ,\wedge }_{n,(i),V',\varDelta _0'}} (\log \infty ) \end{aligned}$$is an isomorphism.
-
(2)
The natural map
$$\begin{array}{l} \pi _{(V',\varDelta _0'),(V,\varDelta _0)}^* \pi _{(U,\varSigma _0),(V,\varDelta _0),*} \varOmega ^1_{T^{(m),\natural ,\wedge }_{n,(i),U,\varSigma _0}/T^{\natural ,\wedge }_{n,(i),V,\varDelta _0}}(\log \infty ) \\ \quad \longrightarrow \pi _{(U',\varSigma _0'),(V',\varDelta _0'),*} \varOmega ^1_{T^{(m),\natural ,\wedge }_{n,(i),U',\varSigma _0'}/T^{\natural ,\wedge }_{n,(i),V',\varDelta _0'}}(\log \infty ) \end{array}$$is an isomorphism.
Similarly we have the following lemma.
Lemma 4.12
Suppose that \(U^p\) is a neat open compact subgroup of \(P_{n,(i)}^{(m),+}({\mathbb {A}}^{\infty ,p})\) and let \((U^p)'\) denote the image of \(U^p\) in \(P_{n,(i)}^+({\mathbb {A}}^{\infty ,p})\). Suppose that \(N_2 \ge N_1 \ge 0\) are integers. Let \(\varDelta _0\) be a smooth admissible cone decomposition for \(X_*({\mathcal {S}}^{{{\text {ord}}},+}_{n,(i),(U^p)'(N_1,N_2)})\) and let \(\varSigma _0\) be a compatible smooth admissible cone decomposition for \(X_*({\mathcal {S}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)})\). Let \(\pi ^\natural =\pi _{(U^p(N_1,N_2),\varSigma _0),((U^p)'(N_1,N_2),\varDelta _0)}^\natural \) denote the map
-
(1)
The map \(\pi _{(U^p(N_1,N_2),\varSigma _0),((U^p)'(N_1,N_2),\varDelta _0)}^\natural \) is proper.
-
(2)
The natural maps
$$\begin{aligned} {\mathcal {O}}_{{\mathcal {T}}^{{{\text {ord}}},\natural ,\wedge }_{n,(i),(U^p)'(N_1,N_2),\varDelta _0}} \longrightarrow \pi _{(U^p(N_1,N_2),\varSigma _0),((U^p)'(N_1,N_2),\varDelta _0),*}^\natural {\mathcal {O}}_{{\mathcal {T}}^{(m),{{\text {ord}}},\natural ,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0}} \end{aligned}$$and
$$\begin{aligned} {\mathcal {I}}^{{{\text {ord}}},\natural ,\wedge }_{\partial ,n,(i),(U^p)'(N_1,N_2),\varDelta _0} \longrightarrow \pi _{(U^p(N_1,N_2),\varSigma _0),((U^p)'(N_1,N_2),\varDelta _0),*}^\natural {\mathcal {I}}^{(m),{{\text {ord}}},\natural ,\wedge }_{\partial ,n,(i),U^p(N_1,N_2),\varSigma _0} \end{aligned}$$are isomorphisms.
-
(3)
The natural map
$$\begin{aligned} {\mathcal {I}}^{{{\text {ord}}},\natural ,\wedge }_{\partial ,n,(i),(U^p)'(N_1,N_2),\varDelta _0} \otimes R^j\pi _{*}^\natural {\mathcal {O}}_{{\mathcal {T}}^{(m),{{\text {ord}}},\natural ,\wedge }_{n,(i),U^p(N_1,N_2),\varSigma _0}} \longrightarrow R^j\pi _{*}^\natural {\mathcal {I}}^{(m),{{\text {ord}}},\natural ,\wedge }_{\partial ,n,(i),U^p(N_1,N_2),\varSigma _0} \end{aligned}$$is an isomorphism.
We finish this section with an important vanishing result.
Lemma 4.13
Suppose that \(R_0\) is an irreducible, noetherian \({\mathbb {Q}}\)-algebra (resp. \({\mathbb {Z}}_{(p)}\)-algebra) with the discrete topology. Suppose also that U (resp. \(U^p\)) is a neat open compact subgroup of \(P^{+}_{n,(i)}({\mathbb {A}}^\infty )\) (resp. \(P^{+}_{n,(i)}({\mathbb {A}}^{\infty ,p})\)), that \(N_2 \ge N_1 \ge 0\) are integers and that \(\varDelta _0\) is a smooth admissible cone decomposition for \(X_*(S_{n,(i),U}^{+})_{\mathbb {R}}^{\succ 0}\) (resp. \(X_*({\mathcal {S}}_{n,(i),U}^{{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\)). Let \(\pi \) denote the map
(resp.
Further suppose that \({\mathcal {E}}\) is a coherent sheaf on the formal scheme \(T^{\natural ,\wedge }_{n,(i),U,\varDelta _0} \times {{\text {Spf}}\,}R_0\) (resp. \({\mathcal {T}}^{{{\text {ord}}},\natural ,\wedge }_{n,(i),U^p(N_1,N_2),\varDelta _0} \times {{\text {Spf}}\,}R_0\)) with an exhaustive separated filtration, such that the pull-back to \(T^{+,\wedge }_{n,(i),U,\varDelta _0} \times {{\text {Spf}}\,}R_0\) (resp. \({\mathcal {T}}^{{{\text {ord}}},+,\wedge }_{n,(i),U^p(N_1,N_2),\varDelta _0} \times {{\text {Spf}}\,}R_0\)) of each
is \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-equivariantly (resp. \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\)-equivariantly) isomorphic to the pull-back to \(T^{+,\wedge }_{n,(i),U,\varDelta _0} \times {{\text {Spf}}\,}R_0\) (resp. \({\mathcal {T}}^{{{\text {ord}}},+,\wedge }_{n,(i),U^p(N_1,N_2),\varDelta _0} \times {{\text {Spf}}\,}R_0\)) of a locally free sheaf \({\mathcal {F}}_i\) with \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-action (resp. \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\)-action) over \(X_{n,(i),U}^+ \times {{\text {Spec}}\,}R_0\) (resp. \({\mathcal {X}}_{n,(i),U}^{{{\text {ord}}},+} \times {{\text {Spec}}\,}R_0\)).
Then for \(i>0\)
(resp.
Proof
We will treat the case of \(T^{\natural ,\wedge }_{n,(i),U,\varDelta _0} \times {{\text {Spf}}\,}R_0\), the other case being exactly similar. We can immediately reduce to the case that the pull-back to \(T^{+,\wedge }_{n,(i),U,\varDelta _0} \times {{\text {Spf}}\,}R_0\) of \({\mathcal {E}}\) is \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-equivariantly isomorphic to the pull-back to \(T^{+,\wedge }_{n,(i),U,\varDelta _0} {{\text {Spf}}\,}R_0\) of a locally free sheaf \({\mathcal {F}}\) with \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\)-action over \(X_{n,(i),U}^+ \times {{\text {Spec}}\,}R_0\).
Let \(\pi ^+\) denote the map
Also write \(\pi ^+=\pi _1^+ \circ \pi _2^+\), where
and
By Lemma 2.22 we have that
Then by Lemma 4.2 (or in the case of \({\mathcal {T}}^{{{\text {ord}}},\natural ,\wedge }_{n,(i),U^p(N_1,N_2),\varDelta _0} \times {{\text {Spf}}\,}R_0\) Lemma 4.3) we deduce that
Finally there is a spectral sequence
and so the present lemma follows on applying Shapiro’s lemma. \(\square \)
Corollary 4.14
Suppose that U (resp. \(U^p\)) is a neat open compact subgroup of \(P^{+}_{n,(i)}({\mathbb {A}}^\infty )\) (resp. \(P^{+}_{n,(i)}({\mathbb {A}}^{\infty ,p})\)), that \(N_2 \ge N_1 \ge 0\) are integers, and that \(\varDelta _0\) is a smooth admissible cone decomposition for \(X_*(S_{n,(i),U}^{+})_{\mathbb {R}}^{\succ 0}\) (resp. \(X_*({\mathcal {S}}_{n,(i),U}^{{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\)). Let \(\pi \) denote the map
(resp.
Also suppose that \(R_0\) is an irreducible noetherian \({\mathbb {Q}}\)-algebra (resp. \({\mathbb {Z}}_{(p)}\)-algebra) with the discrete topology and that \(\rho \) is a representation of \(R_{n,(n),(i)}\) on a finite locally free \(R_0\)-module.
Then for \(i>0\)
(resp.
6 Compactification of Shimura varieties
We now turn to the compactification of the \(X_{n,U}\) and the \(A^{(m)}_{n,U}\).
6.1 The minimal compactification
There is a canonically defined system of normal projective schemes with \(G_n({\mathbb {A}}^\infty )\)-action, \(\{ X_{n,U}^{{\text {min}}}/{{\text {Spec}}\,}{\mathbb {Q}}\}\) (for \(U \subset G_n({\mathbb {A}}^\infty )\) a neat open compact subgroup), together with compatible, \(G_n({\mathbb {A}}^\infty )\)-equivariant, dense open embeddings
These schemes are referred to as the minimal (or sometimes ‘Baily–Borel’) compactifications. (The introduction to [49] asserts that the scheme \(X_{n,U}^{{\text {min}}}\) is the minimal normal compactification of \(X_{n,U}\), although we won’t need this fact.) For \(g \in G_n({\mathbb {A}}^\infty )\) and \(g^{-1}Ug \subset U'\) the maps
are finite.
Write
There is a family of closed subschemes
such that each
is smooth of dimension \((n-i)^2[F^+: {\mathbb {Q}}]\). The families \(\{ \partial _i X_{n,U}^{{\text {min}}}\}\) and \(\{ \partial _i^0X_{n,U}^{{\text {min}}}\}\) are families of schemes with \(G_n({\mathbb {A}}^\infty )\)-action. Moreover we have a decomposition
If \(g \in G_n({\mathbb {A}}^\infty )\) and if \(g^{-1} U g \subset U'\) then the map
is the coproduct of the maps
where \(hg=g'h'\) with \(g' \in P_{n,(i)}^+({\mathbb {A}}^\infty )\). We will write \(X_{n,U,i}^{{{\text {min}}},\wedge }\) for the completion of \(X_{n,U}^{{\text {min}}}\) along \(\partial _i^0X_{n,U}^{{\text {min}}}\). (See theorem 1.3.1.5 and proposition 1.3.1.14 of [44].)
There is also a canonically defined system of normal quasi-projective schemes with \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\)-action, \(\{ {\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}/{{\text {Spec}}\,}{\mathbb {Z}}_{(p)} \}\), together with compatible, dense open embeddings
which are \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\)-equivariant. Suppose that \(g \in G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\) and that
then
is quasi-finite. If \(p^{N_2-N_2'}\nu (g) \in {\mathbb {Z}}_p^\times \) and either \(N_2'=N_2\) or \(N_2'>0\), then it is also finite. On \({\mathbb {F}}_p\)-fibres \(\varsigma _p\) acts as absolute Frobenius composed with the forgetful map. (See theorem 6.2.1.1, proposition 6.2.2.1 and corollary 6.2.2.9 of [44]. We remark that we are, perhaps unfortunately, following a different convention from [44]. According to our convention
In contrast Lan [44] works with an open subscheme which only contains the ‘ordinary’ part of the boundary. Our \({\mathcal {X}}^{{{\text {ord}}},{{\text {min}}}}_{n,U^p(N_1,N_2)}\) is the union of this open subscheme with \(X^{{\text {min}}}_{n,U^p(N_1,N_2)}\).)
Write
There is a family of closed subschemes
such that each
is smooth over \({\mathbb {Z}}_{(p)}\) of relative dimension \((n-i)^2[F^+: {\mathbb {Q}}]\). Then
and
are families of schemes with \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\)-action. We will write \({\mathcal {X}}_{n,U^p(N_1,N_2),i}^{{{\text {ord}}},{{\text {min}}},\wedge }\) for the completion of \({\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}\) along \(\partial _i^0{\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}\). We have a decomposition
where the second coproduct runs over
(Again see theorems 6.2.1.1 and proposition 6.2.2.1 of [44].)
[We explain why the map
is injective. It suffices to check that
or even that
where
This is clear.]
If \(g \in G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\) and if \(g^{-1} U^p(N_1,N_2) g \subset (U^p)'(N_1',N_2')\) then the map
is the coproduct of the maps
where \(hg=g'h'\) with \(g' \in P_{n,(i)}^+({\mathbb {A}}^\infty )^{{\text {ord}}}\), and of the maps
where \(hg=g'h'\) with \(g' \in P_{n,(i)}^+({\mathbb {A}}^\infty )\). (Again see theorems 6.2.1.1 and proposition 6.2.2.1 of [44].)
If \(N_2' \ge N_2\ge N_1\) then the natural map
is etale in a Zariski neighbourhood of the \({\mathbb {F}}_p\)-fibre, and the natural map
between formal completions along the \({\mathbb {F}}_p\)-fibres is an isomorphism. (See corollary 6.2.2.8 and example 3.4.4.5 of [44].) We will denote this p-adic formal scheme
and will denote its reduced subscheme
We will also write
The families \(\{ {\mathfrak {X}}_{n,U^p(N)}^{{{\text {ord}}},{{\text {min}}}} \}\) and \(\{ \overline{{X}}_{n,U^p(N)}^{{{\text {ord}}},{{\text {min}}}} \}\) and \(\{ \partial \overline{{X}}_{n,U^p(N)}^{{{\text {ord}}},{{\text {min}}}} \}\) have \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\)-actions. There is a family of closed subschemes
such that each
is smooth of dimension \((n-i)^2[F^+: {\mathbb {Q}}]\). Then \(\{ \partial _i \overline{{X}}_{n,U^p(N)}^{{\text {min}}}\}\) and \(\{ \partial _i^0\overline{{X}}_{n,U^p(N)}^{{\text {min}}}\}\) are families of schemes with \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\)-action. Moreover we have a decomposition
If \(g \in G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\) and if \(g^{-1} U^p(N) g \subset (U^p)'(N')\) then the map
is the coproduct of the maps
where \(hg=g'h'\) with \(g' \in P_{n,(i)}^+({\mathbb {A}}^\infty )^{{\text {ord}}}\). In particular \(\varsigma _p\) acts as absolute Frobenius.
The schemes \({\mathcal {X}}^{{{\text {ord}}},{{\text {min}}}}_{n,U^p(N_1,N_2)}\) are not proper. There are proper integral models of the schemes \(X_{n,U}^{{\text {min}}}\), but we have less control over them.
More specifically suppose that \(U \subset G_n({\mathbb {A}}^{\infty ,p} \times {\mathbb {Z}}_p)\) is an open compact subgroup whose projection to \(G_n({\mathbb {A}}^{\infty ,p})\) is neat. Then there is a normal, projective, flat \({\mathbb {Z}}_{(p)}\)-scheme \({\mathcal {X}}_{n,U}^{{\text {min}}}\) with generic fibre \(X_{n,U}^{{\text {min}}}\). If \(g \in G_n({\mathbb {A}}^{\infty ,p} \times {\mathbb {Z}}_p)\) and if
then there is a map
extending the map \(g: X_{n,U}^{{\text {min}}}\rightarrow X_{n,U'}^{{\text {min}}}\). This gives the system \(\{ {\mathcal {X}}^{{\text {min}}}_{n,U}\}\) an action of \(G_n({\mathbb {A}}^{\infty ,p} \times {\mathbb {Z}}_p)\). We set
On \({\mathcal {X}}_{n,U}^{{\text {min}}}\) there is an ample line bundle \(\omega _{U}\), and the system of line bundles \(\{ \omega _{U}\}\) over \(\{ {\mathcal {X}}_{n,U}^{{\text {min}}}\}\) has an action of \(G_n({\mathbb {A}}^{\infty ,p}\times {\mathbb {Z}}_p )\). The pull-back of \(\omega _U\) to \(X_{n,U}\) is \(G_n({\mathbb {A}}^{\infty ,p}\times {\mathbb {Z}}_p)\)-equivariantly identified with \(\wedge ^{n[F: {\mathbb {Q}}]} \varOmega _{n,U}\). (See propositions 2.2.1.2 and 2.2.3.1 of [44]. The rough idea is to take \({\mathcal {X}}_{n,U}^{{\text {min}}}\) as the normalization of \(X_{n,U}^{{\text {min}}}\) over the minimal compactification of some Siegel moduli of genus \(n[F: {\mathbb {Q}}]\) and a suitable neat level away from p, and take \(\omega _U\) to be the pull-back of the corresponding Hodge invertible sheaf there. It is not easy to describe such normalizations in as much detail as in [17, 41], but we can still verify the assertions in this paragraph using the corresponding assertions for the minimal compactifications of Siegel moduli.)
Moreover there are canonical sections
such that
whenever \(g \in G_n({\mathbb {A}}^{\infty ,p}\times {\mathbb {Z}}_p)\) and \(U' \supset g^{-1}Ug\). We will write \(\overline{{X}}_{n,U}^{{{\text {min}}},{{\text {n-ord}}}}\) for the zero locus in \(\overline{{X}}_{n,U}^{{\text {min}}}\) of \({{\text {Hasse}}}_{U}\). (See corollaries 6.3.1.7 and 6.3.1.8 of [44]. The rough idea is to take \({{\text {Hasse}}}_{U}\) to be the pull-back of the corresponding section over the minimal compactifications of some Siegel moduli.) Then \(\overline{{X}}_{n,U}^{{\text {min}}}- \overline{{X}}_{n,U}^{{{\text {min}}},{{\text {n-ord}}}}\) is relatively affine over \(\overline{{X}}_{n,U}^{{\text {min}}}\) associated to the sheaf of algebras
for any \(a \in {\mathbb {Z}}_{>0}\). It is also affine over \({\mathbb {F}}_p\) associated to the algebra
for any \(a \in {\mathbb {Z}}_{>0}\).
There are \(G_n({\mathbb {A}}^{\infty })^{{{\text {ord}}},\times }\)-equivariant open embeddings
These induce maps
on \({\mathbb {F}}_p\)-fibres which are open and closed embeddings. (See proposition 6.3.2.2 of [44].) In the case \(N_1=N_2=0\) this map is in fact an isomorphism. (See lemmas 6.3.2.7 and 6.3.2.9 of [44].) We remark that for \(N_2>0\) this map is not an isomorphism: The definition of \(\overline{{X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}}}\) requires not only that the universal abelian scheme is ordinary, the condition that defines \(\overline{{X}}_{n,U^p(N_1,N_2)}-\overline{{X}}_{n,U^p(N_1,N_2)}^{{{\text {min}}},{{\text {n-ord}}}}\), but also that the universal subgroup \(C^{{\text {univ}}}\subset {\mathcal {A}}^{{{\text {univ}}}}[p^{N_2}]\) is connected above each geometric point.
Also the pull-back of \(\omega _{U^p(N_1,N_2)}\) to \({\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}\) is \(G_n({\mathbb {A}}^{\infty })^{{{\text {ord}}},\times }\)-equivariantly identified with the sheaf \(\wedge ^{n[F: {\mathbb {Q}}]} \varOmega ^{{\text {ord}}}_{n,U^p(N_1,N_2)}\). If \(g \in G_n({\mathbb {A}}^{\infty })^{{{\text {ord}}},\times }\) and
then the commutative square
is a pull-back square. (See theorem 6.2.1.1 and proposition 6.2.2.1 of [44].)
At the referee’s suggestion we include a few remarks about the construction of \({\mathcal {X}}^{{{\text {ord}}},{{\text {min}}}}_{U^p(N_1,N_2)}\) and \({\mathcal {X}}^{{{\text {min}}}}_{U}\) in [44]. If p is unramified in F then one has good control of certain integral toroidal compactifications \({\mathcal {X}}_{U^p(0,0),\varDelta }\) and of the integral minimal compactification \({\mathcal {X}}^{{\text {min}}}_{U^p(0,0)}\). Moreover over \({\mathcal {X}}_{U^p(0,0),\varDelta }\) there is a ‘universal’ semi-abelian scheme and ‘ordinarity’ and \(U^p(N_1,N_2)\)-level structure can be defined for this semi-abelian scheme. For \(U=U^pU_p\) with \(U_p \subset G_n({\mathbb {Z}}_p)\) one can then define \({\mathcal {X}}^{{\text {min}}}_U\) as a normalization of \({\mathcal {X}}^{{\text {min}}}_{U^p(0,0)}\) in \(X^{{\text {min}}}_U\). One can then define \({\mathcal {X}}_{U,\varDelta '}\) as a suitable normalized blow-up of \({\mathcal {X}}_U^{{\text {min}}}\). We don’t have very much control of \({\mathcal {X}}_{U,\varDelta '}\) or \({\mathcal {X}}_U^{{\text {min}}}\). One can also define \({\mathcal {X}}^{{\text {ord}}}_{U^p(N_1,N_2),\varDelta }\), and it solves a relative moduli problem over \({\mathcal {X}}^{{\text {ord}}}_{U^p(0,0),\varDelta } \subset {\mathcal {X}}_{U^p(0,0),\varDelta }\). There are maps
The key point is to show that this map has open image, which we define to be \({\mathcal {X}}^{{{\text {ord}}},{{\text {min}}}}_{U^p(N_1,N_2)}\) and that \({\mathcal {X}}^{{\text {ord}}}_{U^p(N_1,N_2),\varDelta }\) is the pre-image of this open set in \({\mathcal {X}}_{U^p(N_1,N_2),\varDelta }\).
In the case that p ramifies in F things are a bit harder, because we don’t even have a good candidate for \({\mathcal {X}}_{U^p(0,0)}^{{\text {min}}}\). In this case one first constructs the corresponding spaces in the Siegel case, which is analogous to the good case discussed in the previous paragraph. One then has a proper map \({\mathcal {X}}^{{\text {ord}}}_{U^p(N_1,N_2),\varDelta } \rightarrow {\mathcal {Z}}^{{{\text {ord}}},{{\text {min}}}}_{V^p(N_1,N_2)}\), where \({\mathcal {Z}}^{{{\text {ord}}},{{\text {min}}}}_{V^p(N_1,N_2)}\) is the integral minimal partial compactification of the ordinary locus of a Siegel variety. One applies Stein factorization to this map and uses this to define \({\mathcal {X}}^{{{\text {ord}}},{{\text {min}}}}_{U^p(N_1,N_2)}\):
Then \({\mathcal {X}}^{{{\text {ord}}},{{\text {min}}}}_{U^p(N_1,N_2)}\) admits a quasi-finite map to \({\mathcal {X}}^{{\text {min}}}_{U^p(N_1,N_2)}\), which is shown to be an open immersion using Zariski’s main theorem.
6.2 Cone decompositions
Let \(U \subset G^{(m)}({\mathbb {A}}^\infty )\) be an open compact subgroup. By a U-admissible cone decomposition \(\varSigma \) of \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\) we shall mean a set of closed subsets \(\sigma \subset G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\) such that
-
(1)
each \(\sigma \) is contained in \(\{(g,\delta ) \} \times {\mathfrak {C}}^{(m),\succ 0}(W)\) for some isotropic subspace \(W \subset V_n\) and some \((g,\delta ) \in G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}}))\) and is the set of \({\mathbb {R}}_{\ge 0}\)-linear combinations of a finite set of elements of \({{\text {Herm}}}_{V/W^\perp } \times W^m\);
-
(2)
no \(\sigma \in \varSigma \) contains a complete line through the origin in any \((g,\delta ) \times {\mathfrak {C}}^{(m)}(W)\);
-
(3)
if \(\sigma \in \varSigma \) then any face of \(\sigma \) also lies in \(\varSigma \);
-
(4)
if \(\sigma , \sigma ' \in \varSigma \) then either \(\sigma \cap \sigma '=\emptyset \) or \(\sigma \cap \sigma '\) is a face of \(\sigma \) and \(\sigma '\);
-
(5)
\(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}=\bigcup _{\sigma \in \varSigma } \sigma \);
-
(6)
\(\varSigma \) is invariant by the diagonal action of \(G_n^{(m)}({\mathbb {Q}})\) on \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\);
-
(7)
\(\varSigma \) is invariant by the right action of U on \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\) (acting only on the first factor);
-
(8)
\(G_n^{(m)}({\mathbb {Q}}) \backslash \varSigma / U\) is a finite set;
-
(9)
if \(\sigma \in \varSigma \) lies in \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m),\succ 0}(V_{n,(i)})\) and if \(h \in P_{n,(i)}^{(m)}({\mathbb {A}})\), then \(h \sigma \in \varSigma \);
-
(10)
if \(\sigma \in \varSigma \) lies in \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m),\succ 0}(V_{n,(i)})\), if \(\gamma \in G_n^{(m)}({\mathbb {Q}})\), if \(u \in U\) and if \(h \in P_{n,(i)}^{(m)}({\mathbb {A}})\) satisfy
$$\begin{aligned} \sigma \cap \gamma h\sigma u \cap (G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m),> 0}(V_{n,(i)})) \ne \emptyset \end{aligned}$$then \(\gamma \in P_{n,(i)}^{(m)}({\mathbb {Q}})\).
(Here we let \(G_n^{(m)}({\mathbb {A}})\) act on \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\) via multiplication on the first two factors. The restriction of this action to \(G_n^{(m)}({\mathbb {Q}})\) does not coincide with the standard action of \(G_n^{(m)}({\mathbb {Q}})\), which we are using.) Note that if \(U' \subset U\) and if \(\varSigma \) is a U-admissible cone decomposition of \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\) then \(\varSigma \) is also \(U'\)-admissible. We will call a set \(\varSigma \) of closed subsets of \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\) an admissible cone decomposition of \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\) if it is U-admissible for some open compact subgroup U.
We remark that different authors use the term ‘U-admissible cone decomposition’ in somewhat different ways.
We call \(\varSigma '\) a refinement of \(\varSigma \) if every element of \(\varSigma \) is a union of elements of \(\varSigma '\). We define a partial order on the set of pairs \((U,\varSigma )\), where \(U \subset G_n^{(m)}({\mathbb {A}}^\infty )\) is an open compact subgroup and \(\varSigma \) is a U-admissible cone decomposition of \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\), as follows: We set
if and only if \(U' \subset U\) and \(\varSigma '\) is a refinement of \(\varSigma \). If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )\) and \(\varSigma \) is a U-admissible cone decomposition of \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\), then
is a \(g^{-1}U g\)-admissible cone decomposition of \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\). The action of \(G_n^{(m)}({\mathbb {A}}^\infty )\) preserves \(\ge \).
There is a natural projection
We will call admissible cone decompositions \(\varSigma \) of \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\) and \(\varDelta \) of \(G_n({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}\) compatible if the image of every \(\sigma \in \varSigma \) is contained in an element of \(\varDelta \). If in addition \(\varSigma \) is U-admissible, \(\varDelta \) is \(U'\)-admissible and \(U'\) contains the image of U in \(G_n({\mathbb {A}}^\infty )\) we will say that \((U,\varSigma )\) and \((U',\varDelta )\) are compatible and write
Now let \(U^p \subset G^{(m)}({\mathbb {A}}^{\infty ,p})\) be an open compact subgroup and let \(N \ge 0\) be an integer and consider \(U^p(N) \subset G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\). By a \(U^p(N)\)-admissible cone decomposition \(\varSigma \) of \((G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)})^{{\text {ord}}}\) we shall mean a set of closed subsets \(\sigma \subset (G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)})^{{\text {ord}}}\) such that
-
(1)
each \(\sigma \) is contained in \(\{(g,\delta ) \} \times {\mathfrak {C}}^{(m),\succ 0}(W)\) for some isotropic subspace \(W \subset V_n\) and some \((g,\delta ) \in G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}}))\) and is the set of \({\mathbb {R}}_{\ge 0}\)-linear combinations of a finite set of elements of \({{\text {Herm}}}_{V/W^\perp } \times W^m\);
-
(2)
no \(\sigma \in \varSigma \) contains a complete line through the origin in any \((g,\delta ) \times {\mathfrak {C}}^{(m)}(W)\);
-
(3)
if \(\sigma \in \varSigma \) then any face of \(\sigma \) also lies in \(\varSigma \);
-
(4)
if \(\sigma , \sigma ' \in \varSigma \) then either \(\sigma \cap \sigma '=\emptyset \) or \(\sigma \cap \sigma '\) is a face of \(\sigma \) and \(\sigma '\);
-
(5)
\((G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)})^{{\text {ord}}}=\bigcup _{\sigma \in \varSigma } \sigma \);
-
(6)
if \(\sigma \in \varSigma \), if \(\gamma \in G_n^{(m)}({\mathbb {Q}})\) and if \(u \in U^p(N,N)\) are such that \(\gamma \sigma u \subset (G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)})^{{\text {ord}}}\), then \(\gamma \sigma u \in \varSigma \);
-
(7)
there is a finite subset of \(\varSigma \) such that any element of \(\varSigma \) has the form \(\gamma \sigma u\) with \(\gamma \in G_n^{(m)}({\mathbb {Q}})\) and \(u \in U^p(N,N)\) and \(\sigma \) in the given finite subset;
-
(8)
if \(\sigma \in \varSigma \) lies in \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m),\succ 0}(V_{n,(i)})\) and meets \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m),> 0}(V_{n,(i)})\), and if \(h \in P_{n,(i)}^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}}, \times } \times P_{n,(i)}^{(m)}({\mathbb {R}})\), then \(h \sigma \in \varSigma \);
-
(9)
if \(\sigma \in \varSigma \) lies in \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m),\succ 0}(V_{n,(i)})\), if \(\gamma \in G_n^{(m)}({\mathbb {Q}})\), if \(u \in U^p(N,N)\) and if \(h \in P_{n,(i)}^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}}, \times } \times P_{n,(i)}^{(m)}({\mathbb {R}})\) satisfy \(\gamma h\sigma u \in \varSigma \) and
$$\begin{aligned} \sigma \cap \gamma h\sigma u \cap (G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m),> 0}(V_{n,(i)})) \ne \emptyset \end{aligned}$$then \(\gamma \in P_{n,(i)}^{(m)}({\mathbb {Q}})\).
Note that if \((U^p)'(N') \subset U^p(N)\) and if \(\varSigma \) is a \(U^p(N)\)-admissible cone decomposition of \((G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)})^{{\text {ord}}}\) then \(\varSigma \) is also \((U^p)'(N')\)-admissible. We will call a set \(\varSigma \) of closed subsets of \((G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)})^{{\text {ord}}}\) an admissible cone decomposition of \((G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)})^{{\text {ord}}}\) if it is \(U^p(N)\)-admissible for some open compact subgroup \(U^p\) and for some N.
If \(\varSigma \) is a \(U^p(N_1,N_2)\)-admissible cone decomposition of \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\) then
is a \(U^p(N_1)\)-admissible cone decomposition for \((G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)})^{{\text {ord}}}\).
We call \(\varSigma '\) a refinement of \(\varSigma \) if every element of \(\varSigma \) is a union of elements of \(\varSigma '\). We define a partial order on the set of pairs \((U^p(N),\varSigma )\), where \(U^p \subset G_n^{(m)}({\mathbb {A}}^{\infty ,p})\) is an open compact subgroup, \(N \in {\mathbb {Z}}_{\ge 0}\) and \(\varSigma \) is a \(U^p(N)\)-admissible cone decomposition of \((G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)})^{{\text {ord}}}\), as follows: We set
if and only if \((U^p)'(N') \subset U^p(N)\) and \(\varSigma '\) is a refinement of \(\varSigma \). If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) and \(\varSigma \) is a \(U^p(N)\)-admissible cone decomposition of \((G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)})^{{\text {ord}}}\), then
is a \(g^{-1}U^p(N) g\)-admissible cone decomposition of
The action of \(G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) preserves \(\ge \).
There is a natural projection
We will call admissible cone decompositions \(\varSigma \) of \((G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)})^{{\text {ord}}}\) and \(\varDelta \) of \((G_n({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}})^{{\text {ord}}}\) compatible if the image of every \(\sigma \in \varSigma \) is contained in an element of \(\varDelta \). If in addition \(\varSigma \) is \(U^p(N)\)-admissible, \(\varDelta \) is \((U^p)'(N')\)-admissible and \((U^p)'(N')\) contains the image of \(U^p(N)\) in \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\) we will say that \((U^p(N),\varSigma )\) and \(((U^p)'(N'),\varDelta )\) are compatible and write
If \(\varSigma \) is a U-admissible cone decomposition of \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\) and if \(h \in G_n^{(m)}({\mathbb {A}}^\infty )\) then we define an admissible cone decomposition \(\varSigma (h)_0\) for
as follows: The cones in \(\varSigma (h)_0\) over an element
are the cones
which meet \({\mathfrak {C}}^{(m),> 0}(V_{n,(i)})\) and satisfy
This does not depend on the representative \(h'\) we choose for y. It also only depends on
If \(h_1 \in L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^\infty )\) then under the natural isomorphism
we see that \(\varSigma (h)_0\) and \(\varSigma (h_1h)_0\) correspond.
Similarly if \(\varSigma \) is a \(U^p(N)\)-admissible cone decomposition of
and if \(h \in G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) then we define an admissible cone decomposition \(\varSigma (h)_0\) for
as follows: The cones in \(\varSigma (h)_0\) over an element y given as
are the cones
which meet \({\mathfrak {C}}^{(m),> 0}(V_{n,(i)})\) and satisfy
This does not depend on the representative \(h'\) we choose for y. It also only depends on
If \(h_1 \in L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) then under the natural isomorphism
we see that \(\varSigma (h)_0\) and \(\varSigma (h_1h)_0\) correspond.
There are sets \({\mathcal {J}}^{(m),{{\text {tor}}}}_n\) (resp. \({\mathcal {J}}^{(m),{{\text {tor}}},{{\text {ord}}}}_n\)) of pairs \((U,\varSigma )\) (resp. \((U^p(N),\varSigma )\)) where \(U \subset G_n^{(m)}({\mathbb {A}}^\infty )\) is a neat open compact subgroup (resp. \(U^p \subset G_n^{(m)}({\mathbb {A}}^{\infty ,p})\) is a neat open compact subgroup and \(N\in {\mathbb {Z}}_{\ge 0}\)) and \(\varSigma \) is a U-admissible (resp. \(U^p(N)\)-admissible) cone decomposition of \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\) (resp. \((G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)})^{{\text {ord}}}\)), with a number of properties which will be listed in this section and the next section. (See [44].)
Firstly we have the following properties:
-
(1)
The sets \({\mathcal {J}}^{(m),{{\text {tor}}}}_n\) (resp. \({\mathcal {J}}^{(m),{{\text {tor}}},{{\text {ord}}}}_n\)) are invariant under the action of \(G_n^{(m)}({\mathbb {A}}^\infty )\) (resp. \(G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)).
-
(2)
If U is any neat open compact subgroup of \(G_n^{(m)}({\mathbb {A}}^\infty )\), then there is some \(\varSigma \) with \((U,\varSigma ) \in {\mathcal {J}}^{(m),{{\text {tor}}}}_n\).
-
(3)
If \(U^p\) is any neat open compact subgroup of \(G_n^{(m)}({\mathbb {A}}^{\infty ,p})\) and if \(N \in {\mathbb {Z}}_{\ge 0}\), then there is some \(\varSigma \) with \((U^p(N),\varSigma ) \in {\mathcal {J}}^{(m),{{\text {tor}}},{{\text {ord}}}}_n\).
-
(4)
If \((U,\varSigma ) \in {\mathcal {J}}^{(m),{{\text {tor}}}}_n\) and if \(U' \subset U\) then there exists \((U',\varSigma ') \in {\mathcal {J}}^{(m),{{\text {tor}}}}_n\) with \((U',\varSigma ') \ge (U,\varSigma )\).
-
(5)
If \((U^p(N),\varSigma ') \in {\mathcal {J}}^{(m),{{\text {tor}}},{{\text {ord}}}}_n\), if \(N' \ge N\) and if \((U^p)'(N') \subset U^p(N)\) then there exists an element \(((U^p)'(N'),\varSigma ') \in {\mathcal {J}}^{(m),{{\text {tor}}},{{\text {ord}}}}_n\) with \(((U^p)'(N'),\varSigma ') \ge (U^p(N),\varSigma )\).
-
(6)
If \((U',\varSigma ') \ge (U,\varSigma )\) are elements of \({\mathcal {J}}^{(m),{{\text {tor}}}}_n\) and if moreover \(U'\) is a normal subgroup of U, then we may choose \((U',\varSigma '')\in {\mathcal {J}}^{(m),{{\text {tor}}}}_n\) such that \(\varSigma ''\) is U-invariant and such that \((U',\varSigma '') \ge (U',\varSigma ')\).
-
(7)
If \(((U^p)'(N'),\varSigma ') \ge (U^p(N),\varSigma )\) are elements of \({\mathcal {J}}^{(m),{{\text {tor}}},{{\text {ord}}}}_n\) with \((U^p)'\) a normal subgroup of \(U^p\), then there is an element \(((U^p)'(N'),\varSigma '')\in {\mathcal {J}}^{(m),{{\text {tor}}}}_n\) such that \(\varSigma ''\) is \(U^p(N)\)-invariant and such that \(((U^p)'(N'),\varSigma '') \ge ((U^p)'(N'),\varSigma ')\).
-
(8)
If \((U,\varSigma )\) and \((U,\varSigma ') \in {\mathcal {J}}^{(m),{{\text {tor}}}}_n\) (resp. if \((U^p(N),\varSigma )\) and \((U^p(N),\varSigma ') \in {\mathcal {J}}^{(m),{{\text {tor}}}.{{\text {ord}}}}_n\)) then there exists \((U,\varSigma '') \in {\mathcal {J}}^{(m),{{\text {tor}}}}_n\) (resp. there exists \((U^p(N),\varSigma '')\) in \({\mathcal {J}}^{(m),{{\text {tor}}},{{\text {ord}}}}_n\)) with \((U,\varSigma '')\ge (U,\varSigma )\) and \((U,\varSigma '')\ge (U,\varSigma ')\) (resp. with \((U^p(N),\varSigma '')\ge (U^p(N),\varSigma )\) and \((U^p(N),\varSigma '')\ge (U^p(N),\varSigma ')\)).
-
(9)
If \((U',\varDelta ) \in {\mathcal {J}}^{{\text {tor}}}_n\) (resp. \(((U^p)'(N'),\varDelta ) \in {\mathcal {J}}^{{{\text {tor}}},{{\text {ord}}}}_n\)) and if U is a neat open compact subgroup of \(G_n^{(m)}({\mathbb {A}}^\infty )\) mapping into \(U'\) (resp. \(U^p\) is a neat open compact subgroup of \(G_n^{(m)}({\mathbb {A}}^{\infty ,p})\) mapping into \((U^p)'\) and \(N \ge N'\)), then there exists \((U,\varSigma ) \in {\mathcal {J}}^{(m),{{\text {tor}}}}_n\) (resp. \((U^p(N),\varSigma ) \in {\mathcal {J}}^{(m),{{\text {tor}}},{{\text {ord}}}}_n\)) compatible with \((U',\varDelta )\) (resp. \(((U^p)'(N'),\varDelta )\)).
-
(10)
If \((U^p(N_1,N_2),\varSigma ) \in {\mathcal {J}}^{(m),{{\text {tor}}}}_n\) then \((U^p(N_1),\varSigma ^{{\text {ord}}}) \in {\mathcal {J}}^{(m),{{\text {tor}}},{{\text {ord}}}}_n\).
-
(11)
If \((U^p(N),\varSigma ') \in {\mathcal {J}}^{(m),{{\text {tor}}},{{\text {ord}}}}_n\) and if \(N' \ge N\), then there exists \((U^p(N,N'),\varSigma ) \in {\mathcal {J}}_n^{(m),{{\text {tor}}}}\) with \(\varSigma ^{{\text {ord}}}=\varSigma '\).
-
(12)
If \((U^p(N_1,N_2),\varSigma )\) and \((U^p(N_1,N_2),\varSigma ') \in {\mathcal {J}}^{(m),{{\text {tor}}}}_n\) with \(\varSigma ^{{\text {ord}}}=(\varSigma ')^{{\text {ord}}}\), then there is an element \((U^p(N_1,N_2),\varSigma '') \in {\mathcal {J}}^{(m),{{\text {tor}}}}_n\) with \((\varSigma '')^{{\text {ord}}}=\varSigma ^{{\text {ord}}}=(\varSigma ')^{{\text {ord}}}\) and with \((U^p(N_1,N_2),\varSigma '') \ge (U^p(N_1,N_2),\varSigma )\) and with \((U^p(N_1,N_2),\varSigma '') \ge (U^p(N_1,N_2),\varSigma ')\).
-
(13)
If \((U^p(N_1,N_2),\varSigma )\) and \(((U^p)'(N_1',N_2'),\varSigma ') \in {\mathcal {J}}^{(m),{{\text {tor}}}}_n\) with \((U^p)'(N_1',N_2') \subset U^p(N_1,N_2)\) and with \((\varSigma ')^{{\text {ord}}}\) refining \(\varSigma ^{{\text {ord}}}\), then there also exists another pair \(((U^p)'(N_1',N_2'),\varSigma '') \in {\mathcal {J}}^{(m),{{\text {tor}}}}_n\) with \(\varSigma ''\) refining both \(\varSigma \) and \(\varSigma '\) and with \((\varSigma '')^{{\text {ord}}}=(\varSigma ')^{{\text {ord}}}\).
-
(14)
If \((U^p(N_1,N_2),\varDelta ) \in {\mathcal {J}}^{{\text {tor}}}_n\) and \(((U^p)'(N_1',N_2'),\varSigma ') \in {\mathcal {J}}^{(m),{{\text {tor}}}}_n\) are such that \((U^p)'(N_1',N_2') \subset U^p(N_1,N_2)\) and \((\varSigma ')^{{\text {ord}}}\) is compatible with \(\varDelta ^{{\text {ord}}}\), then there exists \(((U^p)'(N_1',N_2'),\varSigma '') \in {\mathcal {J}}^{(m),{{\text {tor}}}}_n\) with \(\varSigma ''\) refining \(\varSigma '\) and compatible with \(\varDelta \) and with \((\varSigma '')^{{\text {ord}}}=(\varSigma ')^{{\text {ord}}}\).
-
(15)
If \((U^p(N_1,N_2),\varSigma ) \in {\mathcal {J}}_n^{(m),{{\text {tor}}}}\) and if \(N_2'\ge N_2\) then there exists a new pair \((U^p(N_1,N_2'),\varSigma ') \in {\mathcal {J}}_n^{(m),{{\text {tor}}}}\) with \((\varSigma ')^{{\text {ord}}}=\varSigma ^{{\text {ord}}}\).
(See propositions 1.2.4.52 and 7.1.1.21 of [44].)
Secondly if \((U,\varSigma )\in {\mathcal {J}}^{(m),{{\text {tor}}}}_n\) (resp. \((U^p(N),\varSigma )\in {\mathcal {J}}^{(m),{{\text {tor}}},{{\text {ord}}}}_n\)) and if \(h \in G_n^{(m)}({\mathbb {A}}^\infty )\) (resp. \(h \in G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)) then \(\varSigma (h)_0\) is smooth.
Thirdly if \((U,\varSigma ) \in {\mathcal {J}}^{(m),{{\text {tor}}}}_n\), then there is a simplicial complex \({\mathcal {S}}(U,\varSigma )\) whose simplices are in bijection with the cones in
which have dimension bigger than 0, and have the same face relations. We will write \({\mathcal {S}}(U,\varSigma )_{\le i}\) for the subcomplex of \({\mathcal {S}}(U,\varSigma )\) consisting of simplices associated to the orbits of cones \((g,\delta ) \times \sigma \in \varSigma \) with \(\sigma \subset {\mathfrak {C}}^{(m), \succ 0}(W)\) for some W with \(\dim _F W \le i\). We will also set
an open subset of \(| {\mathcal {S}}(U,\varSigma )_{\le i} |\). Then one sees that
and
(See Sect. 1.4.)
If \((U^p(N),\varSigma ) \in {\mathcal {J}}^{(m),{{\text {tor}}},{{\text {ord}}}}_n\) then there is a simplicial complex \({\mathcal {S}}(U^p(N),\varSigma )^{{\text {ord}}}\) whose simplices are in bijection with equivalence classes of cones of dimension greater than 0 in \(\varSigma \), where \(\sigma \) and \(\sigma '\) are considered equivalent if \(\sigma '=\gamma \sigma u\) for some \(\gamma \in G_n^{(m)}({\mathbb {Q}})\) and some \(u\in U^p(N,N)\). We will write \({\mathcal {S}}(U^p(N),\varSigma )^{{\text {ord}}}_{\le i}\) for the subcomplex of \({\mathcal {S}}(U^p(N),\varSigma )^{{\text {ord}}}\) consisting of simplices associated to the orbits of cones \((g,\delta ) \times \sigma \in \varSigma \) with \(\sigma \subset {\mathfrak {C}}^{(m), \succ 0}(W)\) for some W with \(\dim _F W \le i\). We will also set
an open subset of \(| {\mathcal {S}}(U^p,\varSigma )^{{\text {ord}}}_{\le i} |\). Then we see that
where
denotes the image of
in
Moreover
(Use the same argument as in the proof of Lemma 1.8.) In particular
6.3 Toroidal compactifications
If \((U,\varSigma ) \in {\mathcal {J}}^{(m),{{\text {tor}}}}_n\), then there is a smooth projective scheme \(A^{(m)}_{n,U,\varSigma }\) and a divisor with simple normal crossings
together with an isomorphism
and a projection
such that
is a commutative pull-back square. (The set \({\mathcal {J}}^{(m),{{\text {tor}}}}_n\) was chosen so that inter alia these properties are true.) The divisor \(\partial A^{(m)}_{n,U,\varSigma }\) induces a log structure \({\mathcal {M}}_{\varSigma }\) on \(A^{(m)}_{n,U,\varSigma }\).
If \((U,\varSigma ) \in {\mathcal {J}}^{(m),{{\text {tor}}}}_n\) and \((U',\varDelta ) \in {\mathcal {J}}^{{\text {tor}}}_n\) with \((U,\varSigma ) \ge (U',\varDelta )\) then there is a log smooth map
over \(X_{n,U'}^{{\text {min}}}\) extending the map
If \((U',\varSigma ')\) and \((U,\varSigma ) \in {\mathcal {J}}^{(m),{{\text {tor}}}}_n\); if \(g \in G_n^{(m)}({\mathbb {A}}^\infty )\); if \(U' \supset g^{-1}Ug\); and if \(\varSigma g\) is a refinement of \(\varSigma '\) then the map \(g: A^{(m)}_{n,U} \rightarrow A^{(m)}_{n,U'}\) extends to a log etale morphism
The collection \(\{ A^{(m)}_{n,U,\varSigma } \}\) becomes a system of schemes with right \(G_n^{(m)}({\mathbb {A}}^\infty )\)-action, indexed by \({\mathcal {J}}^{(m),{{\text {tor}}}}_n\). The maps \(j_{U,\varSigma }^{(m)}\) and \(\pi _{A^{(m),{{\text {tor}}}}/X^{{\text {min}}}}\) and \(\pi _{A^{(m),{{\text {tor}}}}/X^{{\text {tor}}}}\) are all \(G_n^{(m)}({\mathbb {A}}^\infty )\)-equivariant. If \((U,\varSigma ) \ge (U',\varSigma ')\) we will write \(\pi _{(U,\varSigma ), (U',\varSigma ')}\) for the map \(1: A^{(m)}_{n,U,\varSigma }\rightarrow A^{(m)}_{n,U',\varSigma '}\). (See theorem 1.3.3.15 of [44] for the assertions of the last three paragraphs.)
Any of the (canonically quasi-isogenous) universal abelian schemes \(A^{{\text {univ}}}/X_{n,U}\) extend uniquely to semi-abelian schemes \(A^{{\text {univ}}}_\varDelta /X_{n,U,\varDelta }\). The quasi-isogenies between the \(A^{{\text {univ}}}\) extend uniquely to quasi-isogenies between the \(A^{{\text {univ}}}_\varDelta \). If \(g \in G_n({\mathbb {A}}^\infty )\) and \((U,\varDelta ) \ge (U',\varDelta ')g\) then \(g^* A_{\varDelta '}^{{\text {univ}}}\) is one of the \(A_{\varDelta }^{{\text {univ}}}\). (See remarks 1.1.2.1 and 1.3.1.4 of [44].)
We will write \(\partial _i A_{n,U,\varSigma }^{(m)}\) for the pre-image under \(\pi _{A^{(m),{{\text {tor}}}}/X^{{\text {min}}}}|_{\partial A_{n,U,\varSigma }^{(m)}}\) of \(\partial _i X_U^{{\text {min}}}\). We also set
We will also write \(A_{n,U,\varSigma ,i}^{(m),\wedge }\) for the formal completion of \(A_{n,U,\varSigma }^{(m)}\) along \(\partial _{i}^0 A_{n,U,\varSigma }^{(m)}\) and \({\mathcal {M}}_{\varSigma ,i}^\wedge \) for the log structure induced on \(A_{n,U,\varSigma ,i}^{(m),\wedge }\) by \({\mathcal {M}}_\varSigma \). There are isomorphisms
Suppose that \(g^{-1} U g \subset U'\) and that \(\varSigma g\) is a refinement of \(\varSigma '\). Suppose also that \(h , h' \in G^{(m)}_n({\mathbb {A}}^\infty )\) with
Then the diagram
commutes, and is compatible with the log structures on each of these formal schemes. (See theorem 1.3.3.15 of [44].)
If \(U'\) is a neat subgroup of \(G_n({\mathbb {A}}^\infty )\) containing the image of U; if \((U',\varDelta ) \in {\mathcal {J}}^{{{\text {tor}}}}_n\); and if \(\varSigma \) and \(\varDelta \) are compatible, then for all \(h \in P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )\) with image \(h' \in P_{n,(i)}^+({\mathbb {A}}^\infty )\) the cone decompositions \(\varSigma (h)_0\) and \(\varDelta (h')_0\) are compatible and we have a diagram
which is commutative as a diagram of topological spaces (but not as a diagram of locally ringed spaces). The top square is commutative as a diagram of formal schemes and is compatible with the log structures. (Again see theorem 1.3.3.15 of [44].)
The pull-back of \(A_\varDelta ^{{\text {univ}}}\) from \(X^\wedge _{n,U',\varDelta ,i}\) to \(T^{+,\wedge }_{n,(i),h' U (h')^{-1} \cap P_{n,(i)}^+({\mathbb {A}}^\infty ), \varDelta (h')_0}\) is canonically quasi-isogenous to the pull-back of \(\widetilde{{G}}^{{\text {univ}}}\) from \(A^{+,\wedge }_{n,(i),h' U (h')^{-1} \cap P_{n,(i)}^+({\mathbb {A}}^\infty ), \varDelta (h')_0}\).
We will write
Then there are compatible identifications
and
and
and the latter is compatible with the identifications
(See theorem 1.3.3.15 of [44].) If \([\sigma ] \in {\mathcal {S}}(U,\varSigma )\) we will write
for the corresponding closed boundary stratum of \(A^{(m)}_{n,U,\varSigma }\).
Similarly if \((U^p(N_1,N_2),\varSigma ) \in {\mathcal {J}}^{(m),{{\text {tor}}}}_n\), then there is a smooth quasi-projective scheme \({\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\varSigma }\) and a divisor with simple normal crossings
together with an isomorphism
and a projection
such that
is a commutative pull-back square. The divisor \(\partial {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\varSigma }\) induces a log structure \({\mathcal {M}}_\varSigma \) on \({\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\varSigma }\).
If \((U^p(N_1,N_2),\varSigma ) \in {\mathcal {J}}^{(m),{{\text {tor}}}}_n\) and \(((U^p)'(N_1,N_2),\varDelta ) \in {\mathcal {J}}^{{\text {tor}}}_n\) satisfy
then there is a log smooth map
over \({\mathcal {X}}_{n,(U^p)'(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}\) extending the map
If \(((U^p)'(N_1,N_2),\varSigma ')\) and \((U^p)'(N_1,N_2),\varSigma ) \in {\mathcal {J}}^{(m),{{\text {tor}}}}_n\); if \(g \in G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\); if \((U^p)'(N_1',N_2') \supset g^{-1}U^p(N_1,N_2)g\); and if \(\varSigma g\) is a refinement of \(\varSigma '\) then the map \(g: {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)} \rightarrow {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,(U^p)'(N_1,N_2)}\) extends to a log etale morphism
Then \(\{ {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\varSigma } \}\) is a system of schemes with right \(G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\)-action, indexed by the subset of \({\mathcal {J}}^{(m),{{\text {tor}}}}_n\) consisting of elements of the form \((U^p(N_1,N_2),\varSigma )\). The maps \(j_{U,\varSigma }^{(m),{{\text {ord}}}}\) and \(\pi _{{\mathcal {A}}^{(m),{{\text {ord}}},{{\text {tor}}}}/{\mathcal {X}}^{{{\text {ord}}},{{\text {min}}}}}\) and \(\pi _{{\mathcal {A}}^{(m),{{\text {ord}}},{{\text {tor}}}}/{\mathcal {X}}^{{{\text {ord}}},{{\text {tor}}}}}\) are \(G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) equivariant. If \((U^p(N_1,N_2),\varSigma ) \ge ((U^p)'(N_1',N_2'),\varSigma ')\), then we will denote the map \(1: {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\varSigma }\rightarrow {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,(U^p)'(N_1,N_2),\varSigma '}\) by \(\pi _{(U^p(N_1,N_2),\varSigma ), ((U^p)'(N_1,N_2),\varSigma ')}\). (See theorem 7.1.4.1 of [44] for the assertions of the last three paragraphs.)
Any of the (canonically prime-to-p quasi-isogenous) universal abelian schemes \({\mathcal {A}}^{{\text {univ}}}/{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}\) extend uniquely to semi-abelian schemes \({\mathcal {A}}^{{\text {univ}}}_\varDelta /{\mathcal {X}}_{n,U^p(N_1,N_2),\varDelta }^{{\text {ord}}}\). The prime-to-p quasi-isogenies between the \({\mathcal {A}}^{{\text {univ}}}\) extend uniquely to prime-to-p quasi-isogenies between the \({\mathcal {A}}^{{\text {univ}}}_\varDelta \). If \(g \in G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) and
then \(g^* {\mathcal {A}}_{\varDelta '}^{{\text {univ}}}\) is one of the \({\mathcal {A}}_{\varDelta }^{{\text {univ}}}\). (See remarks 3.4.2.8 and 5.2.1.5 of [44].)
We will write \(\partial _i {\mathcal {A}}_{n,U^p(N_1,N_2),\varSigma }^{(m),{{\text {ord}}}}\) for the pre-image of \(\partial _i {\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}\) under the map
and set
We will also write \({\mathcal {A}}_{n,U^p(N_1,N_2),\varSigma ,i}^{(m),{{\text {ord}}},\wedge }\) for the formal completion of \({\mathcal {A}}_{n,U^p(N_1,N_2),\varSigma }^{(m),{{\text {ord}}}}\) along \(\partial _{i}^0 {\mathcal {A}}_{n,U^p(N_1,N_2),\varSigma }^{(m),{{\text {ord}}}}\), and \({\mathcal {M}}_{\varSigma ,i}^\wedge \) for the log structure induced on \({\mathcal {A}}_{n,U^p(N_1,N_2),\varSigma ,i}^{(m),{{\text {ord}}},\wedge }\) by \({\mathcal {M}}_\varSigma \). There are isomorphisms
Suppose that \(g \in G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) and \(g^{-1} U^p(N_1,N_2) g \subset (U^p)'(N_1',N_2')\) and that \(\varSigma g\) is a refinement of \(\varSigma '\). Suppose also that \(h , h' \in G^{(m)}_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) with
Then the diagram
commutes, where
and
Moreover this is compatible with the log structures defined on each of the four formal schemes. (See theorem 7.1.4.1 of [44].)
If \([\sigma ]\in {\mathcal {S}}(U^p(N_1,N_2),\varSigma )\) we will write
for the closure of \(\partial _{[\sigma ]} A^{(m)}_{n,U^p(N_1,N_2),\varSigma }\) in \({\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\varSigma }\). The special fibre
is non-empty if and only if \([\sigma ] \in {\mathcal {S}}(U^p(N_1),\varSigma ^{{\text {ord}}})^{{\text {ord}}}\). (We remind the reader that the first superscript \({}^{{\text {ord}}}\) associates the ‘ordinary’ cone decomposition \(\varSigma ^{{\text {ord}}}\) to the cone decomposition \(\varSigma \), while the second superscript \({}^{{\text {ord}}}\) is the notation we are using for the simplicial complex associated to an ‘ordinary’ cone decomposition.) We will write
This only depends on \(\varSigma ^{{\text {ord}}}\).
If \((U^p)'\) is a neat subgroup of \(G_n({\mathbb {A}}^{\infty ,p})\) containing the image of \(U^p\); if the pair \(((U^p)'(N_1,N_2),\varDelta ) \in {\mathcal {J}}_n^{{{\text {tor}}}}\); and if \(\varSigma \) and \(\varDelta \) are compatible, then for all \(h \in P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) with image \(h' \in P_{n,(i)}^+({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) the cone decompositions \(\varSigma ^{{\text {ord}}}(h)_0\) and \(\varDelta ^{{\text {ord}}}(h')_0\) are compatible and we have a diagram
which is commutative as a diagram of topological spaces (but not as a diagram of locally ringed spaces). The top square is commutative as a diagram of formal schemes and is compatible with the log structures. (See theorem 7.1.4.1 of [44].)
The pull-back of \({\mathcal {A}}^{{\text {univ}}}_\varDelta \) to \({\mathcal {T}}^{{{\text {ord}}},+,\wedge }_{n,(i),(h' (U^p)' (h')^{-1} \cap P_{n,(i)}^+({\mathbb {A}}^{\infty ,p}))(N_1,N_2), \varDelta ^{{\text {ord}}}(h')_0}\) is canonically quasi-isogenous to the pull-back of \({\widetilde{{\mathcal {G}}}}^{{\text {univ}}}\) from
All this is compatible with passage to the generic fibre and our previous discussion. (Again see theorem 7.1.4.1 of [44].)
If \(N_2' \ge N_2\ge N_1\), if \(\varSigma '\) is a refinement of \(\varSigma \) and if \(\varSigma ^{{\text {ord}}}=(\varSigma ')^{{\text {ord}}}\) then the natural map
is etale in a neighbourhood of the \({\mathbb {F}}_p\)-fibre of \({\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2'),\varSigma '}\) and induces an isomorphism between the formal completions of these schemes along their \({\mathbb {F}}_p\)-fibres. (See theorem 7.1.4.1(4) of [44].) We will denote this p-adic formal scheme
and will denote its reduced subscheme
(In the case \(m=0\) we could also write \({\mathfrak {X}}^{{{\text {ord}}}}_{n,U^p(N_1),\varSigma ^{{\text {ord}}}}\) and \(\overline{{x}}^{{{\text {ord}}}}_{n,U^p(N_1),\varSigma ^{{\text {ord}}}}\).) We will also write
The family \(\{ {\mathfrak {A}}_{n,U^p(N),\varSigma ^{{\text {ord}}}}^{(m),{{\text {ord}}}} \}\) (resp. \(\{ \overline{{A}}_{n,U^p(N),\varSigma ^{{\text {ord}}}}^{(m),{{\text {ord}}}} \}\), resp. \(\{ \partial \overline{{A}}_{n,U^p(N),\varSigma ^{{\text {ord}}}}^{(m),{{\text {ord}}}} \}\)) is a family of formal schemes (resp. schemes, resp. schemes) indexed by \({\mathcal {J}}^{(m),{{\text {tor}}},{{\text {ord}}}}_n\) with \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\) action. Let
denote the pre-image of \(\partial _i\overline{{X}}_{n,U^p(N)}^{{{\text {ord}}},{{\text {min}}}}\) in \(\partial \overline{{A}}_{n,U^p(N),\varSigma ^{{\text {ord}}}}^{(m),{{\text {ord}}}}\), and set
The families \(\{ \partial _i \overline{{A}}_{n,U^p(N),\varSigma ^{{\text {ord}}}}^{(m),{{\text {ord}}}} \}\) and \(\{ \partial _i^0\overline{{A}}_{n,U^p(N),\varSigma ^{{\text {ord}}}}^{(m),{{\text {ord}}}} \}\) have actions of \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\). Moreover we have a decomposition
If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\), if \(g^{-1} U^p(N) g \subset (U^p)'(N')\) and if \(\varSigma ^{{\text {ord}}}g\) is a refinement of \((\varSigma ')^{{\text {ord}}}\), then the map
is the coproduct of the maps
where \(hg=g'h'\) with \(g' \in P_{n,(i)}^+({\mathbb {A}}^\infty )^{{\text {ord}}}\).
The map
is finite flat of degree \(p^{(2m+n)n[F^+: {\mathbb {Q}}]}\) and on \({\mathbb {F}}_p\)-fibres it is identified with absolute Frobenius.
If \(N_2' \ge N_2\ge N_1\), if \(\varSigma '\) is a refinement of \(\varSigma \) and if \(\sigma \in \varSigma ^{{\text {ord}}}=(\varSigma ')^{{\text {ord}}}\) then the natural map
is etale in a neighbourhood of the \({\mathbb {F}}_p\)-fibre of \(\partial _{[\sigma ]} {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2'),\varSigma '}\) and so induces an isomorphism of the formal completions of these schemes along their \({\mathbb {F}}_p\)-fibres. We will denote this p-adic formal scheme
and will denote its reduced subscheme
For \(s>0\) we will write
and
The maps
are finite flat of degree \(p^{(2m+n)n[F^+: {\mathbb {Q}}]-s}\).
Then \(\partial \overline{{A}}_{n,U^p(N_1),\varSigma ^{{\text {ord}}}}^{(m),{{\text {ord}}}}\) is stratified by the \(\partial _{[\sigma ]} \overline{{A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1),\varSigma ^{{\text {ord}}}}\) with \([\sigma ]\) running over \({\mathcal {S}}(U^p(N_1),\varSigma ^{{\text {ord}}})^{{\text {ord}}}\). If \(\sigma \in \varSigma ^{{\text {ord}}}\) but \(\sigma \) is not contained in
then \(\partial _{[\sigma ]} \overline{{A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1),\varSigma ^{{\text {ord}}}}\) is irreducible. (Because \(\partial _{[\sigma ]} \overline{{A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1),\varSigma ^{{\text {ord}}}}\) is a toric variety over \({\mathbb {F}}_p\). It is presumably also true that \(\partial _{[\sigma ]} \overline{{A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1),\varSigma ^{{\text {ord}}}}\) is irreducible for any \(\sigma \), but to prove it one would need an irreducibility statement about the special fibre of a Shimura variety. In many cases such a theorem has been proved by Hida [33], but not in the full generality in which we are working here.)
We will write
an open subset of \(|{\mathcal {S}}(\partial \overline{{A}}_{n,U^p(N),\varSigma ^{{\text {ord}}}}^{(m),{{\text {ord}}}}-\partial _{ i+1} \overline{{A}}_{n,U^p(N),\varSigma ^{{\text {ord}}}}^{(m),{{\text {ord}}}})|\). Then there are natural surjections
which restrict to surjections
This gives rise to surjections
In the case \(n=i\) this is actually a homeomorphism
This is compatible with the identifications
6.4 Vector bundles
We will write \({\mathcal {I}}_{\partial X_{n,U}^{{\text {min}}}}\) (resp. \({\mathcal {I}}_{\partial X_{n,U,\varDelta }}\), resp. \({\mathcal {I}}_{\partial A_{U,\varSigma }^{(m)}}\)) for the ideal sheaf in \({\mathcal {O}}_{X_{n,U}^{{\text {min}}}}\) (resp. \({\mathcal {O}}_{X_{n,U,\varDelta }}\), resp. \({\mathcal {O}}_{A_{U,\varSigma }^{(m)}}\)) defining the boundary \(\partial X_{n,U}^{{\text {min}}}\) (resp. \({\partial X_{n,U,\varDelta }}\), resp. \({\partial A_{U,\varSigma }^{(m)}}\)). More generally we will write \({\mathcal {I}}_{\partial X_{n,U}^{{\text {min}}}\times {{\text {Spec}}\,}R_0}\) (resp. \({\mathcal {I}}_{\partial X_{n,U,\varDelta } \times {{\text {Spec}}\,}R_0}\), resp. \({\mathcal {I}}_{\partial A_{U,\varSigma }^{(m)} \times {{\text {Spec}}\,}R_0}\)) for the correspondingly defined sheaves on \(X_{n,U}^{{\text {min}}}\times {{\text {Spec}}\,}R_0\) (resp. \(X_{n,U,\varDelta }\times {{\text {Spec}}\,}R_0\), resp. \( A_{U,\varSigma }^{(m)}\times {{\text {Spec}}\,}R_0\)).
Lemma 5.1
Suppose that \(R_0\) is an irreducible, noetherian \({\mathbb {Q}}\)-algebra.
-
(1)
If \(i>0\) then
$$\begin{aligned} R^i\pi _{(U,\varSigma ),(U',\varSigma '),*} {\mathcal {O}}_{A^{(m)}_{n,U,\varSigma } \times {{\text {Spec}}\,}R_0}=(0) \end{aligned}$$and
$$\begin{aligned} R^i\pi _{(U,\varSigma ),(U',\varSigma '),*} {\mathcal {I}}_{\partial A^{(m)}_{n,U,\varSigma } \times {{\text {Spec}}\,}R_0}=(0). \end{aligned}$$ -
(2)
If \((U,\varSigma ) \ge (U',\varSigma ')\) and U is a normal subgroup of \(U'\), then the natural maps
$${\mathcal {O}}_{A^{(m)}_{U',\varSigma '} \times {{\text {Spec}}\,}R_0} \longrightarrow \left( \pi _{(U,\varSigma ),(U',\varSigma '),*}{\mathcal {O}}_{A^{(m)}_{U,\varSigma }\times {{\text {Spec}}\,}R_0}\right) ^{U'}$$and
$${\mathcal {I}}_{\partial A^{(m)}_{U',\varSigma '}\times {{\text {Spec}}\,}R_0} \longrightarrow \left( \pi _{(U,\varSigma ),(U',\varSigma '),*}{\mathcal {I}}_{\partial A^{(m)}_{U,\varSigma }\times {{\text {Spec}}\,}R_0}\right) ^{U'}$$are isomorphisms.
-
(3)
If \(U'\) is the image in \(G_n({\mathbb {A}}^\infty )\) of \(U \subset G_n^{(m)}({\mathbb {A}}^\infty )\) and if \(\varSigma \) and \(\varDelta \) are compatible, then
$$\begin{aligned} \pi _{A^{(m),{{\text {tor}}}}/X^{{\text {tor}}},*} {\mathcal {O}}_{A^{(m)}_{n,U,\varSigma }}={\mathcal {O}}_{X_{n,U',\varDelta }}. \end{aligned}$$
Proof
If \(\varSigma \) is \(U'\)-invariant the first two parts follow from Lemma 4.6. In the general case we choose \((U,\varSigma '') \ge (U,\varSigma )\) with \(\varSigma ''\) being \(U'\)-invariant, and apply the cases of the lemma already proved to the pairs \(((U,\varSigma ''),(U',\varSigma '))\) and \(((U,\varSigma ''),(U,\varSigma ))\).
The third part follows from Lemma 4.8. \(\square \)
Similarly we will write \({\mathcal {I}}_{\partial {\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}}\) (resp. \({\mathcal {I}}_{\partial {\mathcal {X}}_{n,U^p(N_1,N_2),\varDelta }^{{\text {ord}}}}\), resp. \({\mathcal {I}}_{\partial {\mathcal {A}}_{n,U^p(N_1,N_2),\varSigma }^{(m),{{\text {ord}}}}}\)) for the ideal sheaf in \({\mathcal {O}}_{{\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}}\) (resp. \({\mathcal {O}}_{{\mathcal {X}}_{n,U^p(N_1,N_2),\varDelta }^{{\text {ord}}}}\), resp. \({\mathcal {O}}_{{\mathcal {A}}_{n,U^p(N_1,N_2),\varSigma }^{(m),{{\text {ord}}}}}\)) defining the boundary \(\partial {\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}\) (resp. \({\partial {\mathcal {X}}_{n,U^p(N_1,N_2),\varDelta }^{{\text {ord}}}}\), resp. \({\partial {\mathcal {A}}_{n,U^p(N_1,N_2),\varSigma }^{(m),{{\text {ord}}}}}\)). More generally we will also write \({\mathcal {I}}_{\partial {\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}\times {{\text {Spec}}\,}R_0}\) (resp. \({\mathcal {I}}_{\partial {\mathcal {X}}_{n,U^p(N_1,N_2),\varDelta }^{{\text {ord}}}\times {{\text {Spec}}\,}R_0}\), resp. \({\mathcal {I}}_{\partial {\mathcal {A}}_{n,U^p(N_1,N_2),\varSigma }^{(m),{{\text {ord}}}}\times {{\text {Spec}}\,}R_0}\)) for the sheaf defined in the corresponding manner on the scheme \({\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}\times {{\text {Spec}}\,}R_0\) (resp. \({\mathcal {X}}_{n,U^p(N_1,N_2),\varDelta }^{{\text {ord}}}\times {{\text {Spec}}\,}R_0\), resp. \({\mathcal {A}}_{n,U^p(N_1,N_2),\varSigma }^{(m),{{\text {ord}}}}\times {{\text {Spec}}\,}R_0\)). The next lemma follows from Lemmas 4.6 and 4.12.
Lemma 5.2
Suppose that \(R_0\) is an irreducible, noetherian \({\mathbb {Z}}_{(p)}\)-algebra.
-
(1)
If \(i>0\) then
$$\begin{aligned} R^i\pi _{(U^p(N_1,N_2),\varSigma ),((U^p)'(N_1',N_2'),\varSigma '),*} {\mathcal {O}}_{{\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\varSigma }\times {{\text {Spec}}\,}R_0}=(0) \end{aligned}$$and
$$\begin{aligned} R^i\pi _{(U^p(N_1,N_2),\varSigma ),((U^p)'(N_1',N_2'),\varSigma '),*} {\mathcal {I}}_{\partial {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\varSigma } \times {{\text {Spec}}\,}R_0}=(0). \end{aligned}$$ -
(2)
If \((U^p(N_1,,N_2),\varSigma ) \ge ((U^p)'(N_1',N_2),\varSigma ')\) and \(U^p\) is a normal subgroup of \((U^p)'\), then the natural maps
$${\mathcal {O}}_{{\mathcal {A}}^{(m),{{\text {ord}}}}_{n,(U^p)'(N_1',N_2'),\varSigma '}\times {{\text {Spec}}\,}R_0} \!\!\! \rightarrow \left( \pi _{(U^p(N_1,N_2),\varSigma ),((U^p)'(N_1',N_2'),\varSigma '),*}{\mathcal {O}}_{{\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\varSigma } \times {{\text {Spec}}\,}R_0}\right) ^{(U^p)'(N_1')}$$and
$${\mathcal {I}}_{\partial {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,(U^p)'(N_1',N_2'),\varSigma '}\times {{\text {Spec}}\,}R_0} \!\!\! \rightarrow \left( \pi _{(U^p(N_1,N_2),\varSigma ),((U^p)'(N_1',N_2'),\varSigma '),*}{\mathcal {I}}_{\partial {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\varSigma }\times {{\text {Spec}}\,}R_0}\right) ^{(U^p)'(N_1')}$$are isomorphisms.
-
(3)
If \((U^p)'\) is the image in \(G_n({\mathbb {A}}^{\infty ,p})\) of \(U^p \subset G_n^{(m)}({\mathbb {A}}^{\infty ,p})\) and if \(\varSigma \) and \(\varDelta \) are compatible, then
$$\begin{aligned} \pi _{{\mathcal {A}}^{(m),{{\text {ord}}},{{\text {tor}}}}/X^{{{\text {ord}}},{{\text {tor}}}},*} {\mathcal {O}}_{{\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\varSigma }}={\mathcal {O}}_{{\mathcal {X}}_{n,(U^p)'(N_1,N_2),\varDelta }}. \end{aligned}$$
The pull-back by the identity section of \(\varOmega ^1_{A^{{\text {univ}}}_\varDelta /X_{n,U,\varDelta }}\) (resp. \(\varOmega ^1_{{\mathcal {A}}^{{\text {univ}}}_\varDelta /{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2),\varDelta }}\)) is a locally free sheaf, which is canonically independent of the choice of \(A^{{\text {univ}}}\) (resp. \({\mathcal {A}}^{{\text {univ}}}\)). We will denote it \(\varOmega _{n,U,\varDelta }\) (resp. \(\varOmega ^{{\text {ord}}}_{n,U^p(N_1,N_2),\varDelta }\)). If \(g \in G_n({\mathbb {A}}^\infty )\) (resp. \(g \in G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)) and \((U,\varDelta )g \ge (U',\varDelta ')\) (resp. \((U^p(N_1,N_2),\varDelta )g \ge ((U^p)'(N_1',N_2'),\varDelta ')\)) then there is a natural isomorphism
(resp.
This gives the inverse system \(\{ \varOmega _{n,U,\varDelta } \}\) (resp. \(\{ \varOmega ^{{\text {ord}}}_{n,U^p(N_1,N_2),\varDelta } \}\)) an action of \(G_n({\mathbb {A}}^\infty )\) (resp. \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)). There is also a natural map
There is a canonical identification
(resp.
We will write
(resp.
The pull-back of \(\varOmega _{n,U,\varDelta }\) to \(T^{\natural ,\wedge }_{n,(i),h U h^{-1} \cap P_{n,(i)}^+({\mathbb {A}}^{\infty }), \varDelta (h)_0}\) is canonically and equivariantly identified with the sheaf \(\widetilde{\varOmega }^{\natural }_{n,(i), h U h^{-1} \cap P_{n,(i)}^+({\mathbb {A}}^{\infty }),\varDelta (h)_0}\). Similarly the pull-back of \(\varOmega ^{{\text {ord}}}_{n,U^p(N_1,N_2),\varDelta }\) to \({\mathcal {T}}^{{{\text {ord}}},\natural ,\wedge }_{n,(i),(h U^p h^{-1} \cap P_{n,(i)}^+({\mathbb {A}}^{\infty ,p}))(N_1,N_2), \varDelta ^{{\text {ord}}}(h)_0}\) is canonically and equivariantly identified with the sheaf \(\widetilde{\varOmega }^{{{\text {ord}}},\natural }_{n,(i), (h U^p h^{-1} \cap P_{n,(i)}^+({\mathbb {A}}^{\infty ,p}))(N_1,N_2),\varDelta ^{{\text {ord}}}(h)_0}\). (See lemmas 1.3.2.41 and 5.2.4.38 of [44].)
We will write
(resp.
for the structure sheaf of \(X_{n,U,\varDelta }\) (resp. \({\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2),\varDelta }\)) with the \(G_n({\mathbb {A}}^\infty )\) (resp. \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\)) action twisted by \(||\nu ||\). If \(g \in G_n({\mathbb {A}}^\infty )\) (resp. \(g \in G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)) then the maps
(resp.
are isomorphisms.
The pull-back of \(\varXi _{n,U,\varDelta }\) to the formal scheme \(T^{\natural ,\wedge }_{n,(i),h U h^{-1} \cap P_{n,(i)}^+({\mathbb {A}}^{\infty }), \varDelta (h)_0}\) equals the pull-back of the sheaf \(\varXi ^{\natural }_{n,(i), h U h^{-1} \cap P_{n,(i)}^+({\mathbb {A}}^{\infty })}\) from \(X^{\natural }_{n,(i),h U h^{-1} \cap P_{n,(i)}^+({\mathbb {A}}^{\infty })}\). Similarly the pull-back of \(\varXi ^{{\text {ord}}}_{n,U^p(N_1,N_2),\varDelta }\) to \({\mathcal {T}}^{{{\text {ord}}},\natural ,\wedge }_{n,(i),(h U^p h^{-1} \cap P_{n,(i)}^+({\mathbb {A}}^{\infty ,p}))(N_1,N_2), \varDelta ^{{\text {ord}}}(h)_0}\) is naturally isomorphic to the pull-back of the sheaf \(\varXi ^{{{\text {ord}}},\natural }_{n,(i), (h U^p h^{-1} \cap P_{n,(i)}^+({\mathbb {A}}^{\infty ,p}))(N_1,N_2)}\) from the scheme \({\mathcal {X}}^{{{\text {ord}}},\natural }_{n,(i),(h U^p h^{-1} \cap P_{n,(i)}^+({\mathbb {A}}^{\infty ,p}))(N_1,N_2)}\) to \({\mathcal {T}}^{{{\text {ord}}},\natural ,\wedge }_{n,(i),(h U^p h^{-1} \cap P_{n,(i)}^+({\mathbb {A}}^{\infty ,p}))(N_1,N_2), \varDelta ^{{\text {ord}}}(h)_0}\).
Let \({\mathcal {E}}^{{\text {can}}}_{U,\varDelta }\) (resp. \({\mathcal {E}}_{U^p(N_1,N_2),\varDelta }^{{{\text {ord}}},{{\text {can}}}}\)) denote the principal \(L_{n,(n)}\)-bundle on \(X_{n,U,\varDelta }\) (resp. on \({\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2),\varDelta }\)) in the Zariski topology defined by setting, for \(W \subset X_{n,U,\varDelta }\) (resp. \({\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2),\varDelta }\)) a Zariski open, \({\mathcal {E}}^{{\text {can}}}_{U,\varDelta }(W)\) (resp. \({\mathcal {E}}^{{{\text {ord}}},{{\text {can}}}}_{U^p(N_1,N_2),\varDelta }(W)\)) to be the set of pairs \((\xi _0,\xi _1)\), where
(resp.
and
(resp.
We define the \(L_{n,(n)}\)-action on \({\mathcal {E}}^{{\text {can}}}_{U,\varDelta }\) (resp. \({\mathcal {E}}_{U^p(N_1,N_2),\varDelta }^{{{\text {ord}}},{{\text {can}}}}\)) by
The inverse system \(\{ {\mathcal {E}}^{{\text {can}}}_{U,\varDelta } \}\) (resp. \(\{ {\mathcal {E}}_{U^p(N_1,N_2),\varDelta }^{{{\text {ord}}},{{\text {can}}}} \}\)) has an action of \(G_n({\mathbb {A}}^\infty )\) (resp. \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)).
Suppose that \(R_0\) is a \({\mathbb {Q}}\)-algebra (resp. \({\mathbb {Z}}_{(p)}\)-algebra) and that \(\rho \) is a representation of \(L_{n,(n)}\) on a finite, locally free \(R_0\)-module \(W_\rho \). We define a locally free sheaf \({\mathcal {E}}^{{\text {can}}}_{U,\varDelta ,\rho }\) (resp. \({\mathcal {E}}_{U^p(N_1,N_2),\varDelta ,\rho }^{{{\text {ord}}},{{\text {can}}}}\)) over \(X_{n,U,\varDelta } \times {{\text {Spec}}\,}R_0\) (resp. \({\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2),\varDelta }\times {{\text {Spec}}\,}R_0\)) by setting \({\mathcal {E}}^{{\text {can}}}_{U,\varDelta ,\rho }(W)\) (resp. \({\mathcal {E}}_{U^p(N_1,N_2),\varDelta ,\rho }^{{{\text {ord}}},{{\text {can}}}}(W)\)) to be the set of \(L_{n,(n)}({\mathcal {O}}_W)\)-equivariant maps of Zariski sheaves of sets
(resp.
Then \(\{ {\mathcal {E}}^{{\text {can}}}_{U,\varDelta ,\rho } \}\) (resp. \(\{ {\mathcal {E}}^{{{\text {ord}}},{{\text {can}}}}_{U^p(N_1,N_2),\varDelta ,\rho } \}\)) is a system of locally free sheaves with \(G_n({\mathbb {A}}^\infty )\)-action (resp. \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-action) over the system of schemes \(\{ X_{n,U,\varDelta } \times {{\text {Spec}}\,}R_0\}\) (resp. \(\{ {\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2),\varDelta } \times {{\text {Spec}}\,}R_0 \} \)).
Note that
and
and
Similarly
and
and
Also note that the pull-back of \({\mathcal {E}}^{{\text {can}}}_{U,\varDelta ,\rho }\) (resp. \({\mathcal {E}}_{U^p(N_1,N_2),\varDelta ,\rho }^{{{\text {ord}}},{{\text {can}}}}\)) to \(X_{n,U} \times {{\text {Spec}}\,}R_0\) (resp. \({\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)} \times {{\text {Spec}}\,}R_0\)) is canonically identified with \({\mathcal {E}}_{U,\rho }\) (resp. \({\mathcal {E}}_{U^p(N_1,N_2),\rho }^{{{\text {ord}}}}\)). These identifications are \(G_n({\mathbb {A}}^\infty )\) (resp. \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)) equivariant.
Moreover note that the pull-back of \({\mathcal {E}}^{{\text {can}}}_{U,\varDelta ,\rho }\) to \(T^{\natural ,\wedge }_{n,(i),h U h^{-1} \cap P_{n,(i)}^+({\mathbb {A}}^{\infty }), \varDelta (h)_0}\) is canonically and equivariantly identified with the sheaf \({\mathcal {E}}^{\natural }_{n,(i), h U h^{-1} \cap P_{n,(i)}^+({\mathbb {A}}^{\infty }),\varDelta (h)_0,\rho |_{R_{n,(n),(i)}}}\). Similarly the pull-back of \({\mathcal {E}}^{{{\text {ord}}},{{\text {can}}}}_{U^p(N_1,N_2),\varDelta ,\rho }\) to \({\mathcal {T}}^{{{\text {ord}}},\natural ,\wedge }_{n,(i),(h U^p h^{-1} \cap P_{n,(i)}^+({\mathbb {A}}^{\infty ,p}))(N_1,N_2), \varDelta ^{{\text {ord}}}(h)_0}\) is canonically and equivariantly identified with
Set
and
Then \(\{ {\mathcal {E}}^{{\text {sub}}}_{U,\varDelta ,\rho } \}\) (resp. \(\{ {\mathcal {E}}^{{{\text {ord}}},{{\text {sub}}}}_{U^p(N_1,N_2),\varDelta ,\rho } \}\)) is also a system of locally free sheaves with \(G_n({\mathbb {A}}^\infty )\)-action (resp. \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-action) over the systems of schemes \(\{ X_{n,U,\varDelta } \times {{\text {Spec}}\,}R_0\}\) (resp. \(\{ {\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2),\varDelta } \times {{\text {Spec}}\,}R_0\} \)).
Lemma 5.3
-
(1)
If \(g \in G_n({\mathbb {A}}^\infty )\) (resp. \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)) and \(g: X_{n,U,\varDelta } \rightarrow X_{n,U',\varDelta '}\) (resp. \(g: {\mathcal {X}}_{n,U^p(N_1,N_2),\varDelta }^{{\text {ord}}}\rightarrow {\mathcal {X}}_{n,(U^p)'(N_1',N_2'),\varDelta '}^{{\text {ord}}}\)) then
$$\begin{aligned} g^* {\mathcal {E}}^{{\text {can}}}_{U',\varDelta ',\rho } \mathop {\longrightarrow }\limits ^{\sim }{\mathcal {E}}^{{\text {can}}}_{U,\varDelta ,\rho } \end{aligned}$$(resp.
$$\begin{aligned} \left. g^* {\mathcal {E}}^{{{\text {ord}}},{{\text {can}}}}_{(U^p)'(N_1',N_2'),\varDelta ',\rho } \mathop {\longrightarrow }\limits ^{\sim }{\mathcal {E}}^{{{\text {ord}}},{{\text {can}}}}_{U^p(N_1,N_2),\varDelta ,\rho } \right) . \end{aligned}$$ -
(2)
If \(i>0\) then
$$\begin{aligned} R^i\pi _{(U,\varDelta ),(U',\varDelta '),*} {\mathcal {E}}^{{\text {can}}}_{U,\varDelta ,\rho }=(0) \end{aligned}$$and
$$\begin{aligned} R^i\pi _{(U,\varDelta ),(U',\varDelta '),*} {\mathcal {E}}_{U,\varDelta ,\rho }^{{\text {sub}}}=(0). \end{aligned}$$Similarly, for \(i>0\) we have
$$\begin{aligned} R^i\pi _{(U^p(N_1,N_2),\varDelta ),((U^p)'(N_1',N_2'),\varDelta '),*} {\mathcal {E}}_{U^p(N_1,N_2),\varDelta ,\rho }^{{{\text {ord}}},{{\text {can}}}}=(0) \end{aligned}$$and
$$\begin{aligned} R^i\pi _{(U^p(N_1,N_2),\varDelta ),((U^p)'(N_1',N_2'),\varDelta '),*} {\mathcal {E}}_{U^p(N_1,N_2),\varDelta ,\rho }^{{{\text {ord}}},{{\text {sub}}}}=(0). \end{aligned}$$ -
(3)
$$ \left( \lim _{\rightarrow (U,\varDelta )} \pi _{(U,\varDelta ),(U',\varDelta '),*} {\mathcal {E}}^{{\text {can}}}_{U,\varDelta ,\rho }\right) ^{U'}={\mathcal {E}}_{U',\varDelta ',\rho }$$
and
$$\left( \lim _{\rightarrow (U,\varDelta )} \pi _{(U,\varDelta ),(U',\varDelta '),*} {\mathcal {E}}_{U,\varDelta ,\rho }^{{\text {sub}}}\right) ^{U'}={\mathcal {E}}_{U',\varDelta ',\rho }^{{\text {sub}}}$$and
$$\begin{array}{l} {\mathcal {E}}_{(U^p)'(N_1',N_2),\varDelta ',\rho }^{{{\text {ord}}},{{\text {can}}}}\\ \quad =\left( \lim _{\rightarrow (U^p(N_1,N_2),\varDelta )} \pi _{(U^p(N_1,N_2),\varDelta ),((U^p)'(N_1',N_2),\varDelta '),*} {\mathcal {E}}_{U^p(N_1,N_2),\varDelta ,\rho }^{{{\text {ord}}},{{\text {can}}}}\right) ^{(U^p)'(N_1')} \end{array}$$and
$$\begin{array}{l} {\mathcal {E}}_{(U^p)'(N_1',N_2),\varDelta ',\rho }^{{{\text {ord}}},{{\text {sub}}}}\\ \quad = \left( \lim _{\rightarrow (U^p(N_1,N_2),\varDelta )} \pi _{(U^p(N_1,N_2),\varDelta ),((U^p)'(N_1',N_2),\varDelta '),*} {\mathcal {E}}_{U^p(N_1,N_2),\varDelta ,\rho }^{{{\text {ord}}},{{\text {sub}}}}\right) ^{(U^p)'(N_1')} . \end{array}$$
Proof
The first part follows easily from the corresponding facts for \(\varOmega _{n,U,\varDelta }\) and \(\varXi _{n,U,\varDelta }\) (resp. \(\varOmega ^{{\text {ord}}}_{n,U^p(N_1,N_2),\varDelta }\) and \(\varXi _{n,U^p(N_1,N_2),\varDelta }^{{\text {ord}}}\)). The second and third parts follow from the first part and parts 1 and 2 of Lemma 5.1 (resp. Lemma 5.2). \(\square \)
We next deduce our first main observation.
Theorem 5.4
If \(i>0\) and U is neat then \(R^i\pi _{X^{{\text {tor}}}/X^{{\text {min}}},*} {\mathcal {E}}_{U,\varDelta ,\rho }^{{\text {sub}}}=(0)\).
Similarly if \(i>0\) and \(U^p\) is neat then \(R^i\pi _{{\mathcal {X}}^{{{\text {ord}}},{{\text {tor}}}}/{\mathcal {X}}^{{{\text {ord}}},{{\text {min}}}},*} {\mathcal {E}}_{U^p(N_1,N_2),\varDelta ,\rho }^{{{\text {ord}}},{{\text {sub}}}} =(0)\).
Proof
The argument is the same in both cases, so we explain the argument only in the first case. Write \(X^\wedge _{n,U,\varDelta ,i,h}\) (resp. \(X^{{{\text {min}}},\wedge }_{n,U,i,h}\)) for the open and closed subset of \(X^\wedge _{n,U,\varDelta ,i}\) (resp. \(X^{{{\text {min}}},\wedge }_{n,U,i}\)) corresponding to \(T^{\natural ,\wedge }_{n,(i),h U h^{-1} \cap P^+_{n,(i)}({\mathbb {A}}^\infty ), \varDelta (h)}\) (resp. \(X^\natural _{n,(i),h U h^{-1} \cap P^+_{n,(i)}({\mathbb {A}}^\infty )}\)). (Recall that \(X^\wedge _{n,U,\varDelta ,i}\) is the completion of a smooth toroidal compactification of the Shimura variety \(X_{n,U}\) along the locally closed subspace of the boundary corresponding to the parabolic subgroup \(P^+_{n,(i)} \subset G_n\). The formal scheme \(X^{{{\text {min}}},\wedge }_{n,U,i}\) is the completion of the minimal (Baily–Borel) compactification of the same Shimura variety along the locally closed subspace of the boundary corresponding to the same parabolic. Each of these formal schemes is a disjoint union of subformal schemes indexed by certain elements \(h \in G_n({\mathbb {A}}^\infty )\).)
We have maps of locally ringed spaces
(Recall that \(T^{\natural ,\wedge }_{n,(i),h U h^{-1} \cap P^+_{n,(i)}({\mathbb {A}}^\infty ), \varDelta (h)}\) is a formal local model for the boundary of the toroidal compactification. It is the quotient by a discrete group of the formal completion of a toroidal embedding over a principal homogeneous space for an abelian scheme over a disjoint union of smaller Shimura varieties. The scheme \(X^\natural _{n,(i),h U h^{-1} \cap P^+_{n,(i)}({\mathbb {A}}^\infty )}\) is a disjoint union of smaller Shimura varieties, and also a locally closed subscheme of the boundary of the minimal compactification of \(X_{n,U}\).)
This diagram is commutative as a diagram of topological spaces (but not of locally ringed spaces) and the lower horizontal map is an isomorphism on the underlying topological spaces. It suffices to show that the higher direct images from the topological space \(T^{\natural ,\wedge }_{n,(i),h U h^{-1} \cap P^+_{n,(i)}({\mathbb {A}}^\infty ), \varDelta (h)}\) to the topological space \(X^\natural _{n,(i),h U h^{-1} \cap P^+_{n,(i)}({\mathbb {A}}^\infty )}\) of the pull-back of \({\mathcal {E}}_{U,\varDelta ,\rho }^{{\text {sub}}}\) vanishes. The theorem follows from Corollary 4.14. \(\square \)
We set
(resp.
a coherent sheaf on \(X_{n,U}^{{\text {min}}}\times {{\text {Spec}}\,}R_0\) (resp. \({\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}} \times {{\text {Spec}}\,}R_0\)). (Note that we do not expect these sheaves to be locally free in general.) These definitions are independent of \(\varDelta \). Note that
and
We will let \({\mathcal {E}}^{{{\text {ord}}},{{\text {can}}}}_{U^p(N),\varDelta ^{{\text {ord}}},\rho }\) (resp. \({\mathcal {E}}^{{{\text {ord}}},{{\text {sub}}}}_{U^p(N),\varDelta ^{{\text {ord}}},\rho }\), resp. \({\mathcal {E}}^{{{\text {ord}}},{{\text {sub}}}}_{U^p(N),\rho }\)) denote the pull-back of \({\mathcal {E}}^{{{\text {ord}}},{{\text {can}}}}_{U^p(N,N'),\varDelta ,\rho }\) (resp. \({\mathcal {E}}^{{{\text {ord}}},{{\text {sub}}}}_{U^p(N,N'),\varDelta ,\rho }\), resp. \({\mathcal {E}}^{{{\text {ord}}},{{\text {sub}}}}_{U^p(N,N'),\rho }\)) to \({\mathfrak {X}}^{{{\text {ord}}}}_{U^p(N),\varDelta ^{{\text {ord}}}}\) (resp. \({\mathfrak {X}}^{{{\text {ord}}}}_{U^p(N),\varDelta ^{{\text {ord}}}}\), resp. \({\mathfrak {X}}^{{{\text {ord}}},{{\text {min}}}}_{U^p(N)}\)). It is independent of the choice of \(N'\) and \(\varDelta \).
If \(\rho \) is a representation of \(L_{n,(n)}\) on a finite \({\mathbb {Q}}\)-vector space, we will set
It is an admissible \(G_n({\mathbb {A}}^\infty )\)-module with
Similarly, if \(\rho \) is a representation of \(L_{n,(n)}\) on a finite free \({\mathbb {Z}}_{(p)}\)-module, we will set
and
They are smooth \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-modules with
and
(Use Lemma 5.3.) Note that there is a \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-equivariant embedding
Finally set
a smooth representation of \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\).
We record the following result from [44].
Lemma 5.5
If \(\rho \) is a representation of \(L_{n,(n)}\) on a finite locally free \({\mathbb {Z}}_{(p)}\)-module then there is a unique system \(\{ {\mathcal {E}}^{{\text {sub}}}_{U^p(N_1,N_2),\rho } \}\) of \({\mathcal {O}}_{{\mathcal {X}}^{{\text {min}}}_{n,U^p(N_1,N_2)}}\)-torsion free coherent sheaves with \(G_n({\mathbb {A}}^{\infty })^{{{\text {ord}}},\times }\)-action over \(\{ {\mathcal {X}}^{{\text {min}}}_{n,U^p(N_1,N_2)} \}\) with the following properties.
-
(1)
\(\{ {\mathcal {E}}_{U^p(N_1,N_2),\rho }^{{\text {sub}}}\}\) pulls back to \(\{ {\mathcal {E}}_{U^p(N_1,N_2),\rho \otimes _{{\mathbb {Z}}_{(p)}} {\mathbb {Q}}}^{{\text {sub}}}\}\) on \(\{ X_{n,U^p(N_1,N_2)}^{{\text {min}}}\}\);
-
(2)
\(\{ {\mathcal {E}}_{U^p(N_1,N_2),\rho }^{{\text {sub}}}\}\) pulls back to \(\{ {\mathcal {E}}_{U^p(N_1,N_2),\rho }^{{{\text {ord}}},{{\text {sub}}}} \}\) on \(\{ {\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}\}\);
-
(3)
if \(U^p\) is a normal subgroup of \((U^p)'\) and if \(g \in (U^p)'(N_1',N_2)\) then
$$\begin{aligned} g: g^*{\mathcal {E}}_{U^p(N_1,N_2),\rho }^{{\text {sub}}}\mathop {\rightarrow }\limits ^{\sim }{\mathcal {E}}_{U^p(N_1,N_2),\rho }^{{\text {sub}}}; \end{aligned}$$ -
(4)
if \(U^p\) is a normal subgroup of \((U^p)'\) then
$${\mathcal {E}}_{(U^p)'(N_1,N_2),\rho }^{{\text {sub}}}\mathop {\rightarrow }\limits ^{\sim }\left( \pi _{U^p(N_1',N_2),(U^p)'(N_1,N_2),*} {\mathcal {E}}_{U^p(N_1',N_2),\rho }^{{\text {sub}}}\right) ^{(U^p)'(N_1,N_2)};$$ -
(5)
\(\{ {\mathcal {E}}_{U^p(N_1,N_2),\rho \otimes \wedge ^{n[F: {\mathbb {Q}}]} {{\text {Std}}}^\vee }^{{\text {sub}}}\} \cong \{ \omega _{U^p(N_1,N_2)}\otimes {\mathcal {E}}_{U^p(N_1,N_2),\rho }^{{\text {sub}}}\}\).
Proof
For the definition of \({\mathcal {E}}^{{\text {sub}}}_{U^p(N_1,N_2),\rho }\) see definition 8.3.5.1 of [44]. For the \({\mathcal {O}}_{{\mathcal {X}}^{{\text {min}}}_{n,U^p(N_1,N_2)}}\)-torsion freeness see corollary 8.3.5.8 of [44]. For the \(G_n({\mathbb {A}}^{\infty })^{{{\text {ord}}},\times }\)-action see corollary 8.3.6.5 of [44]. For part one of the lemma see lemma 8.3.5.2 of [44]. For the second part see corollary 8.3.5.4 of [44]. The third part is clear. For the fourth part see proposition 8.3.6.9 of [44], and for the final part see lemma 8.3.5.10 of [44]. \(\square \)
We will write \(\varOmega ^i_{A_{n,U,\varSigma }^{(m)}}(\log \infty )\) (resp. \(\varOmega ^i_{A_{n,U,\varSigma }^{(m)}/X_{n,U',\varDelta }}(\log \infty )\)) as shorthand for the sheaf \(\varOmega ^i_{A_{n,U,\varSigma }^{(m)}/{{\text {Spec}}\,}{\mathbb {Q}}}(\log {\mathcal {M}}_\varSigma )\) (resp. \(\varOmega ^i_{A_{n,U,\varSigma }^{(m)}/X_{n,U',\varDelta }}(\log {\mathcal {M}}_\varSigma /{\mathcal {M}}_\varDelta )\)). Then the collection \(\{ \varOmega ^1_{A_{n,U,\varSigma }^{(m)}}(\log \infty )\}\) (resp. \(\{\varOmega ^1_{A_{n,U,\varSigma }^{(m)}/X_{n,U',\varDelta }}(\log \infty )\}\)) is a system of locally free sheaves (for the Zariski topology) with \(G_n^{(m)}({\mathbb {A}}^\infty )\)-action.
There are natural differentials
(resp.
making \(\varOmega ^\bullet _{A_{n,U,\varSigma }^{(m)}}(\log \infty )\) (resp. \(\varOmega ^\bullet _{A_{n,U,\varSigma }^{(m)}/X_{n,U',\varDelta }}(\log \infty )\)) a complex. The tensor product \(\varOmega ^\bullet _{A_{n,U,\varSigma }^{(m)}}(\log \infty ) \otimes {\mathcal {I}}_{\partial A_{n,U,\varSigma }^{(m)}}\) (resp. \(\varOmega ^\bullet _{A_{n,U,\varSigma }^{(m)}/X_{n,U',\varDelta }}(\log \infty ) \otimes {\mathcal {I}}_{\partial A_{n,U,\varSigma }^{(m)}}\)) is a subcomplex.
Lemma 5.6
-
(1)
If \((U,\varSigma ) \ge (U',\varDelta ) \ge (U'',\varDelta ')\) then the natural morphism
$$\varOmega ^1_{A_{n,U,\varSigma }^{(m)}/X_{n,U'',\varDelta '}}(\log \infty ) \mathop {\rightarrow }\limits ^{\sim }\varOmega ^1_{A_{n,U,\varSigma }^{(m)}/X_{n,U',\varDelta }}(\log \infty )$$is an isomorphism, so we will simply write \(\varOmega ^1_{A_{n,U,\varSigma }^{(m)}/X}(\log \infty )\) for this sheaf.
-
(2)
If \((U',\varSigma ')\ge (U,\varSigma )\) then
$$\pi _{(U',\varSigma '),(U,\varSigma )}^* \varOmega ^1_{A^{(m)}_{n,U,\varSigma }}(\log \infty ) \mathop {\rightarrow }\limits ^{\sim }\varOmega ^1_{A^{(m)}_{n,U',\varSigma '}}(\log \infty )$$and
$$\pi _{(U',\varSigma '),(U,\varSigma )}^* \varOmega ^1_{A^{(m)}_{n,U,\varSigma }/X}(\log \infty ) \mathop {\rightarrow }\limits ^{\sim }\varOmega ^1_{A^{(m)}_{n,U',\varSigma '}/X}(\log \infty ).$$ -
(3)
If \((U,\varSigma ) \ge (U',\varDelta )\) then there is an exact sequence
$$\begin{aligned} (0) \rightarrow \pi _{(U,\varSigma ),(U',\varDelta )}^* \varOmega ^1_{X_{n,U',\varDelta }}(\log \infty ) \rightarrow \varOmega ^1_{A^{(m)}_{n,U,\varSigma }}(\log \infty ) \rightarrow \varOmega ^1_{A^{(m)}_{n,U,\varSigma }/X}(\log \infty ) \rightarrow (0). \end{aligned}$$ -
(4)
Suppose that \((U_1,\varSigma _1) \ge (U_2,\varSigma _2) \ge (U',\varDelta )\) and that \(U'\) is the image of both \(U_1\) and \(U_2\) in \(G_n({\mathbb {A}}^\infty )\). Then the natural maps
$$\begin{aligned} R^i\pi _{A^{(m),{{\text {tor}}}}/X^{{\text {tor}}},*} \varOmega ^j_{A_{n,U_2,\varSigma _2}^{(m)}/X}(\log \infty ) \longrightarrow R^i\pi _{A^{(m),{{\text {tor}}}}/X^{{\text {tor}}},*} \varOmega ^j_{A_{n,U_1,\varSigma _1}^{(m)}/X}(\log \infty ) \end{aligned}$$and
$$\begin{array}{l} R^i\pi _{A^{(m),{{\text {tor}}}}/X^{{\text {tor}}},*} \left( \varOmega ^j_{A_{n,U_2,\varSigma _2}^{(m)}/X}(\log \infty ) \otimes {\mathcal {I}}_{\partial A^{(m)}_{n,U_2,\varSigma _2}}\right) \\ \quad \longrightarrow R^i\pi _{A^{(m),{{\text {tor}}}}/X^{{\text {tor}}},*} \left( \varOmega ^j_{A_{n,U_1,\varSigma _1}^{(m)}/X}(\log \infty ) \otimes {\mathcal {I}}_{\partial A^{(m)}_{n,U_1,\varSigma _1}}\right) \end{array}$$on \(X_{n,U',\varDelta }\) are isomorphisms. We will write simply
$$\begin{aligned} \left( R^i\pi _*\varOmega ^j_{A^{(m)}/X}(\log \infty )\right) _{(U',\varDelta )} \end{aligned}$$and
$$\left( R^i\pi _*\left( \varOmega ^j_{A^{(m)}/X}(\log \infty ) \otimes {\mathcal {I}}_{\partial A^{(m)}}\right) \right) _{(U',\varDelta )}$$for these sheaves.
-
(5)
\(\{ (R^i\pi _*\varOmega ^j_{A^{(m)}/X}(\log \infty ))_{(U',\varDelta )} \}\) and \(\{ (R^i\pi _*(\varOmega ^j_{A^{(m)}/X}(\log \infty ) \otimes {\mathcal {I}}_{\partial A^{(m)}}))_{(U',\varDelta )} \}\) are systems of coherent sheaves with \(G_n^{(m)}({\mathbb {A}}^\infty )\)-action over \(\{ X_{n,U',\varDelta } \}\). Moreover the maps
$$g: g^* \left( R^i\pi _*\varOmega ^j_{A^{(m)}/X}(\log \infty )\right) _{(U',\varDelta )} \longrightarrow \left( R^i\pi _*\varOmega ^j_{A^{(m)}/X}(\log \infty )\right) _{(U'',\varDelta ')}$$are isomorphisms.
-
(6)
The \(G_n^{(m)}({\mathbb {A}}^\infty )\)-actions on both the systems
$$\left\{ \left( R^i\pi _*\varOmega ^j_{A^{(m)}/X}(\log \infty )\right) _{(U',\varDelta )} \right\} $$and
$$\left\{ \left( R^i\pi _*\left( \varOmega ^j_{A^{(m)}/X}(\log \infty ) \otimes {\mathcal {I}}_{\partial A^{(m)}}\right) \right) _{(U',\varDelta )} \right\} $$factor through \(G_n({\mathbb {A}}^\infty )\).
-
(7)
The pull-back of \((\pi _* \varOmega ^1_{A^{(m)}/X}(\log \infty ))_{(U,\varDelta )}\) to \(T^{\natural ,\wedge }_{n,(i),h U h^{-1} \cap P_{n,(i)}^+({\mathbb {A}}^\infty ), \varDelta (h)_0}\) is isomorphic to
$$\pi _{(U',\varSigma _0),\left( h U h^{-1} \cap P_{n,(i)}^+({\mathbb {A}}^\infty ), \varDelta (h)_0\right) ,*} \varOmega ^1_{T^{(m),\natural ,\wedge }_{n,(i),U',\varSigma _0}/T^{\natural ,\wedge }_{n,(i),h U h^{-1} \cap P_{n,(i)}^+({\mathbb {A}}^\infty ), \varDelta (h)_0}} (\log \infty )$$for some \(U'\) and \(\varSigma _0\).
Proof
This follows from the properties of log differentials for log smooth maps (see Sect. 2.2). For part 4 we also use Lemma 5.1. For part 6 we also use the discussion of Sect. 3.4 and a density argument. \(\square \)
The next lemma follows from Lemma 4.10.
Lemma 5.7
-
(1)
The natural maps
$$\begin{aligned} \left( \pi _* \varOmega ^1_{A^{(m)}/X}(\log \infty )\right) _{(U',\varDelta )} \otimes _{{\mathcal {O}}_{X_{n,U',\varDelta }}} {\mathcal {O}}_{A^{(m)}_{n,U,\varSigma }} \longrightarrow \varOmega ^1_{A^{(m)}_{n,U,\varSigma } /X}(\log \infty ) \end{aligned}$$are \(G^{(m)}_n({\mathbb {A}}^\infty )\)-equivariant isomorphisms.
-
(2)
The natural maps
$$ \left( \wedge ^j \left( \pi _*\varOmega ^1_{A^{(m)}/X}(\log \infty )\right) _{(U',\varDelta )}\right) \otimes (R^i\pi _*{\mathcal {O}}_{A^{(m)}})_{(U',\varDelta )} \longrightarrow \left( R^i\pi _{*} \varOmega ^j_{A^{(m)}/X}(\log \infty )\right) _{(U',\varDelta )}$$and
$$\begin{array}{l} \left( \wedge ^j \left( \pi _*\varOmega ^1_{A^{(m)}/X}(\log \infty )\right) _{(U',\varDelta )}\right) \otimes \left( R^i\pi _*{\mathcal {O}}_{A^{(m)}}\right) _{(U',\varDelta )}\otimes {\mathcal {I}}_{\partial X_{n,U',\varDelta }}\\ \quad \longrightarrow \left( R^i\pi _{*}\left( \varOmega ^j_{A^{(m)}/X}(\log \infty ) \otimes {\mathcal {I}}_{\partial A^{(m)}}\right) \right) _{(U',\varDelta )} \end{array}$$are \(G_n({\mathbb {A}}^\infty )\)-equivariant isomorphisms.
-
(3)
\((\pi _* \varOmega ^1_{A^{(m)}/X}(\log \infty ))_{(U,\varDelta )}\) is a flat coherent \({\mathcal {O}}_{X_{n,U,\varDelta }}\)-module, and hence locally free of finite rank.
Next we record some results from [42, 44].
Lemma 5.8
-
(1)
There are natural \(G_n({\mathbb {A}}^\infty )\)-equivariant isomorphisms
$${{\text {Hom}}}_F\left( F^m , \varOmega _{n,U',\varDelta }\right) \mathop {\longrightarrow }\limits ^{\sim }\left( \pi _* \varOmega ^1_{A_n^{(m)}/X_n}(\log \infty )\right) _{(U',\varDelta )}.$$ -
(2)
The cup product maps
$$\wedge ^i \left( R^1\pi _* {\mathcal {O}}_{A^{(m)}}\right) _{(U',\varDelta )} \longrightarrow \left( R^i\pi _* {\mathcal {O}}_{A^{(m)}}\right) _{(U',\varDelta )}$$are \(G_n({\mathbb {A}}^\infty )\)-equivariant isomorphisms.
-
(3)
There is a unique embedding
$$\varXi _{n,U',\varDelta } \hookrightarrow \left( R^1\pi _* \varOmega ^1_{A^{(m)}/X}(\log \infty )\right) _{(U',\varDelta )}$$extending
$$\begin{aligned} \varXi _{n,U'} \hookrightarrow \left( R^1\pi _* \varOmega ^1_{A^{(m)}/X}\right) _{U'}. \end{aligned}$$It is \(G_n({\mathbb {A}}^\infty )\)-equivariant.
-
(4)
The composite maps
$$\begin{array}{rl} &{} {{\text {Hom}}}\left( \left( \pi _* \varOmega ^1_{A_n^{(m)}/X_n}(\log \infty )\right) _{(U',\varDelta )}, \varXi _{n,U',\varDelta }\right) \\ &{}\quad \longrightarrow {{\text {Hom}}}\left( \left( \pi _* \varOmega ^1_{A_n^{(m)}/X_n}(\log \infty )\right) _{(U',\varDelta )}, \right. \\ &{}\quad \quad \left. \left( \pi _* \varOmega ^1_{A_n^{(m)}/X_n}(\log \infty )\right) _{(U',\varDelta )} \otimes \left( R^1\pi _* {\mathcal {O}}_{A^{(m)}}\right) _{(U',\varDelta )}\right) \\ &{}\quad \mathop {\longrightarrow }\limits ^{{{\text {tr}}}} (R^1\pi _* {\mathcal {O}}_{A^{(m)}})_{(U',\varDelta )}\end{array}$$are \(G_n({\mathbb {A}}^\infty )\)-equivariant isomorphisms.
-
(5)
The boundary maps
$$\begin{array}{rcl} \varOmega _{n,U',\varDelta } &{}\longrightarrow &{}R^1\pi _{A^{(1)}/X,*} \left( \pi _{A^{(1)}/X}^* \varOmega ^1_{X_{n,U',\varDelta }}(\log \infty )\right) \\ &{} \cong &{}\varOmega ^1_{X_{n,U',\varDelta }}(\log \infty ) \otimes {{\text {Hom}}}\left( \varOmega _{n,U',\varDelta }, \varXi _{n,U',\varDelta }\right) \end{array}$$associated to the short exact sequence of part 3 of Lemma 5.6, give rise to isomorphisms
$$\begin{aligned} S(\varOmega _{n,U',\varDelta }) \mathop {\longrightarrow }\limits ^{\sim }\varOmega ^1_{X_{n,U',\varDelta }}(\log \infty ) \otimes \varXi _{n,U',\varDelta }. \end{aligned}$$ -
(6)
There are \(G_n({\mathbb {A}}^{\infty ,p} \times {\mathbb {Z}}_p)\)-equivariant identifications between the pull-back of \(\omega _U\) from \({\mathcal {X}}^{{\text {min}}}_{n,U}\) to \(X_{n,U,\varDelta }\) and \(\omega _{U,\varDelta }\).
Proof
For the first four parts see theorem 2.15 and proposition 6.9 of [42] and theorem 1.3.3.15 of [44]. For the fifth part see theorem 1.3.1.3(4) of [44]. For the sixth part see propositions 2.2.1.2 and 2.2.3.1 of [44]. \(\square \)
Corollary 5.9
There are equivariant isomorphisms \({\mathcal {E}}_{U,\varDelta ,{{\text {KS}}}}^{{\text {can}}}\cong \varOmega ^1_{X_{n,U,\varDelta }}(\log \infty )\). (See Sect. 1.2 for the definition of the representation \({{\text {KS}}}\).)
Lemma 5.10
Suppose that U is a neat open compact subgroup of \(G^{(m)}_n({\mathbb {A}}^\infty )\) with image \(U'\) in \(G_n({\mathbb {A}}^\infty )\). The coherent sheaf \(\varOmega ^r_{A^{(m)}_{n,U,\varSigma }}(\log \infty )\) admits a decreasing filtration by local direct summands \({{\text {Fil}}}^j \varOmega ^r_{A^{(m)}_{n,U,\varSigma }}(\log \infty )\) with
-
\({{\text {Fil}}}^0 \varOmega ^r_{A^{(m)}_{n,U,\varSigma }}(\log \infty )=\varOmega ^r_{A^{(m)}_{n,U,\varSigma }}(\log \infty )\),
-
\({{\text {Fil}}}^{r+1} \varOmega ^r_{A^{(m)}_{n,U,\varSigma }}(\log \infty )=(0)\),
-
and \({{\text {gr}}}^j \varOmega ^r_{A^{(m)}_{n,U,\varSigma }}(\log \infty ) \cong (\pi _{A^{(m),{{\text {tor}}}}/X^{{\text {tor}}}}^* \varOmega ^j_{X_{n,U',\varDelta }}(\log \infty ))\otimes \varOmega ^{r-j}_{A_{n,U,\varSigma }^{(m)}/X}(\log \infty )\).
This filtration is \(G_n({\mathbb {A}}^\infty )\)-equivariant.
Moreover there are representations \(\rho _{m,r}^{i,j}\) of \(L_{n,(n)}\) such that there are \(G_n({\mathbb {A}}^\infty )\)-equivariant isomorphisms
Thus there is a spectral sequence with first page
This spectral sequence is \(G_n({\mathbb {A}}^\infty )\)-equivariant.
Proof
Using part 2 of Corollary 5.6 and parts 1 and 2 of Lemma 5.1, we may reduce to the case that there is a cone decomposition \(\varDelta \) compatible with \(\varSigma \). The first assertion now follows from part 3 of Lemma 5.6.
For the second assertion, note that by Lemma 5.7 we have that
Combining this with parts 1, 2, 4 and 5 of Lemma 5.8 we find representations \(\rho _{m,r}^{i,j}\) such that there are \(G_n({\mathbb {A}}^\infty )\)-equivariant isomorphisms
The second assertion now follows from Theorem 5.4.
The third assertion follows from the first two. \(\square \)
6.5 Connection to the complex theory
Lemma 5.11
Suppose that
satisfies
for all \(\tau \in {{\text {Hom}}}(F,{\mathbb {C}})\). Then \(H^0(X^{{\text {min}}},{\mathcal {E}}_{\rho _{(n),b}}^{{\text {sub}}})\) is a semi-simple \(G_n({\mathbb {A}}^\infty )\)-module. If \(\pi \) is an irreducible subquotient of \(H^0(X^{{\text {min}}},{\mathcal {E}}_{\rho _{(n),b}}^{{\text {sub}}})\), then \(\pi \) is the finite part of a cohomological, cuspidal automorphic representation of \(G_n({\mathbb {A}})\).
Proof
According to proposition 5.4.2 and lemma 5.2.3 of [28] and theorems 4.1.1, 5.1.1 and 5.2.12 of [43] we have an isomorphism
where \(\varPi \) runs over cuspidal automorphic representations of \(G_n({\mathbb {A}})\) taken with their multiplicity in the space of cuspidal automorphic forms.
Thus \(\pi \cong \varPi ^\infty \) for some cuspidal automorphic representation \(\varPi \) of \(G_n({\mathbb {A}})\) with
It follows from theorem 2.6 of [21] that the Harish-Chandra parameter of the infinitesimal character of \(\varPi _\infty \) equals
As we have assumed that
we see that \(\varPi _\infty \) has the same infinitesimal character as \(\rho _{b - 2(\varrho _n-\varrho _{n,(n)})}^\vee \). Moreover proposition 4.5 of [28] tells us that
We deduce that
However \(\rho _{(n),-2(\varrho _n-\varrho _{n,(n)})}\) is the representation of \(U_{n,\infty }^0A_n({\mathbb {R}})^0\) on \(\wedge ^{[F^+: {\mathbb {Q}}]n^2} {\mathfrak {p}}^+\). Thus
Proposition II.3.1 of [15] then tells us that
and the lemma follows. \(\square \)
Corollary 5.12
Suppose that
satisfies
for all \(\tau \in {{\text {Hom}}}(F,\overline{{{\mathbb {Q}}}}_p)\). If \(\varPi \) is an irreducible subquotient of \(H^0(X_n^{{\text {min}}},{\mathcal {E}}_{\rho _{(n),b}}^{{\text {sub}}})\), then there is a continuous representation
which is de Rham above p and has the following property: Suppose that v is a prime of F above a rational prime \(q \ne p\) such that
-
either q splits in \(F_0\),
-
or F and \(\varPi \) are unramified above q;
then
where q is the rational prime below v.
Proof
By the lemma \(\imath \varPi \) is the finite part of a cohomological, square integrable, automorphic representation of \(G_n({\mathbb {A}})\). The result now follows from Corollary 1.3. \(\square \)
7 The ordinary locus
We will now fairly systematically drop the subscript n, as it will be fixed throughout this section.
7.1 P-adic automorphic forms
Let U be a neat open compact subgroup of \(G_n({\mathbb {A}}^{\infty ,p} \times {\mathbb {Z}}_p)\). Zariski locally on \({\mathcal {X}}_{U}^{{\text {min}}}\) we may lift \({{\text {Hasse}}}_{U}\) to a (non-canonical) section \(\widetilde{{{\text {Hasse}}}}_{U}\) of \(\omega ^{\otimes (p-1)}\) over (an open subset of) \({\mathcal {X}}_{U}^{{\text {min}}}\). Although \(\widetilde{{{\text {Hasse}}}}_{U}\) is non-canonical,
is canonical, and so these glue to give a canonical element
Again if \(g \in G_n({\mathbb {A}}^{\infty ,p}\times {\mathbb {Z}}_p)\) and \(U' \supset g^{-1}Ug\) then
We will denote by \(\omega _{U^p(N)}\) the line bundle on \({\mathfrak {X}}_{U^p(N)}^{{{\text {ord}}},{{\text {min}}}}\) induced by \(\omega _{U^p(N,N')}\) on \({\mathcal {X}}^{{\text {min}}}_{U^p(N,N')}\), and by \({{\text {Hasse}}}_{M,U^p(N)}\) the restriction of \({{\text {Hasse}}}_{M,U^p(N,N')}\) to
This is independent of \(N'\).
If \(\rho \) is a representation of \(L_{n,(n)}\) on a finite free \({\mathbb {Z}}_{p}\)-module then, for any integer i, there is a natural map
which sends f to
These maps are \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-equivariant.
Lemma 6.1
For any r the induced map
is surjective.
Proof
The proof here follows standard lines. As far as we know the argument originated in [35]. For the properties of \({\mathcal {X}}_{U}^{{\text {min}}}\) see Sect. 5.1.
To simplify the formulae in this proof, for the duration of the proof we will write U for \(U^p(N_1,N_2)\).
Multiplying by a power of \({{\text {Hasse}}}_{M,U}\) we may replace \(\rho \) by
and r by \(r-t\) for any t. Thus, using the ampleness of \(\omega _{U}\) over \({\mathcal {X}}^{{\text {min}}}_{U}\), we may suppose that
for all \(i > 0\) and \(j\ge 0\). We may also suppose that \(r\le 0\). Then we may replace r by 0.
Because \({\mathcal {X}}^{{{\text {ord}}},{{\text {min}}}}_{U} \times {{\text {Spec}}\,}{\mathbb {Z}}/p^M{\mathbb {Z}}\) is a union of connected components of
it suffices to replace \({\mathcal {X}}^{{{\text {ord}}},{{\text {min}}}}_{U} \times {{\text {Spec}}\,}{\mathbb {Z}}/p^M{\mathbb {Z}}\) by \({\mathcal {Y}}\).
Now we need to show that
under the assumption that
for all \(i > 0\) and \(j\ge 0\).
The scheme \({\mathcal {Y}}\) is relatively affine over \({\mathcal {X}}^{{{\text {min}}}}_{U}\) corresponding to the sheaf of algebras
Hence
and the map
is induced by the map
of sheaves over \({\mathcal {X}}_U^{{\text {min}}}\).
Because
for all \(i > 0\) and \(j\ge 0\), we see that
for all \(j\ge 0\), and
for all \(i>0\) and \(j \ge 0\). Thus it suffices to check that
is surjective. This follows using the long exact sequence in cohomology associated to the short exact sequence
and the vanishing
\(\square \)
Let S denote a finite set of rational primes containing p and all rational primes q which are both non-split in \(F_0\) and ramified in F. Also choose a neat open compact subgroup
Suppose that v is a place of F above a rational prime \(q \not \in S\) and let \(i \in {\mathbb {Z}}\). There is a unique element \({\mathfrak {t}}_v^{(i)}\) in the Bernstein centre of \(G_n({\mathbb {Q}}_q)\) such that
-
\({\mathfrak {t}}_v^{(i)}\) acts as 0 on any irreducible smooth representation of \(G_n({\mathbb {Q}}_q)\) over \({\mathbb {C}}\) which is not a subquotient of an unramified principal series;
-
on an unramified representation \(\varPi _q\) of \(G_n({\mathbb {Q}}_q)\) the eigenvalue of \({\mathfrak {t}}_v^{(i)}\) on \(\varPi _q\) equals \({{\text {tr}}}\,{\text {rec}}_{F_v}({{\text {BC}}}(\varPi _q)_v|\det |_v^{(1-2n)/2})({{\text {Frob}}}_v^i)\).
(See [4].) Multiplying \({\mathfrak {t}}_v^{(i)}\) by the characteristic function of \(G_n({\mathbb {Z}}_q)\) we obtain a unique element \(T_v^{(i)} \in {\mathbb {C}}[G_n({\mathbb {Z}}_q)\backslash G_n({\mathbb {Q}}_q) /G_n({\mathbb {Z}}_q)]\) such that if \(\varPi _q\) is an unramified representation of \(G_n({\mathbb {Q}}_q)\) and if \(T_v^{(i)}\) has eigenvalue \(t_v^{(i)}(\varPi _q)\) on \(\varPi _q^{G_n({\mathbb {Z}}_q)}\) then
(See [29].) If \(\sigma \in {{\text {Aut}}}({\mathbb {C}})\) we see that \({}^\sigma T_v^{(i)}=T_v^{(i)}\). (Use the fact that
Thus
Choose \(d_v^{(i)} \in {\mathbb {Q}}^\times \) such that
Suppose that \(q\not \in S\) is a rational prime. Let \(u_1,\ldots ,u_r\) denote the primes of \(F^+\) above \({\mathbb {Q}}\) which split \(u_i=w_i\,{}^cw_i\) in F, and let \(v_1,\ldots ,v_s\) denote the primes of \(F^+\) above q which do not split in F. Then under the identification
of Sect. 1.3, the Hecke operator \(T_{w_i}^{(1)}\) is identified with the double coset
where \(a_i \in GL_n(F_{w_i})\) is the diagonal matrix \({{\text {diag}}}(1,\ldots ,1,\varpi _{w_i})\), and we may take \(d_{w_i}^{(1)}=1\).
We will call a topological \({\mathbb {Z}}_p[G_n({\widehat{{\mathbb {Z}}}}^S) \backslash G_n({\mathbb {A}}^S) /G_n({\widehat{{\mathbb {Z}}}}^S)]\)-algebra \({\mathbb {T}}\) of Galois type if there is a continuous pseudo-representation (see [55])
such that
for all \(v|q \not \in S\) and all \(i \in {\mathbb {Z}}\).
Let \({\mathbb {T}}_{U^p(N_1,N_2),\rho }^{S}\) denote the image of \({\mathbb {Z}}_p[G_n({\widehat{{\mathbb {Z}}}}^S)\backslash G_n({\mathbb {A}}^S) /G_n({\widehat{{\mathbb {Z}}}}^S)]\) in the endomorphism algebra \({{\text {End}}}(H^0({\mathcal {X}}^{{{\text {min}}}}_{U^p(N_1,N_2)}, {\mathcal {E}}_\rho ^{{{\text {sub}}}}))\), which is also the image in the endomorphism algebra \({{\text {End}}}(H^0(X^{{{\text {min}}}}_{U^p(N_1,N_2)}, {\mathcal {E}}_\rho ^{{{\text {sub}}}}))\).
Lemma 6.2
For t sufficiently large \({\mathbb {T}}_{U^p(N_1,N_2),\rho \otimes (\wedge ^{n[F: {\mathbb {Q}}]} {{\text {Std}}}^\vee )^{\otimes t}}^{S}\) is of Galois type.
Proof
Write
It suffices to show that there is a continuous pseudo-representation
which is unramified outside S and satisfies
for all \(v |q \not \in S\) and all \(i \in {\mathbb {Z}}\). (Because T will then automatically be valued in \({\mathbb {T}}_{U^p(N_1,N_2),\rho _t}^{S}\), by the Cebotarev density theorem. Note that if v is a prime of F split over \(F^+\) and lying above a rational prime \(q \not \in S\), then
We may then reduce to the case that \(\rho \otimes \overline{{{\mathbb {Q}}}}_p\) is irreducible. Let
denote the highest weight of \(\rho \otimes \overline{{{\mathbb {Q}}}}_p\).
Suppose that t satisfies the inequality
By Lemma 5.11,
where the sum runs over irreducible admissible representations of \(G_n({\mathbb {A}}^{S,\infty })\) for which there exists an irreducible admissible representation \(\varPi _S\) of \(\prod _{v \in S}G_n({\mathbb {Q}}_v)\) such that \(\varPi ^S \otimes \varPi _S\) occurs in \(H^0(X^{{\text {min}}}\times {{\text {Spec}}\,}\overline{{{\mathbb {Q}}}}_p,{\mathcal {E}}_{\rho _t }^{{\text {sub}}})\) and \((\varPi ^S \otimes \varPi _S)^{U^p(N_1,N_2)} \ne (0)\). Further, from Corollary 5.12, we deduce that there is a continuous representation
such that if \(v|q \not \in S\) then r is unramified at v and
for all \(i \in {\mathbb {Z}}\). Taking \(T={{\text {tr}}}\,r\) completes the proof of the lemma. \(\square \)
If
(resp.
is a finitely generated \({\mathbb {Z}}_p\)-submodule invariant under the action of the Hecke algebra
then we will let \({\mathbb {T}}_{U^p(N),\rho }^{{{\text {ord}}},S}(W)\) (resp. \({\mathbb {T}}_{U^p(N_1,N_2),\rho }^{{{\text {ord}}},S}(W)\)) denote the image of the abstract Hecke algebra \({\mathbb {Z}}_p[G_n({\widehat{{\mathbb {Z}}}}^S)\backslash G_n({\mathbb {A}}^S) /G_n({\widehat{{\mathbb {Z}}}}^S)]\) in \({{\text {End}}}_{{\mathbb {Z}}_p}(W)\). The next corollary follows from Lemmas 6.1 and 6.2.
Corollary 6.3
If
is a finitely generated \({\mathbb {Z}}_p\)-submodule invariant under the action of the Hecke algebra
then \({\mathbb {T}}_{U^p(N_1,N_2),\rho }^{{{\text {ord}}},S}(W)\) is of Galois type.
We deduce from this the next corollary.
Corollary 6.4
If
is a finitely generated \({\mathbb {Z}}_p\)-submodule invariant under the action of the Hecke algebra
then \({\mathbb {T}}_{U^p(N),\rho }^{{{\text {ord}}},S}(W)\) is of Galois type.
Finally we deduce the following proposition.
Proposition 6.5
Suppose that \(\rho \) is a representation of \(L_{n,(n)}\) over \({\mathbb {Z}}_{(p)}\). Suppose also that \(\varPi \) is an irreducible quotient of an admissible \(G_n({\mathbb {A}}^{\infty })^{{{\text {ord}}},\times }\)-submodule \(\varPi '\) of \(H^0({\mathfrak {X}}^{{{\text {ord}}},{{\text {min}}}}, {\mathcal {E}}_\rho ^{{{\text {ord}}},{{\text {sub}}}})_{\overline{{{\mathbb {Q}}}}_p}\). Then there is a continuous semi-simple representation
with the following property: If \(\varPi \) is unramified at a rational prime \(q\ne p\) (in the sense of Sect. 1.3) and if v|q is a prime of F, then
Proof
Let S denote the set of rational primes consisting of p and the primes where F or \(\varPi \) ramifies. Also choose a neat open compact subgroup
and integer N such that
As \((\varPi ')^{U^p(N)}\) is a finite dimensional, and hence closed, subspace of the topological vector space \(H^0({\mathfrak {X}}^{{{\text {ord}}},{{\text {min}}}}, {\mathcal {E}}_\rho ^{{{\text {ord}}},{{\text {sub}}}})_{\overline{{{\mathbb {Q}}}}_p}\) preserved by \({\mathbb {Z}}_p[G_n({\widehat{{\mathbb {Z}}}}^S)\backslash G_n({\mathbb {A}}^S) /G_n({\widehat{{\mathbb {Z}}}}^S)]\) and, as there is a \({\mathbb {Z}}_p[G_n({\widehat{{\mathbb {Z}}}}^S)\backslash G_n({\mathbb {A}}^S) /G_n({\widehat{{\mathbb {Z}}}}^S)]\)-equivariant map \((\varPi ')^{U^p(N)} \twoheadrightarrow \varPi ^{U^p(N)}\), there is a continuous homomorphism
which for \(v|q \not \in S\) sends \(T_v^{(i)}\) to its eigenvalue on \(\varPi ^{G_n({\mathbb {Z}}_q)}\). Proposition 6.5 now follows from the above corollary and the main theorem on pseudo-representations (see [55]). \(\square \)
We remark that we don’t know how to prove this proposition for a general irreducible subquotient of \(H^0({\mathfrak {X}}^{{{\text {ord}}},{{\text {min}}}}, {\mathcal {E}}_\rho ^{{{\text {ord}}},{{\text {sub}}}})_{\overline{{{\mathbb {Q}}}}_p}\) (or indeed whether the corresponding statement remains true).
7.2 Interlude concerning linear algebra
Suppose that K is an algebraic extension of \({\mathbb {Q}}_p\). For \(a \in {\mathbb {Q}}\), we say that a polynomial \(P(X) \in K(X)\) has slopes \(\le a\) if \(P(X) \ne 0\) and every root of P(X) in \(\overline{{K}}\) has p-adic valuation \(\le a\). (We normalize the p-adic valuation so that p has valuation 1.) If V is a K-vector space and T is an endomorphism of V, then we say that V admits slope decompositions for T, if for each \(a \in {\mathbb {Q}}\) there is a decomposition
with the following properties:
-
T preserves \(V_{\le a}\) and \(V_{>a}\);
-
\(V_{\le a}\) is finite dimensional;
-
if \(P(X) \in K[X]\) has slopes \(\le a\) then the endomorphism P(T) restricts to an automorphism of \(V_{>a}\);
-
there is a nonzero polynomial \(P(X) \in K[X]\) with slopes \(\le a\) such that the endomorphism P(T) restricts to 0 on \(V_{\le a}\).
In this case \(V_{\le a}\) and \(V_{>a}\) are unique, and we refer to them as the slope a decomposition of V with respect to T.
Lemma 6.6
-
(1)
If V is finite dimensional then it always admits slope decompositions.
-
(2)
If K is a finite extension of \({\mathbb {Q}}_p\), if V is a K-Banach space and if T is a completely continuous (see [50]) endomorphism of V then V admits slope decompositions for T.
-
(3)
Suppose that L / K is an algebraic extension and that V is a K vector space which admits slope decompositions with respect to an endomorphism T. Then \(V \otimes _KL\) also admits slope decompositions with respect to T.
-
(4)
Suppose that \(V_1\) admits slope decompositions with respect to \(T_1\); that \(V_2\) admits a slope decomposition with respect to \(T_2\); and that \(d: V_1 \rightarrow V_2\) is a linear map such that
$$\begin{aligned} d \circ T_1=T_2 \circ d. \end{aligned}$$Then for all \(a \in {\mathbb {Q}}\) we have
$$\begin{aligned} d V_{1,\le a} \subset V_{2,\le a} \end{aligned}$$and
$$\begin{aligned} d V_{1,> a} \subset V_{2,>a}. \end{aligned}$$Moreover \(\ker d\) admits slope decompositions for \(T_1\), while \({\text {Im}}d\) and \({{\text {coker}}\,}d\) admit slope decompositions for \(T_2\). More specifically
$$\begin{aligned} (\ker d)_{\le a}=(\ker d) \cap V_{1,\le a} \end{aligned}$$and
$$\begin{aligned} (\ker d)_{> a}=(\ker d) \cap V_{1,> a} \end{aligned}$$and
$$\begin{aligned} ({\text {Im}}d)_{\le a}=V_{1,\le a}/(\ker d)_{\le a} \end{aligned}$$and
$$\begin{aligned} ({\text {Im}}d)_{> a}=V_{1,> a}/(\ker d)_{> a} \end{aligned}$$and
$$\begin{aligned} ({{\text {coker}}\,}d)_{\le a}=V_{2,\le a}/({\text {Im}}d)_{\le a} \end{aligned}$$and
$$\begin{aligned} ({{\text {coker}}\,}d)_{> a}=V_{2,> a}/({\text {Im}}d)_{> a}. \end{aligned}$$ -
(5)
Suppose that
$$\begin{aligned} V_1 \subset V_2 \subset V_3 \subset \cdots \subset V_\infty \end{aligned}$$are vector spaces with
$$\begin{aligned} V_\infty =\bigcup _{i=1}^\infty V_i. \end{aligned}$$Suppose also that T is an endomorphism of \(V_\infty \) such that for all \(i>1\)
$$\begin{aligned} T V_i \subset V_{i-1}. \end{aligned}$$If for each i the space \(V_i\) admits slope decompositions for i, then \(V_\infty \) admits slope decompositions for T.
-
(6)
Suppose that
$$\begin{aligned} (0) \longrightarrow V_1 \longrightarrow V \longrightarrow V_2 \longrightarrow (0) \end{aligned}$$is an exact sequence of K-vector spaces and that T is an endomorphism of V that preserves \(V_1\). If \(V_1\) and \(V_2\) both admit slope decompositions with respect to T, then so does V. Moreover we have short exact sequences
$$\begin{aligned} (0) \longrightarrow V_{1,\le a} \longrightarrow V_{\le a} \longrightarrow V_{2,\le a} \longrightarrow (0) \end{aligned}$$and
$$\begin{aligned} (0) \longrightarrow V_{1,> a} \longrightarrow V_{> a} \longrightarrow V_{2,> a} \longrightarrow (0) \end{aligned}$$
Proof
The first and third and fourth parts are straightforward. The second part follows from [50].
For the fifth part one checks that \(V_{i, \le a}\) is independent of i. If we set
for any i, and
then these provide the slope a decomposition of \(V_\infty \) with respect to T.
Finally we turn to the sixth part. Choose nonzero polynomials \(P_i(X) \in K[X]\) with slopes \(\le a\) such that \(P_i(T)V_{i,\le a}=(0)\), for \(i=1,2\). Set \(P(X)=P_1(X)P_2(X)\). Also set \(V_{\le a}=\ker P(T)\) and \(V_{>a}= {\text {Im}}P(T)\). We have complexes
and
It suffices to show that these complexes are both short exact sequences. For then we see that, if \(Q(X) \in K[X]\) has slopes \(\le a\), then the restriction of Q(T) to \(V_{>a}\) is an automorphism of \(V_{>a}\). Applying this to P(T), we see that \(V_{\le a} \cap V_{>a}=(0)\). Moreover \(V_{\le a}+V_{>a}\) contains \(V_1\) and maps onto \(V_2\), so that \(V=V_{\le a}+V_{>a}\).
To show the first complex is short exact we need only check that \(V_{1,>a}=V_{>a} \cap V_1\), i.e. that \(V_{1,\le a} \cap V_{>a} =(0)\). So suppose that \(v \in V_{1,\le a} \cap V_{>a}\). Then \(v=P(T)v'\) and \(P_1(T)v=0\). Thus \(P_1(T)^2P_2(T)v'=0\) so the image of \(v'\) in \(V_2\) lies in \(V_{2,\le a}\) and so \(P_2(T)v' \in V_1\), and in fact \(P_2(T)v' \in V_{1,\le a}\). Finally we see that \(v=P_1(T)P_2(T)v'=0\), as desired.
To show the second complex is short exact we have only to show that \(V_{\le a} \rightarrow V_{2,\le a}\) is surjective. So suppose that \(\overline{{v}}\in V_{2,\le a}\) and suppose that \(v \in V\) lifts \(\overline{{v}}\). Then \(P(T) v\in V_{1,>a}\). Set
Then \(v'\) maps to \(\overline{{v}}\in V_2\), while
so that \(v' \in V_{\le a}\). \(\square \)
We warn the reader that to the best of our knowledge it is not in general true that, if V admits a slope decomposition for T and \(V_1 \subset V\) is T-invariant, then either \(V_1\) or \(V/V_1\) admits slope decompositions for T.
7.3 The ordinary locus of a toroidal compactification as a dagger space
7.3.1 Review of dagger spaces
We first review some general facts about dagger spaces. We refer to [27] for the basic facts.
Suppose that \(K/{\mathbb {Q}}_p\) is a finite extension with ring of integers \({\mathcal {O}}_K\) and residue field k. Suppose also that \({\mathcal {Y}}/{\mathcal {O}}_K\) is quasi-projective. Let Y denote the generic fibre \({\mathcal {Y}}\times {{\text {Spec}}\,}K\), let \(\overline{{Y}}\) denote the special fibre \({\mathcal {Y}}\times {{\text {Spec}}\,}k\), and let \({\mathcal {Y}}^\wedge \) denote the formal completion of \({\mathcal {Y}}\) along \(\overline{{Y}}\). Let \(Y^{{\text {an}}}\) (resp. \(Y^\dag \)) denote the rigid analytic (resp. dagger) space associated to Y. (For the latter see section 3.3 of [27].) Thus \(Y^{{\text {an}}}\) and \(Y^\dag \) share the same underlying G-topological space, and in fact the completion \((Y^\dag )'\) (see theorem 2.19 of [27]) of \(Y^\dag \) equals \(Y^{{\text {an}}}\). Let \({\mathcal {Y}}^\wedge _\eta \) denote the rigid analytic space associated to \({\mathcal {Y}}^\wedge \), its ‘generic fibre’. Then \({\mathcal {Y}}^\wedge _\eta \) is identified with an admissible open subset \(]\overline{{Y}}[ \subset Y^{{\text {an}}}\). We will denote by \({\mathcal {Y}}^\dag \) the admissible open dagger subspace of \(Y^\dag \) with the same underlying topological space as \(]\overline{{Y}}[\).
To a coherent sheaf \({\mathcal {F}}/Y\) one can associate a coherent sheaf \({\mathcal {F}}^\dag /Y^\dag \) and hence \({\mathcal {F}}^\dag /{\mathcal {Y}}^\dag \). The functor \({\mathcal {F}}\mapsto {\mathcal {F}}^\dag \) from coherent sheaves on Y to coherent sheaves on \({\mathcal {Y}}^\dag \) is exact.
Lemma 6.7
If \({\mathcal {Y}}\) and \({\mathcal {Y}}'\) are two quasi-projective \({\mathcal {O}}_K\)-schemes as described in the previous paragraph and if \(f: {\mathcal {Y}}\rightarrow {\mathcal {Y}}'\) is a morphism, then there is an induced map \(f^\dag : {\mathcal {Y}}^\dag \rightarrow ({\mathcal {Y}}')^\dag \).
If further \(f: \overline{{Y}}\mathop {\rightarrow }\limits ^{\sim }\overline{{Y}}'\) and f is etale in a neighbourhood of \(\overline{{Y}}\) then \(f^\dag \) is an isomorphism.
Proof
The first part of the lemma is clear.
For the second part, let \({\mathcal {Y}}\hookrightarrow {\mathbb {P}}_{{\mathcal {O}}_K}^M\) and \({\mathcal {Y}}' \hookrightarrow {\mathbb {P}}_{{\mathcal {O}}_K}^{M'}\) be closed embeddings. Let \({\mathcal {P}}'\) denote the closure of \({\mathcal {Y}}'\) in \({\mathbb {P}}_{{\mathcal {O}}_K}^{M'}\). Also let \({\mathcal {P}}\) denote the closure of \({\mathcal {Y}}\) in \({\mathbb {P}}_{{\mathcal {O}}_K}^{M} \times {\mathbb {P}}_{{\mathcal {O}}_K}^{M'}\). Then f extends to a map \({\mathcal {P}}\rightarrow {\mathcal {P}}'\). The second part of the lemma follows from theorem 1.3.5 of [6] applied to \(\overline{{Y}}\subset {\mathcal {P}}\) and \(\overline{{Y}}' \subset {\mathcal {P}}'\). \(\square \)
We will let \(H^i_{{\text {rig}}}(\overline{{Y}})\) denote the rigid cohomology of \(\overline{{Y}}\) in the sense of Berthelot—see, for instance, [46].
Lemma 6.8
-
(1)
If \({\mathcal {Y}}/{\mathcal {O}}_K\) is a smooth and quasi-projective scheme, then there is a canonical isomorphism
$$\begin{aligned} H^i_{{\text {rig}}}(\overline{{Y}}) \cong {\mathbb {H}}^i\left( {\mathcal {Y}}^\dag , \varOmega ^\bullet _{{\mathcal {Y}}^\dag }\right) . \end{aligned}$$ -
(2)
If \(f: {\mathcal {Y}}\rightarrow {\mathcal {Z}}\) is a morphism of smooth quasi-projective schemes over \({\mathcal {O}}_K\) then the following diagram is commutative:
$$\begin{array}{ccc} H^i_{{\text {rig}}}(\overline{{Z}}) &{}\mathop {\longrightarrow }\limits ^{f^*} &{} H^i_{{\text {rig}}}(\overline{{Y}}) \\ ||\wr &{}&{} || \wr \\ {\mathbb {H}}^i\left( {\mathcal {Z}}^\dag , \varOmega ^\bullet _{{\mathcal {Z}}^\dag }\right) &{} \mathop {\longrightarrow }\limits ^{f^*}&{} {\mathbb {H}}^i\left( {\mathcal {Y}}^\dag , \varOmega ^\bullet _{{\mathcal {Y}}^\dag }\right) . \end{array}$$
Proof
For the first part apply theorem 5.1 of [27] to the closure of \({\mathcal {Y}}\) in some projective space over \({\mathcal {O}}_K\). For the second part choose embeddings \(i: {\mathcal {Y}}\hookrightarrow {\mathbb {P}}_{{\mathcal {O}}_K}^{M}\) and \(i': {\mathcal {Z}}\hookrightarrow {\mathbb {P}}_{{\mathcal {O}}_K}^{M'}\). Let \({\mathcal {P}}'\) denote the closure of \({\mathcal {Z}}\) in \({\mathbb {P}}_{{\mathcal {O}}_K}^{M'}\) and \({\mathcal {P}}\) the closure of \({\mathcal {Y}}\) in \({\mathbb {P}}_{{\mathcal {O}}_K}^M \times {\mathcal {P}}'\), so that f extends to a map \({\mathcal {P}}\rightarrow {\mathcal {P}}'\). The desired result again follows from theorem 5.1 of [27], because the isomorphisms of theorem 5.1 of [27] are functorial under morphisms of the setup in that theorem. \(\square \)
[It is unclear to us whether this functoriality is supposed to be implied by the word ‘canonical’ in the statement of theorem 5.1 of [27]. For safety’s sake we sketch the argument for this functoriality. More precisely if \(f: {\mathcal {X}}_1\rightarrow {\mathcal {X}}_2\) is a morphism of proper admissible formal \({{\text {Spf}}\,}R\)-schemes which takes \(Y_1 \subset {\mathcal {X}}_{1,s}\) to \(Y_2 \subset {\mathcal {X}}_{2,s}\), then we will show that the isomorphisms of theorem 5.1 of [27] are compatible with the maps in cohomology induced by f. For part (a) we also suppose that we are given a map \(f^*: f^* {\mathcal {F}}_2 \rightarrow {\mathcal {F}}_1\).
Using the notation of part (a) of theorem 5.1 of [27], it suffices to show that the diagram
commutes. (The functoriality of parts (b) and (c) follow easily from the functoriality of part (a).) The vertical morphisms arise from maps \(L_k^\bullet \rightarrow K_k^\bullet \) of resolutions of the sheaves \(Ri_* {\mathcal {F}}_{k,X_k}\) and \(j_k^\dag {\mathcal {F}}_k'\), respectively. To define these resolutions one needs to choose affine covers \(\{ Y_{k,i} \}\) of \(Y_k\). We may suppose these are chosen so that f carries \(Y_{1,i}\) to \(Y_{2,i}\) for all i. Then \(L_k^\bullet \) and \(K_k^\bullet \) are the Cech complexes with
and
The maps \(L_k^\bullet \rightarrow K_k^\bullet \) arise from maps
Here V runs over strict neighbourhoods of \(U \cap ]Y_{k,J}[_{{\mathcal {X}}_k}\) in \(]\overline{{Y}}_k[_{{\mathcal {X}}_k}\) and \(V'\) runs over strict neighbourhoods of \(]Y_{k,J}[_{{\mathcal {X}}_k}\) in \(]\overline{{Y}}_k[_{{\mathcal {X}}_k}\). The first isomorphism is justified in section 2.23 of [27]. The second morphism arises because, for every V, we can find a \(V'\) so that
It suffices to show that if \(fU_1 \subset U_2\), then the diagrams
are commutative. But this is now clear.]
Lemma 6.9
Suppose that \(f: X \rightarrow Y\) is a proper morphism between \({\mathbb {Q}}_p\)-schemes of finite type and that \({\mathcal {F}}/X\) is a coherent sheaf. Denote by \(f^\dag : X^\dag \rightarrow Y^\dag \) the corresponding map of dagger spaces and by \({\mathcal {F}}^\dag \) the coherent sheaf on \(X^\dag \) corresponding to \({\mathcal {F}}/X\). Suppose also that V is an admissible open subset of \(Y^\dag \) and that U is its pre-image in \(X^\dag \). Then
where \((R^if_* {\mathcal {F}})^\dag \) denotes the coherent sheaf on \(Y^\dag \) corresponding to \((R^if_* {\mathcal {F}})/Y\).
Proof
It suffices to check this in the case \(V=Y^\dag \). There is a chain of isomorphisms
The first arrow is the transitivity of dagger and rigid analytification. The second arrow is theorem 6.5 of [38]. The third arrow is theorem 3.5 of [27]. Since \(Y^\dag \) is partially proper, theorem 2.26 of [27] implies that there is a unique isomorphism \((R^i f_*{\mathcal {F}})^\dag \cong R^i f^\dag _*{\mathcal {F}}^\dag \) which recovers the above map after passage to rigid spaces. \(\square \)
7.3.2 The ordinary locus as a dagger space
Now we return to our Shimura and Kuga–Sato varieties.
If \(U^p\) is a neat open compact subgroup of \(G_n^{(m)}({\mathbb {A}}^{\infty ,p})\), if \(N_2 \ge N_1 \ge 0\) and if \((U^p(N_1,N_2),\varSigma ) \in {\mathcal {J}}^{(m),{{\text {tor}}}}\), we will write
(resp.
resp.
for \({[\sigma ]} \in {\mathcal {S}}(U^p(N_1,N_2),\varSigma )\)) for the dagger space associated to \({\mathcal {A}}^{(m),{{\text {ord}}}}_{U^p(N_1,N_2),\varSigma }\) (resp. \(\partial {\mathcal {A}}^{(m),{{\text {ord}}}}_{U^p(N_1,N_2),\varSigma }\), resp. \(\partial _{[\sigma ]} {\mathcal {A}}^{(m),{{\text {ord}}}}_{U^p(N_1,N_2),\varSigma }\)) as described in the paragraph before Lemma 6.7. For \(s>0\) also write
and \(i^{(s)}\) for the finite map
We set
and
Then the various systems of dagger spaces \(\{ {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N_1,N_2),\varSigma } \}\) and \(\{ \partial {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N_1,N_2),\varSigma } \}\) and \(\{ \partial ^{(s)} {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N_1,N_2),\varSigma } \}\) have compatible actions of \(G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\).
If \(N_2' \ge N_2\) and if \(\varSigma '\) is a refinement of \(\varSigma \) with \(\varSigma ^{{\text {ord}}}=(\varSigma ')^{{\text {ord}}}\) then the natural map
restricts to an isomorphism
and is etale in a neighbourhood of \(\overline{{A}}^{(m),{{\text {ord}}}}_{U^p(N_1,N_2'),\varSigma '}\). It follows from Lemma 6.7 that
is an isomorphism. We will denote this dagger space simply
Similarly \(\partial {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N_1,N_2),\varSigma }\) and \(\partial _{[\sigma ]} {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N_1,N_2),\varSigma }\) and \(\partial ^{(s)} {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N_1,N_2),\varSigma }\) depend only on the group \(U^p(N_1)\) and \(\varSigma ^{{\text {ord}}}\) and we will denote them \(\partial {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N_1),\varSigma ^{{\text {ord}}}}\) and \(\partial _{[\sigma ]} {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N_1),\varSigma ^{{\text {ord}}}}\) and \(\partial ^{(s)} {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N_1),\varSigma ^{{\text {ord}}}}\), respectively. If \({[\sigma ]} \not \in {\mathcal {S}}(U^p(N_1),\varSigma ^{{\text {ord}}})^{{\text {ord}}}\) then
Thus for \(s>0\)
The three projective systems of dagger spaces \(\{ {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\varSigma ^{{\text {ord}}}} \}\) and \(\{ \partial {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\varSigma ^{{\text {ord}}}} \}\) and \(\{ \partial ^{(s)} {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\varSigma ^{{\text {ord}}}} \}\) have actions of \(G^{(m)}_n({\mathbb {A}}^\infty )^{{\text {ord}}}\).
We will write \({\mathfrak {X}}^{{{\text {ord}}},\dag }_{(U^p)'(N),\varDelta }\) for \({\mathfrak {A}}^{(0),{{\text {ord}}},\dag }_{(U^p)'(N),\varDelta }\). If \((U^p)'\) contains the projection of \(U^p\) and if \(\varDelta ^{{\text {ord}}}\) and \(\varSigma ^{{\text {ord}}}\) are compatible, then there are maps
These maps are \(G_n^{(m)}({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-equivariant (as \(U^p\), \((U^p)'\) and N vary).
We will write \(\varOmega ^j_{{\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\varSigma }}(\log \infty )\) (resp. \(\varOmega ^j_{{\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\varSigma }}(\log \infty ) \otimes {\mathcal {I}}_{\partial {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\varSigma }}\)) for the locally free sheaf on \({\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\varSigma }\) induced by \(\varOmega ^j_{{\mathcal {A}}^{(m),{{\text {ord}}}}_{U^p(N,N'),\varSigma '}}(\log \infty )\) (resp. \(\varOmega ^j_{{\mathcal {A}}^{(m),{{\text {ord}}}}_{U^p(N,N'),\varSigma '}}(\log \infty )\otimes {\mathcal {I}}_{\partial {\mathcal {A}}^{(m),{{\text {ord}}}}_{U^p(N,N'),\varSigma '}}\)) for any \(N' \ge N\) and \(\varSigma ' \in {\mathcal {J}}_n^{(m),{{\text {tor}}}}\) with \((\varSigma ')^{{\text {ord}}}=\varSigma \). This is canonically independent of the choices of \(N'\) and \(\varSigma '\). The systems of sheaves \(\{ \varOmega ^j_{{\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\varSigma }}(\log \infty )\}\) and \(\{ \varOmega ^j_{{\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\varSigma }}(\log \infty )\otimes {\mathcal {I}}_{\partial {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\varSigma }} \}\) over \(\{ {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\varSigma }\}\) have actions of \(G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\). For \(g \in G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) the map
is an isomorphism.
We will also write \(\varOmega ^j_{ \partial ^{(s)} {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\varSigma } }\) for the sheaf of j-forms on \(\partial ^{(s)} {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\varSigma }\). The system \(\{ \varOmega ^j_{\partial ^{(s)} {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\varSigma }} \}\) over \(\{ \partial ^{(s)} {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\varSigma }\}\) has an action of \(G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\).
Furthermore if \(\rho \) is a representation of \(L_{n,(n)}\) on a finite dimensional \({\mathbb {Q}}_p\)-vector space, there is a locally free sheaf \({\mathcal {E}}^{{{\text {can}}},\dag }_{U^p(N),\varDelta ,\rho }\) (resp. \({\mathcal {E}}^{{{\text {sub}}},\dag }_{U^p(N),\varDelta ,\rho }\)) on \({\mathcal {X}}^{{{\text {ord}}},\dag }_{U^p(N),\varDelta }\) induced by \({\mathcal {E}}^{{\text {can}}}_{U^p(N,N'),\varDelta ',\rho }\) (resp. \({\mathcal {E}}^{{\text {sub}}}_{U^p(N,N'),\varDelta ',\rho }\)) for any \(N' \ge N\) and \(\varDelta ' \in {\mathcal {J}}_n^{{{\text {tor}}}}\) with \((\varDelta ')^{{\text {ord}}}=\varDelta \). This is canonically independent of the choices of \(N'\) and \(\varDelta '\). The systems of sheaves \(\{ {\mathcal {E}}^{{{\text {can}}},\dag }_{U^p(N),\varDelta ,\rho }\}\) and \(\{ {\mathcal {E}}^{{{\text {sub}}},\dag }_{U^p(N),\varDelta ,\rho }\}\) over \(\{ {\mathcal {X}}^{{{\text {ord}}},\dag }_{U^p(N),\varDelta }\}\) have actions of \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\). There are equivariant identifications
where \({\mathcal {I}}_{\partial {\mathcal {X}}^{{{\text {ord}}},\dag }_{U^p(N),\varDelta }}\) denotes the sheaf of ideals in \({\mathcal {O}}_{{\mathcal {X}}^{{{\text {ord}}},\dag }_{U^p(N),\varDelta }}\) defining \(\partial {\mathcal {X}}^{{{\text {ord}}},\dag }_{U^p(N),\varDelta }\). For \(g \in G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\) the map
is an isomorphism. (Because the same is true over \(X_{U^p(N,N'),\varDelta '}\) and hence over \(X_{U^p(N,N'),\varDelta '}^\dag \).)
We define \(H^i({\mathcal {A}}^{(m),{{\text {ord}}},\dag } , \varOmega ^j(\log \infty )\otimes {\mathcal {I}})\) to be
and \(H^i(\partial ^{(s)}{\mathcal {A}}^{(m),{{\text {ord}}},\dag } , \varOmega ^j)\) to be
and \(H^0({\mathcal {X}}^{{{\text {ord}}},\dag }, {\mathcal {E}}^{{{\text {sub}}}}_\rho )\) to be
They are all smooth \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\)-modules.
Lemma 6.10
There are natural isomorphisms
and
Proof
Use Lemmas 5.1, 5.6, 5.7, 5.3 and 6.9. \(\square \)
7.3.3 The Frobenius lift \(\varsigma _p\) and \(\hbox {tr}_F\)
The inverse of \(\varsigma _p^*\) gives maps
and
The maps
-
\(\varsigma _p: {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\varSigma ^{{\text {ord}}}} \rightarrow {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\varSigma ^{{\text {ord}}}}\),
-
and \(\varsigma _p: \partial ^{(s)} {\mathcal {A}}_{U^p(N),\varSigma ^{{\text {ord}}}}^{(m),{{\text {ord}}},\dag } \rightarrow \partial ^{(s)} {\mathcal {A}}_{U^p(N),\varSigma ^{{\text {ord}}}}^{(m),{{\text {ord}}},\dag }\)
are finite, flat of degrees \(p^{(2m+n)n[F^+: {\mathbb {Q}}]}\) and \(p^{(2m+n)n[F^+: {\mathbb {Q}}]-s}\), respectively. (Use the finite flatness of
and
(see Sect. 5.3), together with theorems 1.7(1) and 1.12 of [27].)
As \(\varsigma _p: {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\varSigma ^{{\text {ord}}}} \rightarrow {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\varSigma ^{{\text {ord}}}}\) is finite and flat we get a trace map
Because \(\partial {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\varSigma }\) has the same support as
this trace map restricts to a map
(This is a consequence of the following fact: If R is a noetherian ring, if S is an R-algebra, finite and free as an R-module, and if I and J are ideals of R and S, respectively, with
then the trace map \({{\text {tr}}}_{S/R}\) maps J to I. To see this we may reduce to the case \(I=0\). In this case every element of J is nilpotent and so has trace 0.)
Composing \((\varsigma _p^*)^{-1}\) with \({{\text {tr}}}_{\varsigma _p}\) we get \(G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-equivariant maps
and
We have
This induces endomorphisms
which commute with the action of \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) and satisfy
We obtain an element
which commutes with the \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-action and satisfies
Similarly the inverse of \(\varsigma _p^*\) gives maps
and
Composing \((\varsigma _p^*)^{-1}\) with \({{\text {tr}}}_{\varsigma _p}\) we get \(G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-equivariant maps
and
We have
This induces compatible endomorphisms
and
which commute with the action of \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) and satisfy
We obtain an element
which commutes with the \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-action and satisfies
We remark that \({{\text {tr}}}_F\) is closely related to the operator often denoted \(U_p\): probably they differ simply by a scalar multiple.
7.4 The ordinary locus of the minimal compactification as a dagger space
7.4.1 The ordinary locus as a dagger space
Suppose that \(U^p\) is a neat open compact subgroup of \(G_n({\mathbb {A}}^{\infty ,p})\) and that \(N_2 \ge N_1 \ge 0\). We will write
for the dagger space associated to \({\mathcal {X}}^{{{\text {ord}}},{{\text {min}}}}_{U^p(N_1,N_2)}\) as described in the paragraph before Lemma 6.7. Then the system of dagger spaces \(\{ {\mathcal {X}}^{{{\text {ord}}},{{\text {min}}},\dag }_{U^p(N_1,N_2)} \}\) has an action of \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\).
If \(U^p\) denotes the image in \(G_n({\mathbb {A}}^{\infty ,p})\) of \((U^p)' \subset G_n^{(m)}({\mathbb {A}}^{\infty ,p})\) then there is a natural map
These maps are \(G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\)-equivariant (as \((U^p)'\), \(N_1\) and \(N_2\) vary).
Recall from Sect. 5.1 that, if \(N_2' \ge N_2\), then the natural map
restricts to an isomorphism
and is etale in a neighbourhood of \(\overline{{X}}^{{{\text {ord}}},{{\text {min}}}}_{U^p(N_1,N_2')}\). It follows from Lemma 6.7 that
is an isomorphism. We will denote this dagger space simply
The system of dagger spaces \(\{ {\mathcal {X}}^{{{\text {ord}}},{{\text {min}}},\dag }_{U^p(N)} \}\) has an action of \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}}}\).
Let \(\overline{{e}}_{U^p(N_1,N_2)}\) denote the idempotent in
which is 1 on \(\overline{{X}}_{U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}\) and 0 on
(The existence of \(\overline{{e}}_{U^p(N_1,N_2)}\) follows from the results recalled in Sect. 5.1.) Multiplying the terms of \(\overline{{e}}_{U^p(N_1,N_2)}\) by suitable powers of \({{\text {Hasse}}}_{U^p(N_1,N_2)}\), we may suppose that \(\overline{{e}}_{U^p(N_1,N_2)}\) lies in \(H^0(\overline{{X}}_{U^p(N_1,N_2)}^{{\text {min}}}, \omega ^{\otimes (p-1)a})\) for any sufficiently large a, and that
Then
For a sufficiently large we have \(H^1({\mathcal {X}}^{{\text {min}}}_{U^p(N_1,N_2)},\omega ^{\otimes (p-1)a})=(0)\). In that case we can lift \(\overline{{e}}_{U^p(N_1,N_2)}\) to a non-canonical element
Let \({\mathcal {X}}^{{\text {min}}}_{U^p(N_1,N_2)}[1/e_{U^p(N_1,N_2)}]\) denote the locus in \({\mathcal {X}}^{{\text {min}}}_{U^p(N_1,N_2)}\) where \(e_{U^p(N_1,N_2)} \ne 0\). As \(\omega ^{\otimes (p-1)a}\) is ample, \({\mathcal {X}}^{{\text {min}}}_{U^p(N_1,N_2)}[1/e_{U^p(N_1,N_2)}]\) is affine and so has the form
for some s and I. It is normal and flat over \({\mathbb {Z}}_{(p)}\).
For \(r \in p^{{\mathbb {Q}}_{\ge 0}}\) let \(||\,\,\,||_r\) denote the norm on \({\mathbb {Z}}_{(p)}[T_1,\ldots ,T_s]\) defined by
where \(\vec {i}\) runs over \({\mathbb {Z}}_{\ge 0}^s\) and \(|(i_1,\ldots ,i_s)|=i_1+\cdots +i_s\). We will write \({\mathbb {Z}}_p\langle T_1,\ldots ,T_s\rangle _r\) for the completion of \({\mathbb {Z}}_{(p)}[T_1,\ldots ,T_s]\) with respect to \(||\,\,\,||_r\). Thus \({\mathbb {Z}}_p\langle T_1,\ldots ,T_s \rangle _1\) is the p-adic completion of \({\mathbb {Z}}_{(p)}[T_1,\ldots ,T_s]\) and also the p-adic completion of \({\mathbb {Z}}_p\langle T_1,\ldots ,T_s\rangle _{r}\) for any \(r\ge 1\). Set \({\mathbb {Q}}_p\langle T_1,\ldots ,T_s\rangle _r={\mathbb {Z}}_p\langle T_1,\ldots ,T_s\rangle _r[1/p]\), the completion of \({\mathbb {Q}}[T_1,\ldots ,T_s]\) with respect to \(||\,\,\,||_r\). In the case \(r=1\) we will drop it from the notation. We will write \({\mathbb {Z}}_p\langle T_1/r,\ldots ,T_s/r\rangle _1\) for the \(||\,\,\,||_r\) unit-ball in \({\mathbb {Q}}_p\langle T_1,\ldots ,T_s \rangle _r\), i.e. for the set of power series
where \(a_{\vec {i}} \in {\mathbb {Q}}_p\), and \(|a_{\vec {i}}|_p \le r^{-|\vec {i}|}\) for all \(\vec {i}\), and \(|a_{\vec {i}}|_p r^{|\vec {i}|} \rightarrow 0\) as \(|\vec {i}|\rightarrow \infty \). We will also write
Let \(\langle I\rangle _r\) denote the ideal of \({\mathbb {Z}}_p\langle T_1,\ldots ,T_s\rangle _r\) generated by I and let \(\langle I \rangle _r'\) denote the intersection of \(\langle I \rangle _1\) with \({\mathbb {Z}}_p\langle T_1,\ldots ,T_s\rangle _r\). Then \(\langle I \rangle _1\) is the p-adic completion of I. Moreover
is normal and flat over \({\mathbb {Z}}_p\), and
Note that
for all \(r \ge 1\). Thus \((\langle I \rangle _r,p)=(\langle I \rangle _r', p)\).
We will also write \(\langle I \rangle _{r,{\mathbb {Q}}_p}\) (resp. \(\langle I \rangle _{r,{\mathbb {Q}}_p}'\)) for the \({\mathbb {Q}}_p\) span of \(\langle I \rangle _r\) (resp. \(\langle I \rangle _r'\)) in \({\mathbb {Q}}_p\langle T_1,\ldots ,T_s\rangle _r\). Then
are all affinoid subdomains of \(X^{{{\text {min}}},{{\text {an}}}}_{U^p(N_1,N_2)}\), the rigid analytic space associated to the scheme \(X_{U^p(N_1,N_2)}^{{\text {min}}}\times {{\text {Spec}}\,}{\mathbb {Q}}_p\). Thus they are normal. Also \({{\text {Sp}}\,}{\mathbb {Q}}_p\langle T_1,\ldots ,T_s\rangle _r/\langle I \rangle _{r,{\mathbb {Q}}_p}' \) and
form an admissible open cover of \({{\text {Sp}}\,}{\mathbb {Q}}_p\langle T_1,\ldots ,T_s\rangle _r/\langle I \rangle _{r,{\mathbb {Q}}_p}\). (\({{\text {Sp}}\,}{\mathbb {Q}}_p\langle T_1,\ldots ,T_s\rangle _r/\langle I \rangle _{r,{\mathbb {Q}}_p}' \) is the union of the connected components of \({{\text {Sp}}\,}{\mathbb {Q}}_p\langle T_1,\ldots ,T_s\rangle _r/\langle I \rangle _{r,{\mathbb {Q}}_p} \) which contain a component of \({{\text {Sp}}\,}{\mathbb {Q}}_p\langle T_1,\ldots ,T_s\rangle _1/\langle I \rangle _{1,{\mathbb {Q}}_p}\). See proposition 8 of section 9.1.4 of [8].) Moreover \({{\text {Sp}}\,}{\mathbb {Q}}_p\langle T_1,\ldots ,T_s\rangle _1/\langle I \rangle _{1,{\mathbb {Q}}_p}\) is Zariski dense in \({{\text {Sp}}\,}{\mathbb {Q}}_p\langle T_1,\ldots ,T_s\rangle _r/\langle I \rangle _{r,{\mathbb {Q}}_p}\). Indeed
is Zariski dense in \({{\text {Sp}}\,}{\mathbb {Q}}_p\langle T_1,\ldots ,T_s\rangle _r/\langle I \rangle _{r,{\mathbb {Q}}_p}\), where \(X^{{{\text {an}}}}_{U^p(N_1,N_2)}\), the rigid analytic space associated to \(X_{U^p(N_1,N_2)} \times {{\text {Spec}}\,}{\mathbb {Q}}_p\).
If \(1 \le r' < r\) then
and
and these are strict neighbourhoods. The natural maps
and
are completely continuous. The latter is an inclusion. Moreover
Also write \(\langle I \rangle ^\dag \) for the ideal of \({\mathbb {Q}}_p\langle T_1,\ldots ,T_s \rangle ^\dag \) generated by I. Thus
Moreover
and
(See, for instance, proposition 3.3.7 of [46]. For the meaning of \({{\text {Sp}}\,}\) in the context of dagger algebras see section 2.11 of [27].) Thus we have the following lemma.
Lemma 6.11
\({\mathcal {X}}^{{{\text {ord}}},{{\text {min}}},\dag }_{U^p(N)}\) is affinoid.
Suppose that \(\rho _0\) is a representation of \(L_{n,(n)}\) on a finite free \({\mathbb {Z}}_p\)-module and let \(\rho \) denote \(\rho _0\) base changed to \({\mathbb {Q}}_p\). There are \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\) equivariant isomorphisms
There are also natural \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-equivariant embeddings
We will set
a smooth \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\)-module. From Lemma 6.10 and the first observation of the last paragraph, we see that
There is a \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-equivariant embedding
The coherent sheaf \({\mathcal {E}}_{U^p(N_1,N_2),\rho }^{{\text {sub}}}\) gives rise to a coherent sheaf \({\mathcal {E}}_{U^p(N_1,N_2),\rho }^{{{\text {sub}}},{{\text {an}}}}\) on the rigid space \(X_{U^p(N_1,N_2)}^{{{\text {min}}},{{\text {an}}}}\). The inverse system \(\{ {\mathcal {E}}_{U^p(N_1,N_2),\rho }^{{{\text {sub}}},{{\text {an}}}} \}\) is a system of coherent sheaves with \(G_n({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-action on \(\{X_{U^p(N_1,N_2)}^{{{\text {min}}},{{\text {an}}}}\}\).
7.4.2 The Frobenius lift \(\varsigma _p\) and \(\hbox {tr}_F\)
We have a map
such that
-
\(\varsigma _p^*(T_j) \equiv (T_j)^p \bmod p\),
-
and there exists an \(r_1 \in p^{{\mathbb {Q}}_{>0}}\) such that for all \(j=1,\ldots ,s\) the element \(\varsigma _p^*(T_j)\) is in the image of \({\mathbb {Q}}_p\langle T_1,\ldots ,T_s\rangle _{r_1}/\langle I \rangle _{r_1}\).
Thus \((\varsigma _p^*(T_j)-T_j^p)/p \in {\mathbb {Z}}_p\langle T_1,\ldots ,T_s\rangle _r/\langle I\rangle _{r_1}'\), and so is the image of some element \(G_j(\vec {T}) \in {\mathbb {Z}}_p\langle T_1,\ldots ,T_s\rangle _{r_1}\). We have
This formula then defines a map \(\varsigma _p^*: {\mathbb {Z}}_p[T_1,\ldots ,T_s] \rightarrow {\mathbb {Z}}_p\langle T_1,\ldots ,T_s\rangle _{r_1}\) such that
commutes. Write \(G_j(\vec {T})=\sum _{\vec {i}} g_{j,\vec {i}}\vec {T}^{\vec {i}}\). Choose \(I_0 \in {\mathbb {Z}}_{>0}\) such that
for all \(j=1,\ldots ,s\) and then choose \(r_2 \in (1,\sqrt{r_1}) \cap p^{\mathbb {Q}}\) with
If \(r \in [1,r_2] \cap p^{\mathbb {Q}}\) we have
(Because if \(|\vec {i}| \ge I_0\) then \(||pg_{j,\vec {i}}\vec {T}^{\vec {i}}||_r\le (1/p)||G_j||_{r_1} (r/r_1)^{I_0} < 1\), while for \(|\vec {i}| \le I_0\) we have \(||pg_{j,\vec {i}}\vec {T}^{\vec {i}}||_r\le (1/p)r^{I_0} < 1\).) If \(r \in (1,r_2] \cap p^{\mathbb {Q}}\) and \(H \in {\mathbb {Z}}_p[ T_1,\ldots ,T_s]\) we deduce that
(We only need check this on monomials. Hence we only need check that if it is true for \(H_1\) and \(H_2\) then it is also true for \(H_1H_2\). For this one uses the formula
Hence, if \(r \in (1,r_2] \cap p^{\mathbb {Q}}\) and \(H \in {\mathbb {Z}}_p[ T_1,\ldots ,T_s]\) we deduce that
and so \(\varsigma _p^*\) extends to an isometric homomorphism
Modulo p this map reduces to the Frobenius, which is finite and so
is finite. (See section 6.3.2 of [8].) Thus we get an isometric, finite homomorphism between normal rings
such that the diagram
commutes.
The map
is compatible with the map
This latter map is finite, and away from the boundary is flat of degree \(p^{n^2[F^+: {\mathbb {Q}}]}\). Thus the pre-image of \({{\text {Sp}}\,}{\mathbb {Q}}_p\langle T_1,\ldots ,T_s\rangle _{r^p}/\langle I \rangle _{r^p,{\mathbb {Q}}_p}'\) has the form \({{\text {Sp}}\,}B\) where B is a normal, finite \({\mathbb {Q}}_p\langle T_1,\ldots ,T_s\rangle _{r^p}/\langle I \rangle _{r^p,{\mathbb {Q}}_p}'\) algebra, and we have a factorization
For \({\mathfrak {m}}\) a maximal ideal of \({\mathbb {Q}}_p\langle T_1,\ldots ,T_s\rangle _{r^p}/\langle I \rangle _{r^p,{\mathbb {Q}}_p}'\) corresponding to a point of the intersection \(X_{U^p(N_1,N_2)}^{{{\text {an}}}} \cap {{\text {Sp}}\,}{\mathbb {Q}}_p\langle T_1,\ldots ,T_s\rangle _{1}/\langle I \rangle _{1}\) we see that
Thus for a Zariski dense set of maximal ideals \({\mathfrak {m}}\in {{\text {Sp}}\,}{\mathbb {Q}}_p\langle T_1,\ldots ,T_s\rangle _{r^p}/\langle I \rangle _{r^p,{\mathbb {Q}}_p}'\) the map
becomes an isomorphism modulo \({\mathfrak {m}}\). Hence for any minimal prime \(\wp \) of the ring \({\mathbb {Q}}_p\langle T_1,\ldots ,T_s\rangle _{r^p}/\langle I \rangle _{r^p,{\mathbb {Q}}_p}'\) we have
(Choose bases over \(A_\wp /\wp \). Then this map being an isomorphism is equivalent to some matrix having full rank. For \({\mathfrak {m}}\) in a dense Zariski open set these bases reduce to bases modulo \({\mathfrak {m}}\). So modulo a Zariski dense set of \({\mathfrak {m}}\) this matrix has full rank, so it has full rank.) As B is normal and \({\mathbb {Q}}_p\langle T_1,\ldots ,T_s\rangle _{r}/\langle I \rangle _{r,{\mathbb {Q}}_p}'\) is finite over B, we see that
i.e.
The sheaf \({\mathcal {E}}_{U^p(N_1,N_2),\rho }^{{\text {sub}}}\) induces a coherent sheaf \({\mathcal {E}}_{U^p(N_1),\rho }^{{{\text {sub}}},\dag }\) on \({\mathcal {X}}_{U^p(N_1)}^{{{\text {ord}}},{{\text {min}}},\dag }\), which does not depend on \(N_2\). It equals the push-forward from any \({\mathcal {X}}^{{{\text {ord}}},\dag }_{U^p(N_1),\varDelta }\) of the sheaf \({\mathcal {E}}_{U^p(N_1),\rho }^{{{\text {sub}}},\dag }\). The inverse system \(\{ {\mathcal {E}}_{U^p(N),\rho }^{{{\text {sub}}},\dag } \}\) is a system of coherent sheaves with \(G_n({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-action on \(\{{\mathcal {X}}_{U^p(N_1)}^{{{\text {ord}}},{{\text {min}}},\dag }\}\). The map
over \({\mathcal {X}}^{{{\text {ord}}},\dag }_{U^p(N_1),\varDelta }\) induces a map
over \({\mathcal {X}}_{U^p(N_1)}^{{{\text {ord}}},{{\text {min}}},\dag }\). This map does not depend on the choice of \(\varDelta \) and is \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-equivariant. It satisfies
It induces a map
also satisfying
We again remark that \({{\text {tr}}}_F\) is closely related to the operator often denoted \(U_p\)—probably they differ simply by a scalar multiple.
The isomorphisms
are \({{\text {tr}}}_F\)-equivariant. Moreover the space \(H^0({\mathcal {X}}^{{{\text {ord}}},{{\text {min}}},\dag },{\mathcal {E}}_\rho ^{{\text {sub}}})_{\overline{{{\mathbb {Q}}}}_p}\) inherits an endomorphism \({{\text {tr}}}_F\), which commutes with \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) and satisfies \({{\text {tr}}}_F \circ \varsigma _p=p^{n^2[F^+: {\mathbb {Q}}]}\).
The sheaf \({\mathcal {E}}_{U^p(N_1,N_2),\rho }^{{{\text {sub}}},{{\text {an}}}}\) restricted to the space \({{\text {Sp}}\,}{\mathbb {Q}}_p\langle T_1,\ldots ,T_s\rangle _r/\langle I \rangle _{r,{\mathbb {Q}}_p}'\) corresponds to a finitely generated module \(E_r\) over the ring \({\mathbb {Q}}_p\langle T_1,\ldots ,T_s\rangle _r/\langle I \rangle _{p,{\mathbb {Q}}_p}'\), which is naturally a Banach module. If \(r' < r\) then
Then the map \(E_r \rightarrow E_{r'}\), which we will also denote \(i_{r,r'}'\), is completely continuous. The map \({{\text {tr}}}_F\) extends to a continuous \({\mathbb {Q}}_p\langle T_1,\ldots ,T_s\rangle _{r^p}/\langle I \rangle _{r^p,{\mathbb {Q}}_p}'\) linear map
for \(r \in [1,r_2] \cap p^{\mathbb {Q}}\). We set
so that
We have that \({{\text {tr}}}_F|_{E_r}=t_r\). As \(t_r\) is continuous and \(i_{r^p,r}'\) is completely continuous we see that
and that this map is completely continuous. Thus each \(E_r\) admits slope decompositions for \({{\text {tr}}}_F\) and hence by Lemma 6.6 so does \(E^\dag \) and \(E^\dag \otimes \overline{{{\mathbb {Q}}}}_p\).
If \(a \in {\mathbb {Q}}\) we thus have a well-defined, finite dimensional subspace
(Defined with respect to \({{\text {tr}}}_F\).) We set
so that there are \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-equivariant embeddings
We have proved the following lemma. (The referee suggests, in politer terms, that we have made a mountain out of a mole hill in proving this lemma and Lemma 6.11. The referee is probably correct. We are not very practiced at these sorts of arguments. Neither lemma will come as any surprise to experts.)
Lemma 6.12
\(H^0({\mathcal {X}}^{{{\text {ord}}},{{\text {min}}},\dag }, {\mathcal {E}}_{\rho }^{{\text {sub}}})_{\overline{{{\mathbb {Q}}}}_p,\le a}\) is an admissible \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-module.
Combining this with Corollary 6.5 we obtain the following result.
Corollary 6.13
Suppose that \(\rho \) is a representation of \(L_{n,(n)}\) over \({\mathbb {Q}}\), that \(a \in {\mathbb {Q}}\) and that \(\varPi \) is an irreducible \(G_n({\mathbb {A}}^{\infty })^{{{\text {ord}}},\times }\)-subquotient of
Then there is a continuous semi-simple representation
with the following property: If \(\varPi \) is unramified at a rational prime \(q\ne p\) (in the sense of Sect. 1.3) and if v|q is a prime of F, then
We will next explain the consequences of these results for sheaves of differentials on \(A^{(m),{{\text {ord}}},\dag }_{U^p(N_1,N_2),\varSigma }\). But we first need to record a piece of commutative algebra.
Lemma 6.14
Suppose that \(A \rightarrow B \rightarrow C\) are reduced noetherian rings, with B a finite flat A module of rank \(r_B\) and C a finite flat A-module of rank \(r_C\). Suppose also that the total ring of fractions of C is finite flat over the total ring of fractions of B. Then \(r_B|r_C\) and
Proof
It suffices to check this after passing to total rings of fractions (i.e. localizations at the set of nonzero divisors). In this case B is free over A and C is free over B, so the lemma is clear. \(\square \)
Proposition 6.15
There are representations \(\rho ^{i,j}_{m,s}\) of \(L_{n,(n)}\) over \({\mathbb {Q}}\) with the following property. If \((U^p(N),\varSigma ) \in {\mathcal {J}}^{(m),{{\text {tor}}},{{\text {ord}}}}_n\) and if \((U^p)'\) denotes the image of \(U^p \) in \(G_n({\mathbb {A}}^{\infty ,p})\), then there is a spectral sequence with first page
These spectral sequences are equivariant for the action of \(G_n({\mathbb {A}}^{\infty })^{{\text {ord}}}\). The map \({{\text {tr}}}_F\) on the \(H^{i+j}({\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\varSigma },\varOmega ^s_{A^{(m),{{\text {ord}}},\dag }_{U^p(N),\varSigma }}(\log \infty ) \otimes {\mathcal {I}}_{\partial {\mathcal {A}}_{U^p(N),\varSigma }^{(m),{{\text {ord}}},\dag }})\) is compatible with the map \(p^{nm[F: {\mathbb {Q}}]}{{\text {tr}}}_F\) on the \(H^0({\mathcal {X}}^{{{\text {ord}}},{{\text {min}}},\dag }_{(U^p)'(N)}, {\mathcal {E}}^{{{\text {sub}}},\dag }_{(U^p)'(N),\rho _{m,s}^{i,j}})\).
Proof
Let \(\pi \) denote the map \({\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\varSigma } \rightarrow {\mathcal {X}}^{{{\text {ord}}},{{\text {min}}},\dag }_{(U^p)'(N)}\). Lemmas 5.10 and 6.9 tell us that there is a spectral sequence of coherent sheaves on \({\mathcal {X}}^{{{\text {ord}}},{{\text {min}}},\dag }_{U^p(N)}\) with first page
The first assertion follows from Lemma 6.11 of this paper and from proposition 3.1 of [27] (which tell us that
for \(k>0\)).
For the last assertion we may replace \(\varSigma \) by a refinement and so reduce to the case that there is a \(\varDelta \) with \(((U^p)',\varDelta ) \in {\mathcal {J}}_n^{{{\text {tor}}},{{\text {ord}}}}\) and \(((U^p)'(N), \varDelta ) \le (U^p(N),\varSigma )\). (Use Lemma 5.6.) To avoid confusion we will write \(\varsigma _{p,{\mathcal {A}}}\) or \(\varsigma _{p,{\mathcal {X}}}\) depending on whether \(\varsigma _p\) is acting on \({\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\varSigma ^{{\text {ord}}}}\) or \({\mathcal {X}}^{{{\text {ord}}},\dag }_{(U^p)'(N),\varDelta ^{{\text {ord}}}}\). We will also factorize \(\varsigma _{p,{\mathcal {A}}}\) as
Write \(\pi '\) for the map
and \(\pi ''\) for the map
The sheaf \({\mathcal {E}}^{{{\text {can}}},\dag }_{\rho _{m,s}^{i,j}}\) on \({\mathcal {X}}^{{{\text {ord}}},\dag }_{(U^p)'(N),\varDelta ^{{\text {ord}}}}\) is \(R^i\pi _*' {\mathcal {F}}_{j}\), where
To prove the last sentence of the lemma it suffices to show that the diagrams
and
commute. In the first diagram the upper horizontal map is the composite
and the lower horizontal map is
We see that the first square tautologically commutes. The second square commutes because the two maps
and
are equal. This in turn follows from Lemma 6.14. \(\square \)
Corollary 6.16
For all i and s the vector space \(H^{i}({\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{ U^p(N),\varSigma },\varOmega ^s_{{\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\varSigma }}(\log \infty ) \otimes {\mathcal {I}}_{\partial {\mathcal {A}}_{U^p(N),\varSigma }^{(m),{{\text {ord}}},\dag }})\) admits slope decompositions for \({{\text {tr}}}_F\).
We write
The next corollary now follows from the proposition and Lemma 6.6.
Corollary 6.17
For any \(a \in {\mathbb {Q}}\) there is a \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-equivariant spectral sequence with first page \( E_1^{i,j}\):
Combining this with Corollary 6.13 we obtain the following corollary.
Corollary 6.18
Suppose that \(\varPi \) is an irreducible \(G_n({\mathbb {A}}^{\infty })^{{{\text {ord}}},\times }\)-subquotient of
for some \(a \in {\mathbb {Q}}\). Then there is a continuous semi-simple representation
with the following property: If \(\varPi \) is unramified at a rational prime \(q\ne p\) (in the sense of Sect. 1.3) and if v|q is a prime of F, then
7.5 Rigid cohomology
Our main object of study will be the groups
where \((U^p(N),\varSigma ) \in {\mathcal {J}}_n^{(m),{{\text {tor}}},{{\text {ord}}}}\). This can be thought of as a sort of rigid cohomology of \(\overline{{A}}^{(m),{{\text {ord}}}}_{U^p(N),\varSigma }\) with compact supports towards the toroidal boundary, but not towards the non-ordinary locus. It seems plausible to us that this can be intrinsically attached to the pair \(\overline{{A}}^{(m),{{\text {ord}}}}_{U^p(N)} \supset \partial \overline{{A}}^{(m),{{\text {ord}}}}_{U^p(N)}\). Hence our notation. However we will not prove this, so the reader is cautioned that our notation is nothing more than a short-hand, and the group \(H^i_{c-\partial }(\overline{{A}}^{(m),{{\text {ord}}}}_{U^p(N),\varSigma })\) must be assumed to depend on the pair \({\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\varSigma } \supset \partial {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\varSigma }\). We will also set
It has a smooth action of \(G_n({\mathbb {A}}^{\infty })^{{\text {ord}}}\). The maps
induce endomorphisms
which commute with the action of \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) and satisfy
Lemma 6.19
There are natural isomorphisms
Proof
Use Lemmas 5.1, 5.6, 5.7 and 6.9. \(\square \)
We will compute the group \(H^i_{c-\partial }(\overline{{A}}^{(m),{{\text {ord}}}}_{U^p(N),\varSigma })\) in two ways. The first way will be in terms of p-adic cusp forms and will allow us to attach Galois representations to irreducible \(G_n({\mathbb {A}}^{\infty })^{{{\text {ord}}},\times }\)-subquotients of \(H^i_{c-\partial }(\overline{{A}}^{(m),{{\text {ord}}}}) \otimes _{{\mathbb {Q}}_p} \overline{{{\mathbb {Q}}}}_p\). The second way will be geometrical, in terms of the stratification of the boundary. In this second approach the cohomology of the locally symmetric spaces associated to \(L_{n,(n),{{\text {lin}}}}^{(m)}\) will appear.
Here is our first calculation.
Lemma 6.20
The vector spaces \(H^i_{c-\partial }(\overline{{A}}^{(m),{{\text {ord}}}}_{U^p(N),\varSigma })\) admit slope decompositions for \({{\text {tr}}}_F\). If moreover we set
then there is a \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-spectral sequence with first page
Proof
This follows from Lemma 6.6, Corollary 6.16 and the spectral sequence
\(\square \)
And here is our second calculation.
Lemma 6.21
There are \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-equivariant spectral sequences with first page
Moreover the action of Frobenius on the left hand side is compatible with the action of \(\varsigma _p\) on the right hand side.
Proof
By Lemmas 2.3 and 6.9 the group \(H^{i}_{c-\partial }(\overline{{A}}^{(m),{{\text {ord}}}}_{U^p(N),\varSigma })\) is isomorphic to the hypercohomology of the double complex
and so there is a spectral sequence with first page
However, by Lemma 6.8 and the quasi-projectivity of \(\partial ^{(j)} {\mathcal {A}}^{(m),{{\text {ord}}}}_{U^p(N),\varSigma }\), we see that there are \(G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\)-equivariant isomorphisms
and that under this identification \(\varsigma _p\) corresponds to Frobenius (because \(\varsigma _p\) equals Frobenius on the special fibre). \(\square \)
Corollary 6.22
\(H^{i}_{c-\partial }(\overline{{A}}^{(m),{{\text {ord}}}}_{U^p(N),\varSigma })\) is finite dimensional. Moreover
for some a, and so
Proof
The first assertion follows from the lemma and theorem 3.1 of [7]. The second assertion follows because \({{\text {tr}}}_F \circ \varsigma _p=p^{n(n+2m)[F^+: {\mathbb {Q}}]}\) and so by the first part \({{\text {tr}}}_F\) must be an automorphism of \(H^{i}_{c-\partial }(\overline{{A}}^{(m),{{\text {ord}}}}_{U^p(N),\varSigma })\).\(\square \)
Combining this with Corollary 6.18 and Lemma 6.20 we obtain the following corollary.
Corollary 6.23
Suppose that \(\varPi \) is an irreducible \(G_n({\mathbb {A}}^{\infty })^{{{\text {ord}}},\times }\)-subquotient of
Then there is a continuous semi-simple representation
with the following property: If \(\varPi \) is unramified at a rational prime \(q\ne p\) (in the sense of Sect. 1.3) and if v|q is a prime of F. Then
Corollary 6.24
The eigenvalues of \(\varsigma _p\) on \(H^{i}_{c-\partial }(\overline{{A}}^{(m),{{\text {ord}}}})_{\overline{{{\mathbb {Q}}}}_p}\) are Weil \(p^w\)-numbers for some \(w \in {\mathbb {Z}}_{\ge 0}\) (depending on the eigenvalue). We will write
for the subspace of \(H^{i}_{c-\partial }(\overline{{A}}^{(m),{{\text {ord}}}})_{\overline{{{\mathbb {Q}}}}_p}\) spanned by generalized eigenspaces of \(\varsigma _p\) with eigenvalue a \(p^0\)-Weil number.
For \(i>0\) there is a \(G_n({\mathbb {A}}^{\infty })^{{\text {ord}}}\)-equivariant isomorphism
(For \(i=0\) there is still a surjection.)
Proof
By theorem 2.2 of [19], the eigenvalues of the Frobenius endomorphism on \(H^i_{{{\text {rig}}}}(\partial ^{(j)} \overline{{A}}^{(m),{{\text {ord}}}}_{U^p(N),\varSigma })\) are all Weil \(p^w\)-numbers for some \(w \in {\mathbb {Z}}_{\ge i}\) (depending on the eigenvalue). The first part of the corollary follows.
It follows moreover that \(W_0 H^{i}_{c-\partial }(\overline{{A}}^{(m),{{\text {ord}}}}_{U^p(N),\varSigma })_{\overline{{{\mathbb {Q}}}}_p}\) is the cohomology of the complex
However by proposition 8.2.15 of [46]
and so the cohomology of the above complex becomes
The last part of the corollary follows. \(\square \)
The discussion at the end of Sect. 5.3 shows that there are \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\)-equivariant open embeddings
Thus the following corollary follows by applying Lemma 1.7 and Corollary 1.6.
Corollary 6.25
For \(i>0\),
is a \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\) subquotient of \(W_0 H^{i+1}_{c-\partial }(\overline{{A}}^{(m),{{\text {ord}}}}_{n})_{\overline{{{\mathbb {Q}}}}_p}\).
Combining this proposition with Corollary 6.23 (and using Lemma 1.1) we deduce the following consequence.
Corollary 6.26
Suppose that \(i>0\) and that \(\pi \) is an irreducible \(L_{n,(n),{{\text {lin}}}}({\mathbb {A}}^\infty )\)-subquotient of \(H^i_{{{\text {Int}}}}({\mathfrak {T}}_{(n)}^{(m)},\overline{{{\mathbb {Q}}}}_p)\). Then there is a continuous semi-simple representation
with the following property: Suppose that \(q \ne p\) is a rational prime which either splits in \(F_0\) or is unramified in F. Suppose further that \(\pi \) is unramified at all primes of F above q. If v|q is a prime of F, then
Combining this with Corollary 1.9 we obtain the following result.
Corollary 6.27
Suppose that \(n>1\), that \(\rho \) is an irreducible algebraic representation of \(L_{n,(n),{{\text {lin}}}}\) on a finite dimensional \({\mathbb {C}}\)-vector space and that \(\pi \) is a cuspidal automorphic representation of \(L_{n,(n),{{\text {lin}}}}({\mathbb {A}})\) so that \(\pi _\infty \) has the same infinitesimal character as \(\rho ^\vee \). Then, for all sufficiently large integers N, there is a continuous, semi-simple representation
with the following property: Suppose that \(q \ne p\) is a rational prime which either splits in \(F_0\) or is unramified in F. Suppose further that \(\pi \) is unramified at all primes of F above q. If v|q is a prime of F, then
Proof
Take
\(\square \)
8 Galois representations
In order to improve upon Corollary 6.27 it is necessary to apply some simple group theory. To this end, let \(\varGamma \) be a topological group and let \({\mathfrak {F}}\) be a dense set of elements of \(\varGamma \). Let k be an algebraically closed, topological field of characteristic 0 and let \(d \in {\mathbb {Z}}_{>0}\).
Let
be a continuous homomorphism such that \(\mu (f)\) has infinite order for all \(f \in {\mathfrak {F}}\). For \(f \in {\mathfrak {F}}\) let \({\mathcal {E}}_f^1\) and \({\mathcal {E}}_f^2\) be two d-element multisets of elements of \(k^\times \). Let \({\mathcal {M}}\) be an infinite subset of \({\mathbb {Z}}\). For \(m \in {\mathcal {M}}\) let
be a continuous semi-simple representation such that for every \(f \in {\mathfrak {F}}\) the multiset of roots of the characteristic polynomial of \(\rho _m(f)\) equals
Suppose that \({\mathcal {M}}'\) is a finite subset of \({\mathcal {M}}\). Let \(G_{{\mathcal {M}}'}\) denote the Zariski closure in \({\mathbb {G}}_m \times GL_{2d}^{{\mathcal {M}}'}\) of the image of
It is a, possibly disconnected, reductive group. There is a natural continuous homomorphism
Note that \(\rho _{{\mathcal {M}}'}({\mathfrak {F}})\) is Zariski dense in \(G_{{\mathcal {M}}'}\). We will use \(\mu \) for the character of \(G_{{\mathcal {M}}'}\) which is projection to \({\mathbb {G}}_m\). For \(m \in {\mathcal {M}}'\) we will let
denote the projection to the factor indexed by m.
Lemma 7.1
For every \(g \in G_{{\mathcal {M}}'}(k)\) there are two d-element multisets \(\varSigma _g^1\) and \(\varSigma _g^2\) of elements of \(k^\times \) such that for every \(m \in {\mathcal {M}}'\) the multiset of roots of the characteristic polynomial of \(R_m(g)\) equals
Proof
It suffices to show that the subset of \(k^\times \times GL_{2d}^{{\mathcal {M}}'}(k)\) consisting of elements \((t,(g_m)_{m \in {\mathcal {M}}'})\) such that there are d-element multisets \(\varSigma ^1\) and \(\varSigma ^2\) of elements of \(k^\times \) such that for all \(m \in {\mathcal {M}}'\) the multiset of roots of the characteristic polynomial of \(g_m\) equals \(\varSigma ^1 \amalg \varSigma ^2t^m\), is Zariski closed. Let \({{\text {Pol}}}_{2d}\) denote the space of monic polynomials of degree 2d. It even suffices to show that the subset X of \(k^\times \times {{\text {Pol}}}_{2d}^{{\mathcal {M}}'}(k)\) consisting of elements \((t,(P_m)_{m \in {\mathcal {M}}'})\) such that there are d-element multisets \(\varSigma ^1\) and \(\varSigma ^2\) of elements of k such that for all \(m \in {\mathcal {M}}'\) the multiset of roots of \(P_m\) equals \(\varSigma ^1 \amalg \varSigma ^2t^m\), is Zariski closed.
There is a natural finite map
If
where \(S_{2d}\) denotes the symmetric group on 2d letters, define \(V_{(\sigma _m)}\) to be the set of
such that, for all \(m,m' \in {\mathcal {M}}'\) we have
if \(i=1,\ldots ,d\) and
if \(i=d+1,\ldots ,2d\). Then \(V_{(\sigma _m)}\) is closed in \({\mathbb {G}}_m \times ({{\text {Aff}}}^{2d})^{{\mathcal {M}}'}\). Moreover
The lemma now follows from the finiteness of \(1 \times \pi ^{{\mathcal {M}}'}\). \(\square \)
Corollary 7.2
If \(\emptyset \ne {\mathcal {M}}' \subset {\mathcal {M}}''\) are finite subsets of \({\mathcal {M}}\) then \(G_{{\mathcal {M}}''} \mathop {\rightarrow }\limits ^{\sim }G_{{\mathcal {M}}'}\).
Proof
Suppose that g is in the kernel of the natural map
Then for all \(m \in {\mathcal {M}}''\) the only eigenvalue of \(R_m(g)\) is 1. Thus g must be unipotent. However \(\ker (G_{{\mathcal {M}}''} \twoheadrightarrow G_{{\mathcal {M}}'})\) is reductive and so must be trivial. \(\square \)
Thus we can write G for \(G_{{\mathcal {M}}'}\) without danger of confusion.
Corollary 7.3
For every \(g \in G(k)\) there are two d-element multisets \(\varSigma _g^1\) and \(\varSigma _g^2\) of elements of \(k^\times \) such that for every \(m \in {\mathcal {M}}\) the multiset of roots of the characteristic polynomial of \(R_m(g)\) equals
Moreover if \(\mu (g)\) has infinite order then the multisets \(\varSigma ^1_g\) and \(\varSigma ^2_g\) are unique.
Proof
Choose non-empty finite subsets
with
For each i we can find two d-element multisets \(\varSigma _{g,i}^1\) and \(\varSigma _{g,i}^2\) of elements of \(k^\times \) such that for every \(m \in {\mathcal {M}}_i'\) the multiset of roots of the characteristic polynomial of \(R_m(g)\) equals
Let \(m_1 \in {\mathcal {M}}_1'\) and let \(\varSigma \) denote the set of eigenvalues of \(R_{m_1}(g)\). Then, for every i, the multiset \(\varSigma ^1_{g,i}\) consists of elements of \(\varSigma \) and the multiset \(\varSigma ^2_{g,i}\) consists of elements of \(\varSigma \mu (g)^{-m_1}\). Thus there are only finitely many possibilities for the pair of multisets \((\varSigma _{g,i}^1,\varSigma _{g,i}^2)\) as i varies. Hence some such pair \((\varSigma _{g}^1,\varSigma _{g}^2)\) occurs infinitely often. This pair satisfies the requirements of the corollary.
For uniqueness suppose that \(\varSigma _g^{1,\prime }\) and \(\varSigma _g^{2,\prime }\) is another such pair of multisets. Choose \(m \in {\mathcal {M}}\) with \(\mu (g)^m \ne \alpha /\beta \) for any \(\alpha ,\beta \in \varSigma _g^1 \amalg \varSigma _g^2 \amalg \varSigma _g^{1,\prime } \amalg \varSigma _g^{2,\prime }\). Then the equality
implies that \(\varSigma ^{1,\prime }_g=\varSigma _g^1\) and \(\varSigma ^{2,\prime }_g=\varSigma _g^2\). \(\square \)
The connected component \(Z(G)^0\) of the centre of G is a torus.
Lemma 7.4
The character \(\mu \) is non-trivial on \(Z(G)^0\).
Proof
If \(\mu \) were trivial on \(Z(G^0)^0\) then it would be trivial on \(G^0\) (because \(G^0/Z(G^0)^0\) is semi-simple), and so \(\mu \) would have finite order, a contradiction. Thus \(\mu |_{Z(G^0)^0}\) is non-trivial.
The space
is a representation of the finite group \(G/G^0\) and we can decompose
where Y is a \({\mathbb {Q}}[G/G^0]\)-module with
But
is non-trivial, and so \(\mu |_{Z(G)^0}\) is non-trivial. \(\square \)
For \(m \in {\mathcal {M}}\) let \({\mathfrak {X}}_m\) denote the 2d-element multiset of characters of \(Z(G)^0\) which occur in \(R_m\) (taken with their multiplicity). If \(g \in G\) then we will write \({\mathfrak {Y}}(g)_m\) for the 2d-element multiset of pairs \((\chi ,a)\), where \(\chi \) is a character of \(Z(G)^0\) and a is a root of the characteristic polynomial of g acting on the \(\chi \) eigenspace of \(Z(G)^0\) in \(R_m\). (The pair \((\chi ,a)\) occurs with the same multiplicity as a has as a root of the characteristic polynomial of g acting on the \(\chi \)-eigenspace of \(R_m\).)
If \({\mathfrak {Y}}\subset {\mathfrak {Y}}(g)_m\) and if \(\psi \in X^*(G)\) then we will set
We warn the reader that this depends on g and not just on the set \({\mathfrak {Y}}\).
Lemma 7.5
Suppose that T / k is a torus and that \({\mathfrak {X}}\) is a finite set of non-trivial characters of T. Let A be a finite subset of \(k^\times \). Then we can find \(t \in T(k)\) such that \(\chi (t) \ne a\) for all \(\chi \in {\mathfrak {X}}\) and \(a \in A\).
Proof
Let \((\,\, ,\,\,)\) denote the usual perfect pairing
We can find \(\nu \in X_*(T)\) such that \((\chi ,\nu ) \ne 0\) for all \(\chi \in {\mathfrak {X}}\). Thus we are reduced to the case \(T={\mathbb {G}}_m\), in which case we may take t to be any element of \(k^\times \) that does not lie in the divisible hull of the subgroup H of \(k^\times \) generated by A. (For example, we can take t to be a rational prime such that all elements of a finite set of generators of \(H \cap {\mathbb {Q}}^\times \) are units at t.) \(\square \)
Corollary 7.6
Suppose that T / k is a torus and that \({\mathfrak {X}}\) is a finite set of characters of T. Then we can find \(t \in T(k)\) such that if \(\chi \ne \chi '\) lie in \({\mathfrak {X}}\) then
Lemma 7.7
If \(m,m',m'' \in {\mathcal {M}}\), then we can decompose
into two d-element multisets, such that
and
If \(\mu ^{m-m'} \ne \chi /\chi '\) for all \(\chi , \chi ' \in {\mathfrak {X}}_m\) then the equation
uniquely determines this decomposition.
Proof
Choose \(t \in Z(G)^0(k)\) such that \(a\chi (t) \ne a'\chi '(t)\) for \((\chi ,a) \ne (\chi ',a')\) with
(Note that it suffices to choose \(t \in Z(G)^0(k)\) such that for
with \(\chi \ne \chi '\) we have \((\chi /\chi ')(t) \ne a'/a\).) We can decompose
into two d-element multisets, such that
and
Then
and
It follows that
and
If \(\mu ^{m-m'} \ne \chi /\chi '\) for all \(\chi , \chi ' \in {\mathfrak {X}}_m\) then
so the uniqueness assertion is clear. \(\square \)
Corollary 7.8
If \(m \in {\mathcal {M}}\), then we can uniquely decompose
into two d-element multisets, such that for all \(m'\in {\mathcal {M}}\) we have
Proof
Choose \(m'\) such that \(\mu ^{m-m'} \ne \chi /\chi '\) for all \(\chi , \chi ' \in {\mathfrak {X}}_m\). Then we see that for all \(m'', m''' \in {\mathcal {M}}\) we have
and
Then we can simply take \({\mathfrak {Y}}(g)_m^i={\mathfrak {Y}}(g)_{m,m',m''}^i\). \(\square \)
Corollary 7.9
For all \(m,m' \in {\mathcal {M}}\) we have
and
Proof
It is immediate from the previous corollary that \({\mathfrak {Y}}(g)_m^1\) and \({\mathfrak {Y}}(g)_m^2 \mu ^{m'-m}\) have the properties that uniquely characterize \({\mathfrak {Y}}(g)_{m'}^1\) and \({\mathfrak {Y}}(g)_{m'}^2\). \(\square \)
Corollary 7.10
For all \(g \in G\) and \(m \in {\mathcal {M}}\) and for \(i=1,2\) we have
Proof
It is again immediate that \(\{ (\chi ,1): \exists a, (\chi ,a) \in {\mathfrak {Y}}(g)_m^1\}\) and \(\{ (\chi ,1): \exists a, (\chi ,a) \in {\mathfrak {Y}}(g)_m^2\}\) have the properties that uniquely characterize \({\mathfrak {Y}}(1)_{m}^1\) and \({\mathfrak {Y}}(1)_{m}^2\). \(\square \)
We set
Note that
and that
Corollary 7.11
For all but finitely many \(m \in {\mathcal {M}}\) the multisets \({\mathfrak {X}}_m^1\) and \({\mathfrak {X}}_m^2\) are disjoint.
Let \({\mathcal {M}}'\) denote the set of \(m \in {\mathcal {M}}\) such that \({\mathfrak {X}}_m^1\) and \({\mathfrak {X}}_m^2\) are disjoint. Then we see that for \(m \in {\mathcal {M}}'\) we have
Moreover for \(m \in {\mathcal {M}}'\) we may decompose
where \(R_m^i\) is the sum of the \(\chi \)-eigenspaces of \(Z(G)^0\) for \(\chi \in {\mathfrak {X}}_m^i\). We see that the multiset of roots of the characteristic polynomial of \(R_m^i(g)\) equals
Thus \(R_m^1\) is independent of \(m \in {\mathcal {M}}'\), as is \(R_m^2 \mu ^{-m}\). Denote these representations of G by \(r_1\) and \(r_2\), so that
for all \(m \in {\mathcal {M}}'\). From Corollary 7.3 (applied to \({\mathcal {M}}'\)) we see that if \(g \in G\) and \(\mu (g)\) has infinite order then \(\varSigma ^i_g\) is the multiset of roots of the characteristic polynomial of \(r^i(g)\). Thus we have proved the following result.
Proposition 7.12
Keep the notation and assumptions of the first two paragraphs of this section. Then there are continuous semi-simple representations
for \(i=1,2\) such that for all \(f \in {\mathfrak {F}}\) the multiset of roots of the characteristic polynomial of \(\rho ^i(f)\) equals \({\mathcal {E}}_f^i\).
This proposition allows us to deduce our main theorem from Corollary 6.27.
Theorem 7.13
Suppose that \(\pi \) is a cuspidal automorphic representation of \(GL_n({\mathbb {A}}_F)\) such that \(\pi _{\infty }\) has the same infinitesimal character as an algebraic representation of \({{\text {RS}}}^F_{\mathbb {Q}}GL_n\). Then there is a continuous semi-simple representation
with the following property: Suppose that \(q \ne p\) is a rational prime which either splits in \(F_0\) or is unramified in F. Suppose further that \(\pi \) is unramified at all primes of F above q. If v|q is a prime of F, then
Proof
We may suppose that \(n>1\), as in the case \(n=1\) the result is well known. Let S denote the set of rational primes above which F or \(\pi \) ramifies together with p; and let \(G_{F,S}\) denote the Galois group over F of the maximal extension of F unramified outside S. Apply Proposition 7.12 to \(\varGamma =G_{F,S}\), and \(k=\overline{{{\mathbb {Q}}}}_p\), and \(\mu =\epsilon _p^{-2}\), and \({\mathcal {M}}\) consisting of all sufficiently large integers, and \(\rho _m=R_{p,\imath }(\pi ,m)\) (as in Theorem 6.27), and \({\mathcal {F}}\) the set of Frobenius elements at primes not above S, and \({\mathcal {E}}_{{{\text {Frob}}}_v}^1\) equal to the multiset of roots of the characteristic polynomial of \( \imath ^{-1}{\text {rec}}_{F_v}(\pi _{v}|\det |_v^{(1-n)/2})({{\text {Frob}}}_v)\), and \({\mathcal {E}}_{{{\text {Frob}}}_v}^2\) equal to the multiset of roots of the characteristic polynomial of \(\imath ^{-1}{\text {rec}}_{F_{{}^cv}}(\pi _{{}^cv}|\det |_{{}^cv}^{(-1+3n)/2})({{\text {Frob}}}_{{}^cv}^{-1})\). \(\square \)
Corollary 7.14
Suppose that E is a totally real or CM field and that \(\pi \) is a cuspidal automorphic representation such that \(\pi _{\infty }\) has the same infinitesimal character as an algebraic representation of \({{\text {RS}}}^E_{\mathbb {Q}}GL_n\). Then there is a continuous semi-simple representation
such that, if \(q \ne p\) is a rational prime above which \(\pi \) is unramified and if v|q is a prime of E, then \(r_{p,\imath }(\pi )\) is unramified at v and
Proof
This can be deduced from Theorem 7.13 by using lemma 1 of [54]. (This is the same argument used in the proof of theorem VII.1.9 of [29].) \(\square \)
Acknowledgements
We are pleased to dedicate this paper to the memory of our friend Robert Coleman, who was both a personal and professional inspiration. This paper owes a lot to his ideas. The origin of this paper was the conviction that one should be able to relate all the cohomology of a Shimura variety to overconvergent p-adic automorphic forms. The source of this conviction was [22].
We would all like to thank the Institute for Advanced Study for its support and hospitality. This project was begun, and the key steps completed, while we were all attending the special IAS special year on ‘Galois representations and automorphic forms’. M.H.’s research received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement Number 290766 (AAMOT).
K.-W.L.’s research was partially supported by NSF Grants DMS-1069154, DMS-1258962 and DMS-1352216 and by an Alfred P. Sloan Research Fellowship. R.T.’s research was partially supported by NSF Grants DMS-0600716,
DMS-1062759 and DMS-1252158 and by the IAS Oswald Veblen and Simonyi Funds. During some of the period when this research was being written up J.T. served as a Clay Research Fellow.
We would like to thank the referee for their helpful comments. We would also like to thank Elly Gustafsson and Anthony Pulido for help in compiling the index of notations.
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Appendices
Appendix A. Guide to notation
As the paper contains a lot of different notations, we will informally summarize some of the principal notations in this section. We hope that this will help the reader. Formal definitions will be given in the main body of the paper.
1.1 Shimura varieties
We fix a totally real field \(F^+\), an imaginary quadratic field \(F_0\) and let F denote their composite. We also fix a prime p which splits in \(F_0\).
We will consider a quasi-split unitary similitude group \(G_n/{\mathbb {Q}}\) defined with respect to (the trace of) the skew-hermitian form on \(V_n=F^{2n}\)
where \(J_n\) is an anti-diagonal matrix with 1’s in the first n rows and \(-1\)’s in the last n rows. The similitude factor \(\nu : G_n \rightarrow {\mathbb {G}}_m\). (See Sect. 1.1.) For \(U \subset G_n({\mathbb {A}}^\infty )\) a neat open compact subgroup we have a corresponding Shimura variety \(X_{n,U}\) and over it a universal abelian scheme \(A^{{\text {univ}}}\), which is unique up to unique quasi-isogeny. The system \(\{ X_{n,U}\}\) has an action of \(G_n({\mathbb {A}}^\infty )\). (See Sect. 3.1.)
We will also consider the group
It has a (left) action of \({{\text {RS}}}^F_{\mathbb {Q}}GL_m\). (See Sect. 1.1.) For \(U \subset G_n^{(m)}({\mathbb {A}}^\infty )\) a neat open compact subgroup we have the corresponding Kuga–Sato variety \(A_{n,U}^{(m)}\) which is smooth and projective over \(X_{n,U'}\), where \(U'\) denotes the projection of U. The system of schemes \(\{ A_{n,U}^{(m)}\}\) has a right action of \(G_n^{(m)}({\mathbb {A}}^\infty )\) and a left action of \(GL_m(F)\), which don’t commute. (See Sect. 3.2.)
We will also need to consider integral models of these varieties over \({\mathbb {Z}}_{(p)}\), but only for certain level structures at p. To define these level structures we give \(G_n\) the integral structure coming from the (self-dual) lattice \(\varLambda _n=({\mathcal {D}}_F^{-1})^n \oplus {\mathcal {O}}_F^n \subset F^{2n}\). We give \(G_n^{(m)}\) a compatible integral structure so that
We will only consider integral models for Shimura varieties with level \(U^p(N_1,N_2)\), where \(N_2 \ge N_1 \ge 0\) are integers and \(U^p\) is a neat open compact subgroup of \(G_n({\mathbb {A}}^{\infty ,p})\) or \(G_n^{(m)}({\mathbb {A}}^{\infty ,p})\). Here \(U^p(N_1,N_2)=U^p \times U_p(N_1,N_2)_n\) where \(U_p(N_1,N_2)_n \subset G_n({\mathbb {Z}}_p)\) is the subgroup of matrices of the form
or \(U^p(N_1,N_2)=U^p \times U_p(N_1,N_2)_n^{(m)}\) where \(U_p(N_1,N_2)_n^{(m)} \subset G_n^{(m)}({\mathbb {Z}}_p)\) is the subgroup of matrices of the form
(See Sect. 1.1.)
We will consider a certain integral model \({\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}\) of \(X_{n,U^p(N_1,N_2)}\). (See Sect. 3.1.) Although its generic fibre is all of \(X_{n,U^p(N_1,N_2)}\), its special fibre consists only of points parameterizing ordinary abelian schemes with a level structure with respect to which the distinguished subgroup scheme of order \(p^{N_2[F: {\mathbb {Q}}]n}\) is connected. Thus, for example, if \(N_2'>N_2\) then \({\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2')} \rightarrow {\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}\) is not finite, because we are only including some of the possible \(U^p(N_1,N_2')\)-level structures extending a given \(U^p(N_1,N_2)\)-level structure. We denote the universal abelian scheme over \({\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}\) by \({\mathcal {A}}^{{\text {univ}}}\). It is uniquely defined up to prime-to-p quasi-isogeny. The action of the whole group \(G_n({\mathbb {A}}^\infty )\) on \(\{ X_{n,U}\}\) does not extend to an action on \(\{ {\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}\}\). However the action of a sub-semigroup
does extend. Here \(P_{n,(n)}^+\) denotes the subgroup of \(G_n\) consisting of elements of the form
(over \({\mathbb {Q}}\) this defines a maximal parabolic subgroup) and \(\varsigma _p\) denotes the element
We write \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) for the maximal subgroup of the semi-group \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\). We will also write
which is independent of \(N_2\). (See Sect. 1.2.)
The formal completion of \({\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}\) along its \({\mathbb {F}}_p\)-fibre only depends on \(U^p(N_1)=U^p(N_1,N_2) \cap G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\), and so we will denote it \({\mathfrak {X}}^{{\text {ord}}}_{n,U^p(N_1)}\). We will also denote its reduced subscheme \(\overline{{X}}^{{\text {ord}}}_{n,U^p(N_1)}\). The systems \(\{ {\mathfrak {X}}^{{\text {ord}}}_{n,U^p(N_1)} \}\) and \(\{ \overline{{X}}^{{\text {ord}}}_{n,U^p(N_1)} \}\) also have actions of \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\). (See Sect. 3.1.)
We will also consider a certain integral model \({\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)}\) of \(A_{n,U^p(N_1,N_2)}^{(m)}\). Note that \({\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)}\) is smooth and projective over \({\mathcal {X}}^{{\text {ord}}}_{n,(U^p)'(N_1,N_2)}\), where \((U^p)'\) denotes the image of \(U^p\) in \(G_n({\mathbb {A}}^{\infty ,p})\). The system of schemes \(\{ {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)} \}\) has an action of the semi-group
where \(P^{(m),+}_{n,(n)}\) denotes the subgroup of \(G_n^{(m)}\) consisting of elements of the form
We also write \(G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) for the maximal subgroup of the semi-group \(G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) and
It also has an action of \(GL_m({\mathcal {O}}_{F,(p)})\). The formal completion of \({\mathcal {A}}_{n,U^p(N_1,N_2)}^{(m)}\) along its \({\mathbb {F}}_p\)-fibre depends only on \(U^p(N_1)\), so we will denote it \({\mathfrak {A}}_{n,U^p(N_1)}^{(m),{{\text {ord}}}}\). We will also denote its special fibre \(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1)}\). (See Sects. 1.2, 3.2.)
We will write \(L_{n,(n)}\) for the subgroup of \(P_{n,(n)}^+\) consisting of matrices of the form
We let \({{\text {Std}}}\) denote the representation of \(L_{n,(n)}\) over \({\mathbb {Z}}\) which sends the above matrix to the lower left \(n \times n\)-block in \({{\text {RS}}}^{{\mathcal {O}}_F}_{\mathbb {Z}}GL_n \subset GL_{n[F: {\mathbb {Q}}]}\). In fact,
We will let \(T_n\) denote the subgroup of \(G_n\) consisting of diagonal matrices and \(B_n\) the subgroup of upper triangular matrices. The isomorphism \(\nu \times {{\text {Std}}}\) allows us to identify \(T_n\) with \({\mathbb {G}}_m \times {{\text {RS}}}^{{\mathcal {O}}_F}_{\mathbb {Z}}{\mathbb {G}}_m^n\) and hence we get an isomorphism
We will denote a typical element of this group \((b_0,(b_{\tau ,i}))\). The set \(X^*(T_n)_{(n)}^+ \subset X^*(T_n)\) of positive elements of \(X^*(T_n)\) with respect to the subgroup \(B_n\cap L_{n,(n)}\) is characterized by
for all \(\tau \). The set \(X^*(T_n)_{(n)}^+ \subset X^*(T_n)\) of positive elements with respect to \(B_n\) is further characterized by
for all \(\tau \). Over \(\overline{{{\mathbb {Q}}}}\) we can decompose
where \({{\text {Std}}}_\tau : L_{n,(n)} \rightarrow GL_n\). There is a representation
over \({\mathbb {Z}}\), such that over \(\overline{{{\mathbb {Q}}}}\)
(See Sect. 1.2.)
If \(R_0\) is an irreducible noetherian \({\mathbb {Q}}\)-algebra and \(\rho \) is a representation of \(L_{n,(n)}\) on a locally free \(R_0\)-module then we may associate a locally free sheaf \({\mathcal {E}}_{U,\rho }/X_{n,U} \times {{\text {Spec}}\,}R_0\) (in the Zariski topology). As examples we have the following.
-
\({\mathcal {E}}_{U,{{\text {Std}}}^\vee } \cong \varOmega _{n,U}\), the pull-back by the identity section of the sheaf of relative differentials \(\varOmega ^1_{A^{{\text {univ}}}/X_{n,U}}\).
-
\({\mathcal {E}}_{U,\wedge ^{n[F: {\mathbb {Q}}]}{{\text {Std}}}^\vee } \cong \wedge ^{n[F: {\mathbb {Q}}]} \varOmega _{n,U}=\omega _{U}\).
-
\({\mathcal {E}}_{U,{{\text {KS}}}} \cong \varOmega ^1_{X_{n,U}/{\mathbb {Q}}}\).
The system of sheaves \(\{ {\mathcal {E}}_{U,\rho }\}\) has an action of \(G_n({\mathbb {A}}^\infty )\). (See Sect. 3.4.)
Similarly if \(R_0\) is an irreducible noetherian \({\mathbb {Z}}_{(p)}\)-algebra and \(\rho \) is a representation of \(L_{n,(n)}\) on a locally free \(R_0\)-module then we may associate a locally free sheaf \({\mathcal {E}}^{{\text {ord}}}_{U^p(N_1,N_2),\rho }/{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)} \times {{\text {Spec}}\,}R_0\) (in the Zariski topology). As examples we have the following.
-
\({\mathcal {E}}_{U_p(N_1,N_2),{{\text {Std}}}^\vee }^{{\text {ord}}}\cong \varOmega _{n,U^p(N_1,N_2)}^{{\text {ord}}}\), the pull-back by the identity section of the sheaf of relative differentials \(\varOmega ^1_{{\mathcal {A}}^{{{\text {univ}}}}/{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}}\).
-
\({\mathcal {E}}_{U^p(N_1,N_2),\wedge ^{n[F: {\mathbb {Q}}]}{{\text {Std}}}^\vee } \cong \wedge ^{n[F: {\mathbb {Q}}]} \varOmega ^{{\text {ord}}}_{n,U^p(N_1,N_2)}=\omega _{U^p(N_1,N_2)}^{{\text {ord}}}\).
-
\({\mathcal {E}}_{U^p(N_1,N_2),{{\text {KS}}}} \cong \varOmega ^1_{{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}/{\mathbb {Z}}_{(p)}}\).
The system of sheaves \(\{ {\mathcal {E}}_{U^p(N_1,N_2),\rho }\}\) has an action of \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\). (See Sect. 3.4.)
The scheme \(X_{n,U}\) has a canonical compactification \(X_{n,U}^{{\text {min}}}\) called the minimal or Baily–Borel compactification with boundary \(\partial X_{n,U}^{{\text {min}}}\). It is a normal projective scheme over \({\mathbb {Q}}\). The line bundle \(\omega _{U}\) extends (uniquely, if \(n>1\) or \(F^+ \ne {\mathbb {Q}}\)) to an ample line bundle \(\omega _{U}\) on \(X_{n,U}^{{\text {min}}}\). However we can not expect the vector bundles \({\mathcal {E}}_{U,\rho }\) to all extend to vector bundles on \(X_{n,U}^{{\text {min}}}\times {{\text {Spec}}\,}R_0\). All these systems of spaces and the line bundles \(\omega _U\) have compatible actions of \(G_n({\mathbb {A}}^\infty )\). (See Sect. 5.1.) We can describe the scheme \(\partial X_{n,U}^{{\text {min}}}\) more precisely, but this will require considerably more notation, so we will come back to this in the next section.
We can also define a normal quasi-projective scheme \({\mathcal {X}}^{{{\text {ord}}},{{\text {min}}}}_{n,U^p(N_1,N_2)}\) over \({\mathbb {Z}}_{(p)}\) with a closed subscheme \(\partial {\mathcal {X}}^{{{\text {ord}}},{{\text {min}}}}_{n,U^p(N_1,N_2)}\) such that
is a dense open subscheme. The scheme \({\mathcal {X}}^{{{\text {ord}}},{{\text {min}}}}_{n,U^p(N_1,N_2)}\) is not proper—informally speaking it is missing points in characteristic p. Its generic fibre is identified with \(X^{{\text {min}}}_{n,U^p(N_1,N_2)}\) and there is a unique line bundle \(\omega _{U^p(N_1,N_2)}^{{\text {ord}}}\) over it which restricts to \(\omega _{U^p(N_1,N_2)}^{{\text {ord}}}\) on \({\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}\) and to \(\omega _{U^p(N_1,N_2)}\) on \(X^{{\text {min}}}_{n,U^p(N_1,N_2)}\). We will write \({\mathfrak {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}\) for the formal completion of \({\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}\) along its \({\mathbb {F}}_p\)-fibre \(\overline{{X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}\). Schemes \(\partial \overline{{X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}\) are defined similarly. All these systems of (formal) schemes have compatible actions of \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\). The system of line bundles \(\{ \omega _{U^p(N_1,N_2)}^{{\text {ord}}}\}\) has an action of \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\). (See Sect. 5.1.) Again we will describe the schemes \(\partial {\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}\) more precisely in the next section.
If \(U \subset G_n({\mathbb {A}}^{\infty ,p} \times {\mathbb {Z}}_p)\) then we can define a normal scheme \({\mathcal {X}}_{n,U}^{{\text {min}}}\) which is projective and flat over \({\mathbb {Z}}_{(p)}\) with generic fibre \(X_{n,U}^{{\text {min}}}\), together with an extension of \(\omega _U\) to an ample line bundle on \({\mathcal {X}}_{n,U}^{{\text {min}}}\), which we will also denote \(\omega _U\). These systems of schemes and line bundles have compatible actions of \(G_n({\mathbb {A}}^{\infty ,p}\times {\mathbb {Z}}_p)\). There are \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\)-equivariant open embeddings \({\mathcal {X}}^{{{\text {ord}}},{{\text {min}}}}_{n,U^p(N_1,N_2)} \hookrightarrow {\mathcal {X}}^{{\text {min}}}_{n,U^p(N_1,N_2)}\). We will write \(\overline{{X}}_{n,U}^{{\text {min}}}\) for the \({\mathbb {F}}_p\)-fibre of \({\mathcal {X}}_{U,n}^{{\text {min}}}\). There is a canonical section
These sections are invariant under the action of \(G_n({\mathbb {A}}^{\infty ,p} \times {\mathbb {Z}}_p)\). We write \(\overline{{X}}_{n,U}^{{{\text {min}}},{{\text {n-ord}}}}\) for the vanishing locus of \({{\text {Hasse}}}_U\). Then \(\overline{{X}}^{{{\text {ord}}},{{\text {min}}}}_{n,U^p(N_1,N_2)}\) is an open and closed subscheme of \(\overline{{X}}^{{\text {min}}}_{n,U^p(N_1,N_2)}-\overline{{X}}^{{{\text {min}}},{{\text {n-ord}}}}_{n,U^p(N_1,N_2)}\). They are equal if \(N_1=N_2=0\). (See Sect. 5.1.)
To certain additional data \(\varDelta \), with \((U,\varDelta )\) in a certain partially ordered set \({\mathcal {J}}_n^{{\text {tor}}}\), which we will describe more carefully in the next section, one can associate a smooth projective scheme \(X_{n,U,\varDelta }/{\mathbb {Q}}\) and a simple normal crossings divisor
such that
We write \((X_{n,U,\varDelta },{\mathcal {M}}_\varDelta )\) for the log scheme associated to \((X_{n,U,\varDelta }, \partial X_{n,U,\varDelta })\). There is a natural map \(X_{n,U,\varDelta } \rightarrow X_{n,U}^{{\text {min}}}\) which is the identity of \(X_{n,U}\) and sends \(\partial X_{n,U,\varDelta }\) to \(\partial X_{n,U}^{{\text {min}}}\). The universal abelian scheme \(A^{{\text {univ}}}/X_{n,U}\) extends uniquely to a semi-abelian scheme \(A^{{\text {univ}}}/X_{n,U,\varDelta }\). The system of schemes \(\{X_{n,U,\varDelta }\}\) has an action of \(G_n({\mathbb {A}}^\infty )\) via log etale maps. Similarly to certain additional data \(\varSigma \), with \((U,\varSigma )\) in a certain partially ordered set \({\mathcal {J}}_n^{(m),{{\text {tor}}}}\), which we will describe more carefully in the next section, one can associate a smooth projective scheme \(A^{(m)}_{n,U,\varSigma }/{\mathbb {Q}}\) and a simple normal crossings divisor \(\partial A^{(m)}_{n,U,\varSigma } \subset A^{(m)}_{n,U,\varSigma }\) such that
We write \((A^{(m)}_{n,U,\varSigma },{\mathcal {M}}_\varSigma )\) for the log scheme associated to \((A^{(m)}_{n,U,\varSigma }, \partial A^{(m)}_{n,U,\varSigma })\). If \((U,\varSigma )\) and \((U',\varDelta )\) are compatible in a suitable sense, then there is a natural log smooth map \(A_{n,U,\varSigma }^{(m)} \rightarrow X_{n,U',\varDelta }\). (See Sects. 5.2, 5.3.) More details on the structure of the boundary will be given in the next section.
If \(R_0\) is an irreducible noetherian \({\mathbb {Q}}\)-algebra and \(\rho \) is a representation of \(L_{n,(n)}\) on a locally free \(R_0\)-module then we may associate a locally free sheaf \({\mathcal {E}}_{U,\varDelta ,\rho }^{{\text {can}}}/X_{n,U,\varDelta } \times {{\text {Spec}}\,}R_0\) (in the Zariski topology) such that
As examples we have the following.
-
\({\mathcal {E}}_{U,\varDelta ,{{\text {Std}}}^\vee }^{{\text {can}}}\cong \varOmega _{n,U,\varDelta }\), the pull-back by the identity section of the sheaf of relative differentials \(\varOmega ^1_{A^{{\text {univ}}}/X_{n,U,\varDelta }}\).
-
\({\mathcal {E}}_{U,\varDelta ,\wedge ^{n[F: {\mathbb {Q}}]}{{\text {Std}}}^\vee }^{{\text {can}}}\cong \wedge ^{n[F: {\mathbb {Q}}]} \varOmega _{n,U,\varDelta }=\omega _{U,\varDelta }\) is naturally identified with the pull-back of \(\omega _U\) from \(X_{n,U}^{{\text {min}}}\).
-
\({\mathcal {E}}_{U,\varDelta ,{{\text {KS}}}}^{{\text {can}}}\cong \varOmega ^1_{X_{n,U,\varDelta }}(\log \infty )\), the sheaf of differentials with log poles along the boundary.
We will write
where \({\mathcal {I}}_{\partial X_{n,U,\varDelta }}\) denotes the ideal sheaf of the boundary. It is again a locally free sheaf. We will also write \({\mathcal {E}}_{U,\rho }^{{\text {sub}}}\) for the coherent sheaf on \(X_{n,U}^{{\text {min}}}\) which is the push-forward of \({\mathcal {E}}_{U,\varDelta ,\rho }^{{\text {sub}}}\) from \(X_{n,U,\varDelta }\). (This is independent of the choice of \(\varDelta \).) The systems of sheaves \(\{ {\mathcal {E}}_{U,\varDelta ,\rho }^{{\text {can}}}\}\), \(\{ {\mathcal {E}}_{U,\varDelta ,\rho }^{{\text {sub}}}\}\), and \(\{ {\mathcal {E}}_{U,\rho }^{{\text {sub}}}\}\) have actions of \(G_n({\mathbb {A}}^\infty )\). (See Sect. 5.4.)
We will write \(\varOmega ^1_{A^{(m)}_{U,\varSigma }}(\log \infty )\) for the sheaf of differentials on \(A^{(m)}_{U,\varSigma }\) with log poles along \(\partial A^{(m)}_{U,\varSigma }\), and \(\varOmega ^i_{A^{(m)}_{U,\varSigma }}(\log \infty )\) for its ith exterior power.
Similarly to certain additional data \(\varDelta \), with \((U^p(N_1,N_2),\varDelta )\in {\mathcal {J}}_n^{{\text {tor}}}\), one can associate a smooth quasi-projective scheme \({\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2),\varDelta }/{\mathbb {Q}}\) and a simple relative normal crossings divisor
such that
The \({\mathbb {Q}}\)-fibre is identified with \(X_{n,U^p(N_1,N_2),\varDelta }\). We write \(({\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2),\varDelta },{\mathcal {M}}_\varDelta )\) for the log scheme associated to \(({\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2),\varDelta }, \partial {\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2),\varDelta })\). There is a natural map \({\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2),\varDelta } \rightarrow {\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}\) which equals the identity of \({\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}\) and sends \(\partial {\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2),\varDelta }\) to \(\partial {\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}\). The universal abelian scheme \({\mathcal {A}}^{{\text {univ}}}/{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2)}\) extends uniquely to a semi-abelian scheme \({\mathcal {A}}^{{\text {univ}}}/{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2),\varDelta }\). The system of schemes \(\{{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2),\varDelta }\}\) has an action of \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\). Also to certain additional data \(\varSigma \), with \((U^p(N_1,N_2),\varSigma )\in {\mathcal {J}}_n^{(m),{{\text {tor}}}}\), one can associate a smooth quasi-projective scheme \({\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\varSigma }/{\mathbb {Z}}_{(p)}\) and a simple normal crossings divisor
such that
The \({\mathbb {Q}}\)-fibre is identified with \(A^{(m)}_{n,U^p(N_1,N_2),\varSigma }\). We write \(({\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\varSigma },{\mathcal {M}}_\varSigma )\) for the log scheme associated to \(({\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\varSigma }, \partial {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\varSigma })\). If the pairs \((U^p(N_1,N_2),\varSigma )\) and \(((U^p)'(N_1',N_2'),\varDelta )\) are compatible in a suitable sense, then there is a natural log smooth map \({\mathcal {A}}_{n,U^p(N_1,N_2),\varSigma }^{(m),{{\text {ord}}}} \rightarrow {\mathcal {X}}^{{\text {ord}}}_{n,(U^p)'(N_1',N_2'),\varDelta }\). (See Sects. 5.2, 5.3.) More details on the structure of the boundary will be given in the next section.
Similarly if \(R_0\) is an irreducible noetherian \({\mathbb {Z}}_{(p)}\)-algebra and \(\rho \) is a representation of \(L_{n,(n)}\) on a locally free \(R_0\)-module then we may associate a locally free sheaf \({\mathcal {E}}_{U^p(N_1,N_2),\varDelta ,\rho }^{{{\text {ord}}},{{\text {can}}}}/{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2),\varDelta } \times {{\text {Spec}}\,}R_0\) (in the Zariski topology) such that
and the pull-back to the \({\mathbb {Q}}\)-fibre is identified with \({\mathcal {E}}_{U^p(N_1,N_2),\varDelta ,\rho }^{{\text {can}}}\). As examples we have the following.
-
\({\mathcal {E}}_{U^p(N_1,N_2),\varDelta ,{{\text {Std}}}^\vee }^{{{\text {ord}}},{{\text {can}}}} \cong \varOmega _{n,U^p(N_1,N_2),\varDelta }^{{\text {ord}}}\), the pull-back by the identity section of the sheaf of relative differentials \(\varOmega ^1_{{\mathcal {A}}^{{\text {univ}}}/{\mathcal {X}}_{n,U^p(N_1,N_2),\varDelta }^{{\text {ord}}}}\).
-
\({\mathcal {E}}_{U^p(N_1,N_2),\varDelta ,\wedge ^{n[F: {\mathbb {Q}}]}{{\text {Std}}}^\vee }^{{{\text {ord}}},{{\text {can}}}} \cong \wedge ^{n[F: {\mathbb {Q}}]} \varOmega _{n,U^p(N_1,N_2),\varDelta }^{{\text {ord}}}=\omega _{U^p(N_1,N_2),\varDelta }^{{\text {ord}}}\) is naturally identified with the pull-back of \(\omega _{U^p(N_1,N_2)}^{{\text {ord}}}\) from \({\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}\).
-
\({\mathcal {E}}_{U^p(N_1,N_2),\varDelta ,{{\text {KS}}}}^{{{\text {ord}}},{{\text {can}}}} \cong \varOmega ^1_{{\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2),\varDelta }}(\log \infty )\), the sheaf of differentials with log poles along the boundary.
We will write
where \({\mathcal {I}}_{\partial {\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2),\varDelta }}\) denotes the ideal sheaf of the boundary. It is again a locally free sheaf. We will also write \({\mathcal {E}}_{U^p(N_1,N_2),\rho }^{{{\text {ord}}},{{\text {sub}}}}\) for the coherent sheaf on \({\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}\) which is the push-forward of \({\mathcal {E}}_{U^p(N_1,N_2),\varDelta ,\rho }^{{{\text {ord}}},{{\text {sub}}}}\) from \({\mathcal {X}}_{n,U^p(N_1,N_2),\varDelta }\). (This is independent of the choice of \(\varDelta \).) The systems of sheaves \(\{ {\mathcal {E}}_{U^p(N_1,N_2),\varDelta ,\rho }^{{{\text {ord}}},{{\text {can}}}}\}\), \(\{ {\mathcal {E}}_{U^p(N_1,N_2),\varDelta ,\rho }^{{{\text {ord}}},{{\text {sub}}}}\}\), and \(\{ {\mathcal {E}}_{U^p(N_1,N_2),\rho }^{{{\text {ord}}},{{\text {sub}}}}\}\) have actions of \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\). (See Sect. 5.4.)
The formal completion of \({\mathcal {X}}^{{\text {ord}}}_{n,U^p(N_1,N_2),\varDelta }\) along its special fibre depends only on \(U^p(N_1)\) and \(\varDelta ^{{\text {ord}}}\), a subset of the data contained in \(\varDelta \), which we will describe in more detail in the next section. Thus we will denote this completion \({\mathfrak {X}}^{{\text {ord}}}_{n,U^p(N_1),\varDelta ^{{\text {ord}}}}\), and its reduced subscheme \(\overline{{X}}^{{\text {ord}}}_{n,U^p(N_1),\varDelta ^{{\text {ord}}}}\). The latter contains a simple normal crossings divisor \(\partial \overline{{X}}^{{\text {ord}}}_{n,U^p(N_1),\varDelta ^{{\text {ord}}}}\) such that \(\overline{{X}}^{{\text {ord}}}_{n,U^p(N_1),\varDelta ^{{\text {ord}}}} - \partial \overline{{X}}^{{\text {ord}}}_{n,U^p(N_1),\varDelta ^{{\text {ord}}}}=\overline{{X}}^{{\text {ord}}}_{n,U^p(N_1)}\). The systems of these (formal) schemes have an action of \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\). The map
is finite flat of degree \(p^{n^2[F^+: {\mathbb {Q}}]}\) and the induced map on reduced subschemes is absolute Frobenius. Similarly the formal completion of \({\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\varSigma }\) along its special fibre depends only on \(U^p(N_1)\) and \(\varSigma ^{{\text {ord}}}\), a subset of the data contained in \(\varDelta \), which we will describe in more detail in the next section. Thus we will denote this completion \({\mathfrak {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1),\varSigma ^{{\text {ord}}}}\), and its reduced subscheme \(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1),\varSigma ^{{\text {ord}}}}\). The latter contains a simple normal crossings divisor \(\partial \overline{{A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1),\varSigma ^{{\text {ord}}}}\) such that \(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1),\varSigma ^{{\text {ord}}}} - \partial \overline{{A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1),\varSigma ^{{\text {ord}}}}=\overline{{A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1)}\). The systems of these (formal) schemes have an action of \(G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\). The map
is finite flat of degree \(p^{(n+2m)n[F^+: {\mathbb {Q}}]}\) and the induced map on reduced subschemes is absolute Frobenius. (See Sect. 5.3.)
We will write \(X^{{{\text {min}}},\dagger }_{U'}\) and \(X^{\dagger }_{U',\varDelta }\) and \(A^{(m),\dagger }_{U,\varSigma }\) for the dagger spaces in the sense of [27] associated to \(X^{{\text {min}}}_{n,U'}/{\mathbb {Q}}_p\) and \(X_{n,U',\varDelta }/{\mathbb {Q}}_p\) and \(A^{(m)}_{n,U,\varSigma }/{\mathbb {Q}}_p\). We will write \({\mathcal {X}}^{{{\text {ord}}},{{\text {min}}},\dagger }_{(U^p)'(N_1)}\) and \({\mathcal {X}}^{{{\text {ord}}},\dagger }_{(U^p)'(N_1), \varDelta ^{{\text {ord}}}}\) and \({\mathcal {A}}^{(m),{{\text {ord}}},\dagger }_{U^p(N_1),\varSigma ^{{\text {ord}}}}\) for the subdagger spaces of \(X^{{{\text {min}}},\dagger }_{(U^p)'(N_1,N_2)}\) and \(X^{\dagger }_{(U^p)'(N_1,N_2),\varDelta }\) and \(A^{(m),\dagger }_{U^p(N_1,N_2),\varSigma }\) corresponding to the admissible opens in the corresponding rigid spaces which are the generic fibres of \({\mathfrak {X}}^{{{\text {ord}}},{{\text {min}}}}_{n,(U^p)'(N_1)}\) and \({\mathfrak {X}}^{{{\text {ord}}}}_{n,(U^p)'(N_1), \varDelta ^{{\text {ord}}}}\) and \({\mathfrak {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1),\varSigma ^{{\text {ord}}}}\), respectively. These depend only on \((U^p)'(N_1)\), \(\varDelta ^{{\text {ord}}}\), \(U^p(N_1)\) and \(\varSigma ^{{\text {ord}}}\). If \(U^p\) maps to \((U^p)'\) then the spaces \({\mathcal {X}}^{{{\text {ord}}},\dagger }_{(U^p)'(N_1), \varDelta ^{{\text {ord}}}}\) and \({\mathcal {A}}^{(m),{{\text {ord}}},\dagger }_{U^p(N_1),\varSigma ^{{\text {ord}}}}\) are the pre-images in \(X^{\dagger }_{(U^p)'(N_1,N_2),\varDelta }\) and \(A^{(m),\dagger }_{U^p(N_1,N_2),\varSigma }\) of
Similarly we define closed subdagger spaces \(\partial {\mathcal {X}}^{{{\text {ord}}},{{\text {min}}},\dagger }_{(U^p)'(N_1)}\) and \(\partial {\mathcal {X}}^{{{\text {ord}}},\dagger }_{(U^p)'(N_1), \varDelta ^{{\text {ord}}}}\) and \(\partial {\mathcal {A}}^{(m),{{\text {ord}}},\dagger }_{U^p(N_1),\varSigma ^{{\text {ord}}}}\). (See Sects. 6.3, 6.4.)
The systems of dagger spaces \(\{ {\mathcal {X}}^{{{\text {ord}}},{{\text {min}}}, \dagger }_{V^p(N_1)} \}\) and \(\{ {\mathcal {X}}^{{{\text {ord}}},\dagger }_{V^p(N_1), \varDelta ^{{\text {ord}}}} \}\) and \(\{{\mathcal {A}}^{(m),{{\text {ord}}},\dagger }_{U^p(N_1),\varSigma ^{{\text {ord}}}}\}\) have actions of \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\) and \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\) and \(G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\), respectively, which respect the boundaries. If \(g \in G_n({\mathbb {A}}^\infty )^{{{\text {ord}}}}\) or \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}}}\) or \(G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}}}\), respectively, then \({\mathcal {X}}^{{{\text {ord}}},{{\text {min}}}}_{V^p(N_1)}\) and \({\mathcal {X}}^{{{\text {ord}}},\dagger }_{V^p(N_1), \varDelta ^{{\text {ord}}}}\) and \({\mathcal {A}}^{(m),{{\text {ord}}},\dagger }_{U^p(N_1),\varSigma ^{{\text {ord}}}}\) are the pre-images of \({\mathcal {X}}^{{{\text {ord}}},{{\text {min}}}}_{(V^p)'(N_1')}\) and \({\mathcal {X}}^{{{\text {ord}}},\dagger }_{(V^p)'(N_1'), \varDelta ^{\prime ,{{\text {ord}}}}}\) and \({\mathcal {A}}^{(m),{{\text {ord}}},\dagger }_{(U^p)'(N_1'),\varSigma ^{\prime ,{{\text {ord}}}}}\) under
and
and
provided that either \({{\text {val}}}_p(\nu (g))=0\) or \(N_2+{{\text {val}}}_p(\nu (g)) >0\). (See Sects. 6.3, 6.4.)
We will write \({\mathcal {E}}^{{{\text {can}}},\dagger }_{U^p(N_1),\varDelta ^{{\text {ord}}}, \rho }\) for the restriction to \({\mathcal {X}}^{{{\text {ord}}},\dagger }_{U^p(N_1),\varDelta ^{{\text {ord}}}}\) of the locally free sheaf on \(X^{\dagger }_{U^p(N_1,N_2),\varDelta }\) associated to \({\mathcal {E}}^{{{\text {can}}}}_{U^p(N_1,N_2),\varDelta , \rho }\). It does not depend on \(N_2\) or \(\varDelta \). We define \({\mathcal {E}}^{{{\text {sub}}},\dagger }_{U^p(N_1),\varDelta ^{{\text {ord}}}, \rho }\) similarly. The systems of sheaves \(\{ {\mathcal {E}}^{{{\text {can}}},\dagger }_{U^p(N_1),\varDelta ^{{\text {ord}}}, \rho } \}\) and \(\{ {\mathcal {E}}^{{{\text {sub}}},\dagger }_{U^p(N_1),\varDelta ^{{\text {ord}}}, \rho } \}\) have actions of \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\). If \(g \in G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\) then the map
is an isomorphism. We have maps, which we will denote \({{\text {tr}}}_F\):
and
where \({\mathcal {I}}_{\partial {\mathcal {X}}^{{{\text {ord}}},\dagger }_{U^p(N_1),\varDelta ^{{\text {ord}}}}} \subset {\mathcal {O}}_{{\mathcal {X}}^{{{\text {ord}}},\dagger }_{U^p(N_1),\varDelta ^{{\text {ord}}}}}\) denotes the ideal sheaf of the boundary. (See Sect. 6.3.)
We will also write \({\mathcal {E}}^{{{\text {sub}}},\dagger }_{U^p(N_1), \rho }\) for the restriction to \({\mathcal {X}}^{{{\text {ord}}},{{\text {min}}},\dagger }_{U^p(N_1)}\) of the locally free sheaf on \(X^{{{\text {min}}},\dagger }_{U^p(N_1,N_2)}\) associated to \({\mathcal {E}}^{{{\text {sub}}}}_{U^p(N_1,N_2), \rho }\). It does not depend on \(N_2\). It can be identified with the push-forward from \({\mathcal {X}}^{{{\text {ord}}},\dagger }_{U^p(N_1),\varDelta ^{{\text {ord}}}}\) to \({\mathcal {X}}^{{{\text {ord}}},{{\text {min}}},\dagger }_{U^p(N_1)}\) of \({\mathcal {E}}^{{{\text {can}}},\dagger }_{U^p(N_1),\varDelta ^{{\text {ord}}}, \rho }\). The system of sheaves \(\{ {\mathcal {E}}^{{{\text {sub}}},\dagger }_{U^p(N_1), \rho } \}\) has an action of \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\). Moreover the map
pushes forward to a map
which does not depend on \(\varDelta ^{{\text {ord}}}\). (See Sect. 6.4.)
We will denote by \(\varOmega ^i_{{\mathcal {A}}^{(m),{{\text {ord}}},\dagger }_{U^p(N_1),\varSigma ^{{\text {ord}}}}}(\log \infty )\) the sheaf on \({\mathcal {A}}^{(m),{{\text {ord}}},\dagger }_{U^p(N_1),\varSigma ^{{\text {ord}}}}\) associated to the sheaf \(\varOmega ^i_{A^{(m)}_{n,U^p(N_1,N_2),\varSigma }}(\log \infty )\). It is independent of \(N_2\) and \(\varSigma \). Also let \({\mathcal {I}}_{\partial {\mathcal {A}}^{(m),{{\text {ord}}},\dagger }_{U^p(N_1),\varSigma ^{{\text {ord}}}}}\) denote the ideal sheaf in \({\mathcal {O}}_{{\mathcal {A}}^{(m),{{\text {ord}}},\dagger }_{U^p(N_1),\varSigma ^{{\text {ord}}}}}\) defining \(\partial {\mathcal {A}}^{(m),{{\text {ord}}},\dagger }_{U^p(N_1),\varSigma ^{{\text {ord}}}}\). The systems of sheaves \(\{ \varOmega ^i_{{\mathcal {A}}^{(m),{{\text {ord}}},\dagger }_{U^p(N_1),\varSigma ^{{\text {ord}}}}}(\log \infty )\}\) and \(\{ \varOmega ^i_{{\mathcal {A}}^{(m),{{\text {ord}}},\dagger }_{U^p(N_1),\varSigma ^{{\text {ord}}}}}(\log \infty ) \otimes {\mathcal {I}}_{\partial {\mathcal {A}}^{(m),{{\text {ord}}},\dagger }_{U^p(N_1),\varSigma ^{{\text {ord}}}}}\}\) both have actions of \(G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\). If \(g \in G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) then the map
is an isomorphism. We have maps, which we will denote \({{\text {tr}}}_F\):
and
(See Sect. 6.3.)
1.2 The boundary
Unfortunately to describe the boundaries of our various compactifications requires significantly more notation. We remind the reader of our convention that, if \(U \subset G\) are groups and \(G \rightarrow H\) is a homomorphism, we will sometimes also use U to denote the image of U in H, where from the context it is clear that we need a subgroup of H.
We will first consider the boundary of \(X_{n,U}^{{\text {min}}}\). For \(i=0,\ldots ,n\) let \(P_{n,(i)}^+\) denote the subgroup of \(G_n\) consisting of matrices of the form
and let \(L_{n,(i)}\) denote the subgroup consisting of block diagonal matrices. Then \(L_{n,(i)}=L_{n,(i),{{\text {lin}}}} \times L_{n,(i),{{\text {herm}}}}\), where \(L_{n,(i),{{\text {lin}}}} \cong {{\text {RS}}}^{{\mathcal {O}}_F}_{\mathbb {Z}}GL_i\) is the set of matrices of the form
with \(h \in {{\text {RS}}}^{{\mathcal {O}}_F}_{\mathbb {Z}}GL_i\), and where \(L_{n,(i){{\text {herm}}}} \cong G_{n-i}\) is the set of matrices of the form
with \(g \in G_{n-i}\). (See Sect. 1.2.)
For \(U \subset L_{n,(i)}({\mathbb {A}}^\infty )\) a neat open compact subgroup we set
This is locally of finite type, but not of finite type over \({\mathbb {Q}}\). We refer to it as a generalized Shimura variety. It has a left action of \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})=GL_i(F)\) such that \(\delta \in L_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\) acts via the coproduct of the identity maps
The inverse system \(\{ X^+_{n,(i),U} \}\) has a commuting right action of \(L_{n,(i)}({\mathbb {A}}^\infty )\) such that \(g =(g_{{\text {lin}}},g_{{\text {herm}}})\in L_{n,(i)}({\mathbb {A}}^\infty )\) acts via the coproduct of the maps
if \(V \supset g^{-1}Ug\). We further define
The inverse system \(\{ X^\natural _{n,(i),U} \}\) has a right action of \(L_{n,(i)}({\mathbb {A}}^\infty )\). (See Sect. 4.1.)
With these definitions we can describe the boundary of \(X_{n,U}^{{\text {min}}}\). There is a family of closed subschemes
which are preserved by the action of \(G_n({\mathbb {A}}^\infty )\). We set
which is smooth over \({\mathbb {Q}}\) of dimension \((n-i)^2[F^+: {\mathbb {Q}}]\) and write \(X_{n,U,i}^{{{\text {min}}},\wedge }\) for the formal completion of \(X_{n,U}^{{\text {min}}}\) along \(\partial _i^0X_{n,U}^{{\text {min}}}\). We can describe \(\partial ^0_i X_{n,U}^{{\text {min}}}\) as
If \(g \in G_n({\mathbb {A}}^\infty )\) and if \(g^{-1} Ug \subset V\) then the map
is the coproduct of the maps
where \(hg=g'h'\) with \(g' \in P_{n,(i)}({\mathbb {A}}^\infty )\). (See Sect. 5.1.)
We write
and
and if \(U^p\) is a neat open compact subgroup of \(L_{n,(i),{{\text {lin}}}}({\mathbb {A}}^{\infty ,p})\) or \(L_{n,(i)}({\mathbb {A}}^{\infty ,p})\) we will write
or
Moreover we write
and
and
(See Sect. 1.2.)
We set
It has a left action of \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})=GL_i({\mathcal {O}}_{F,(p)})\) such that \(\delta \in L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\) acts via the coproduct of the identity maps
The inverse system \(\{ {\mathcal {X}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)} \}\) has a commuting right action of \(L_{n,(i)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) such that \(g =(g_{{\text {lin}}},g_{{\text {herm}}})\in L_{n,(i)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) acts via the coproduct of the maps
if \(V^p(N_1',N_2') \supset g^{-1}U^p(N_1,N_2)g\). We further define
The inverse system \(\{ {\mathcal {X}}^{{{\text {ord}}},\natural }_{n,(i),U^p(N_1,N_2)} \}\) has a right action of \(L_{n,(i)}({\mathbb {A}}^\infty )^{{\text {ord}}}\). (See Sect. 4.1.)
There is a family of closed subschemes
which are preserved by the action of \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\). We set
which is smooth over \({\mathbb {Z}}_{(p)}\) of relative dimension \((n-i)^2[F^+: {\mathbb {Q}}]\). We can describe it as
where the second coproduct runs over
The action of \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\) can be described as in the case of \(\partial ^0_i X_{n,U}^{{\text {min}}}\). (See Sect. 5.1.)
We will now turn to the boundary of \(X_{n,U,\varDelta }\) and \(A^{(m)}_{n,U,\varSigma }\). The former is the special case of the latter in which \(m=0\), so we will discuss only the case of \(A^{(m)}_{n,U,\varSigma }\) and allow the reader to specialize to the case \(m=0\). We will first describe more precisely what is the data encoded in \(\varSigma \).
If X is an F-vector space we let \({{\text {Herm}}}_X\) denote the space of symmetric \({\mathbb {Q}}\)-bilinear forms
such that
for \(x,y \in X\) and \(a \in F\). If \(W \subset V_n\) is an isotropic F-subspace we set
If \(V_{n,(i)}\) denotes the subspace of \(V_n\) with the last \(2n-i\) entries zero then \({\mathfrak {C}}^{(m)}(V_{n,(i)})\) can be identified with \(Z(N^{(m)}_{n,(i)})({\mathbb {R}})\), where \(Z(N_{n,(i)}^{(m)})\) is the subgroup of \(G_n^{(m)}\) consisting of matrices of the following shape:
If \(W' \subset W\) then \(V_n/W^\perp \twoheadrightarrow V_n/(W')^\perp \) and so
If \(\gamma \in G_n^{(m)}({\mathbb {Q}})\) then there is a natural map
We define \({\mathfrak {C}}^{(m),>0}(W)\) (resp. \({\mathfrak {C}}^{(m),\ge 0}(W)\)) to be the set of pairs \(( (\,\,\,,\,\,\,),f)\) such that \((\,\,\,,\,\,\,)\) is positive definite (positive semi-definite) on \((V_n/W^\perp ) \otimes _{\mathbb {Q}}{\mathbb {R}}\), and set
Alternatively \({\mathfrak {C}}^{(m), \succ 0}(W)\) can be described as the set of \(((\,\,\,,\,\,\,),f) \in {\mathfrak {C}}^{(m), \ge 0}(W)\) such that the kernel (i.e. radical) of \((\,\,\,,\,\,\,)\) is defined over \({\mathbb {Q}}\). We then define a topological space \({\mathfrak {C}}^{(m)}\) by
where \(\sim \) is the equivalence relation generated by the identification of \({\mathfrak {C}}^{(m),\succ 0}(W')\) with its image in \({\mathfrak {C}}^{(m),\succ 0}(W)\) whenever \(W' \subset W\). Thus as a set
The space \({\mathfrak {C}}^{(m)}\) has a continuous action of \(G_n^{(m)}({\mathbb {Q}})\) and of \({\mathbb {R}}^\times _{>0}\), the latter acting by scalar multiples on each \({\mathfrak {C}}^{(m),\succ 0}(W)\). The natural projections \({\mathfrak {C}}^{(m),\succ 0}(W) \rightarrow {\mathfrak {C}}^{(0),\succ 0}(W)\) give rise to a projection \({\mathfrak {C}}^{(m)} \rightarrow {\mathfrak {C}}={\mathfrak {C}}^{(0)}\). (See Sect. 1.4.)
By a U -admissible cone decomposition \(\varSigma \) of \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\) we shall mean a set of closed subsets \(\sigma \subset G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\) such that
-
(1)
each \(\sigma \) is contained in \(\{(g,\delta ) \} \times {\mathfrak {C}}^{(m),\succ 0}(W)\) for some isotropic subspace \(W \subset V_n\) and some \((g,\delta ) \in G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}}))\) and is the set of \({\mathbb {R}}_{\ge 0}\)-linear combinations of a finite set of elements of \({{\text {Herm}}}_{V/W^\perp } \times W^m\);
-
(2)
no \(\sigma \in \varSigma \) contains a complete line through the origin in any \((g,\delta ) \times {\mathfrak {C}}^{(m)}(W)\);
-
(3)
if \(\sigma \in \varSigma \) then any face of \(\sigma \) also lies in \(\varSigma \);
-
(4)
if \(\sigma , \sigma ' \in \varSigma \) then either \(\sigma \cap \sigma '=\emptyset \) or \(\sigma \cap \sigma '\) is a face of \(\sigma \) and \(\sigma '\);
-
(5)
\(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}=\bigcup _{\sigma \in \varSigma } \sigma \);
-
(6)
\(\varSigma \) is invariant by the diagonal action of \(G_n^{(m)}({\mathbb {Q}})\) on \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\);
-
(7)
\(\varSigma \) is invariant by the right action of U on \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\) (acting only on the first factor);
-
(8)
\(G_n^{(m)}({\mathbb {Q}}) \backslash \varSigma / U\) is a finite set;
-
(9)
if \(\sigma \in \varSigma \) lies in \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m),\succ 0}(V_{n,(i)})\) and if \(h \in P_{n,(i)}^{(m)}({\mathbb {A}})\), then \(h \sigma \in \varSigma \);
-
(10)
if \(\sigma \in \varSigma \) lies in \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m),\succ 0}(V_{n,(i)})\), if \(\gamma \in G_n^{(m)}({\mathbb {Q}})\), if \(u \in U\) and if \(h \in P_{n,(i)}^{(m)}({\mathbb {A}})\) satisfy
$$\sigma \cap \gamma h\sigma u \cap \left( G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m),> 0}(V_{n,(i)})\right) \ne \emptyset $$then \(\gamma \in P_{n,(i)}^{(m)}({\mathbb {Q}})\).
(Here we let \(G_n^{(m)}({\mathbb {A}})\) act on \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\) via multiplication on the first two factors. The restriction of this action to \(G_n^{(m)}({\mathbb {Q}})\) does not coincide with the standard action of \(G_n^{(m)}({\mathbb {Q}})\), which we are using.) We call \(\varSigma \) an admissible cone decomposition if it is U-admissible for some U. The group \({\mathbb {G}}_n^{(m)}({\mathbb {A}}^\infty )\) acts on admissible cone decompositions. We call \(\varSigma '\) a refinement of \(\varSigma \) if every element of \(\varSigma \) is a union of elements of \(\varSigma '\). We write \((U',\varSigma ') \ge (U,\varSigma )\) if \(U' \subset U\) and \(\varSigma '\) is a refinement of \(\varSigma \). We say that \(\varSigma \) is compatible with \(\varDelta \), an admissible cone decomposition of \(G_n({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}\), if the projection of each element of \(\varSigma \) is contained in an element of \(\varDelta \). We write \((U,\varSigma ) \ge (V,\varDelta )\) if U maps to V and \(\varSigma \) is compatible with \(\varDelta \). For each m there is a cofinal collection \({\mathcal {J}}_n^{(m),{{\text {tor}}}}\) of pairs \((U,\varSigma )\) of a neat open compact subgroup \(U \subset G_n^{(m)}({\mathbb {A}}^\infty )\) and a U-admissible cone decomposition \(\varSigma \) of \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\) with various natural properties, some of which are listed in Sect. 5.2. In particular it is preserved by the action of \(G_n^{(m)}({\mathbb {A}}^\infty )\). (See Sect. 5.2.)
We define \((G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)})^{{\text {ord}}}\) to be the subset of \((g,\delta ,x)\) such that for some W we have \(x \in {\mathfrak {C}}^{(m),\succ 0}(W)\) and \(W \otimes {\mathbb {Q}}_p=g_p(V_{n,(n)} \otimes {\mathbb {Q}}_p)\). It is invariant under the left action of \(G_n^{(m)}({\mathbb {Q}})\), under the right action of \(G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) and under the action of \({\mathbb {R}}_{>0}^\times \). (See Sect. 1.4.) For \(U^p(N) \subset G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\), by a \(U^p(N)\)-admissible cone decomposition of \((G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)})^{{\text {ord}}}\) we mean a collection \(\varSigma \) of closed subsets \(\sigma \subset (G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)})^{{\text {ord}}}\) satisfying analogous properties to those listed in the previous paragraph. Notions of ‘refinement’ and ‘compatibility’ are defined just as in the previous paragraph. In the same manner we also define a partial ordering on pairs \((U^p(N),\varSigma )\). If \(\varSigma \) is a \(U^p(N_1,N_2)\)-admissible cone decomposition of \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}\), then the collection \(\varSigma ^{{\text {ord}}}\) of elements of \(\varSigma \) which are contained in \((G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)})^{{\text {ord}}}\) is a \(U^p(N_1)\)-admissible cone decomposition of \((G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)})^{{\text {ord}}}\). For each m we define \({\mathcal {J}}_n^{(m),{{\text {tor}}},{{\text {ord}}}}\) to be the set of \((U^p(N_1),\varSigma ^{{\text {ord}}})\), where \((U^p(N_1,N_2),\varSigma ) \in {\mathcal {J}}_n^{(m),{{\text {tor}}}}\) for some \(N_2\) and \(\varSigma \). It has various natural properties listed in Sect. 5.2. In particular it is preserved by the action of \(G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\). (See Sect. 5.2.)
As was already mentioned, to each \((U,\varSigma ) \in {\mathcal {J}}_n^{(m).{{\text {tor}}}}\) we can associate a smooth projective variety \(A^{(m)}_{n,U,\varSigma }/X_{n,U}^{{\text {min}}}\) together with a simple normal crossings divisor \(\partial A^{(m)}_{n,U,\varSigma } \subset A^{(m)}_{n,U,\varSigma }\) such that
and \(\partial A^{(m)}_{n,U,\varSigma }\) is the reduced pre-image of \(\partial X_{n,U}^{{\text {min}}}\). We write \(\partial _i A^{(m)}_{n,U,\varSigma }\) for the reduced pre-image of \(\partial _i X_{n,U}^{{\text {min}}}\);
The irreducible components of \(\partial A^{(m)}_{n,U,\varSigma }\) are in bijection with the one-dimensional cones in \(G_n^{(m)}({\mathbb {Q}})\backslash \varSigma /U\). A collection of such irreducible components have a non-empty intersection if and only if there is a cone \(\sigma \in \varSigma \) such that the given irreducible components correspond to the one-dimensional faces of \(\sigma \). In that case we write \(\partial _{[\sigma ]}A^{(m)}_{n,U,\varSigma }\) for this intersection. We introduce the simplicial complex \({\mathcal {S}}(\partial A^{(m)}_{n,U,\varSigma })\) whose vertices are the irreducible components of \(\partial A^{(m)}_{n,U,\varSigma }\) and whose simplices correspond to collections of such irreducible components with nonzero intersection. Then the topological realization \(|{\mathcal {S}}(\partial A^{(m)}_{n,U,\varSigma })|\) of \({\mathcal {S}}(\partial A^{(m)}_{n,U,\varSigma })\) can be identified with
Moreover
is identified with
where
is a dense open subset. Moreover \(G_n^{(m)}({\mathbb {Q}}) \backslash (G_n^{(m)}({\mathbb {A}}^\infty )/U \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}_{=n}/R^\times _{>0})\) can be identified with
where \(U_{n,\infty }^0\) is a certain maximal connected compact subgroup of \(G_n({\mathbb {R}})\) defined in Sect. 1.1, and \(A_{n,(n)}\) denotes the maximal split torus in the centre of \(L_{n,(n)}\). (See Sect. 5.3.)
Similarly to each \((U^p(N_1,N_2),\varSigma ) \in {\mathcal {J}}_n^{(m).{{\text {tor}}}}\) we can associate a smooth quasi-projective variety \({\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\varSigma }/{\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}\) together with a simple normal crossings divisor
such that
and \(\partial {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\varSigma }\) is the reduced pre-image of \(\partial {\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}\). We will also write \(\partial _i {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\varSigma }\) for the reduced pre-image of \(\partial _i {\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}\) and set
The irreducible components of \(\partial {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\varSigma }\) are in bijection with the one-dimensional cones in \(G_n^{(m)}({\mathbb {Q}})\backslash \varSigma /U^p(N_1,N_2)\). A collection of such irreducible components have a non-empty intersection if and only if there is a cone \(\sigma \in \varSigma \) such that the given irreducible components correspond to the one-dimensional faces of \(\sigma \). In that case we write \(\partial _{[\sigma ]}{\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\varSigma }\) for this intersection. The \({\mathbb {F}}_p\)-fibre of \(\partial _{[\sigma ]}{\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\varSigma }\) is non-empty if and only if \(G_n^{(m)}({\mathbb {Q}})\sigma U\) contains an element of \(\varSigma ^{{\text {ord}}}\). (See Sect. 5.3.)
We let \(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1),\varSigma ^{{\text {ord}}}}\) denote the \({\mathbb {F}}_p\)-fibre of \({\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\varSigma }\). It is independent of \(N_2\ge N_1\) and \(\varSigma \) inducing \(\varSigma ^{{\text {ord}}}\). Similarly we define \(\partial \overline{{A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1),\varSigma ^{{\text {ord}}}}\) and \(\partial _i \overline{{A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1),\varSigma ^{{\text {ord}}}}\) and, if \(\sigma \in \varSigma ^{{\text {ord}}}\), also \(\partial _{[\sigma ]} \overline{{A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1),\varSigma ^{{\text {ord}}}}\). If \(\sigma \) meets \(G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}_{=n}\) then \(\partial _{[\sigma ]} \overline{{A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1),\varSigma ^{{\text {ord}}}}\) is irreducible. (It probably is in all cases, but we don’t know that.) We also define
where the disjoint union is over s-dimensional cones \(\sigma \in \varSigma ^{{\text {ord}}}\) taken up to equivalence, where two cones are considered equivalent if they have the same image in \(G_n^{(m)}({\mathbb {Q}})\backslash \varSigma /U^p(N_1,N_2)\). We have an identification of the topological spaces associated to the simplicial complex recording the intersections of the irreducible components of the boundary:
with
(See Sect. 5.3.) We write
an \((S^1)^{nm[F: {\mathbb {Q}}]}\)-bundle over a locally symmetric space associated to \(L_{n,(n),{{\text {lin}}}}\). With this notation
(See Sect. 1.4.)
We also define \({\mathfrak {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1),\varSigma ^{{\text {ord}}}}\) to be the formal completion of \({\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\varSigma }\) along its \({\mathbb {F}}_p\)-fibre. It is independent of \(N_2 \ge N_1\) and \(\varSigma \) inducing \(\varSigma ^{{\text {ord}}}\). We similarly define \(\partial {\mathfrak {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1),\varSigma ^{{\text {ord}}}}\) and \(\partial _i {\mathfrak {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1),\varSigma ^{{\text {ord}}}}\) and \(\partial _{[\sigma ]} {\mathfrak {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1),\varSigma ^{{\text {ord}}}}\) and \(\partial ^{(s)} {\mathfrak {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1),\varSigma ^{{\text {ord}}}}\). (See Sect. 5.3.) We will write \(\partial _{[\sigma ]} {\mathcal {A}}^{(m),{{\text {ord}}},\dagger }_{n,U^p(N_1),\varSigma ^{{\text {ord}}}}\) and \(\partial ^{(s)} {\mathcal {A}}^{(m),{{\text {ord}}},\dagger }_{n,U^p(N_1),\varSigma ^{{\text {ord}}}}\) for the tube over the corresponding schemes in characteristic p inside the dagger spaces associated to \(\partial _{[\sigma ]} A^{(m)}_{n,U^p(N_1,N_2),\varSigma }\) and \(\partial ^{(s)} A^{(m)}_{n,U^p(N_1,N_2),\varSigma }\). Again this is independent of \(N_2\) and \(\varSigma \). (See Sect. 6.3.)
We write \(A^{(m),\wedge }_{n,U,\varSigma ,i}\) for the formal completion of \(A^{(m)}_{n,U,\varSigma }\) along \(\partial _i^0 A^{(m)}_{n,U,\varSigma }\). There is an explicit description of \(A^{(m),\wedge }_{n,U,\varSigma ,i}\), but it will require considerable extra notation, which we now explain. (See Sect. 5.3.)
The group \(P_{n,(i)}^{(m),+}\) is the subgroup of \(G_n^{(m)}\) consisting of elements of the form
It is the semi-direct product of the subgroup \(N_{n,(i)}^{(m)}\) consisting of elements of the form
by \(L_{n,(i)}^{(m)}=L_{n,(i),{{\text {herm}}}} \times L_{n,(i),{{\text {lin}}}}^{(m)}\), where \(L_{n,(i),{{\text {lin}}}}^{(m)}\) is the subgroup consisting of elements of the form
Thus
We also write \(P_{n,(i)}^{(m)}\) for the subgroup of \(P_{n,(i)}^{(m),+}\) consisting of matrices of the form
Then
and
(See Sect. 1.2.)
For \(U \subset L_{n,(i)}^{(m)}({\mathbb {A}}^\infty )\) a neat open compact subgroup we set
It has a left action of \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}})=GL_i(F) \ltimes {{\text {Hom}}}_F(F^m,F^i)\) and the inverse system \(\{ X^{(m),+}_{n,(i),U} \}\) has a commuting right action of \(L_{n,(i)}^{(m)}({\mathbb {A}}^\infty )\). These actions are defined exactly similarly to those on \(X^+_{n,(i),U}\). We have
(See Sect. 4.1.)
If U is a neat open compact subgroup of \((P^{(m),+}_{n,(i)}/Z(N_{n,(i)}^{(m)}))({\mathbb {A}}^\infty )\) we set
The scheme \(A_{n,(i),U}^{(m),+}\) has a left action of \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}})\) such that \(\delta \in L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}})\) acts via the coproduct of the maps
The system of schemes \(\{ A_{n,(i),U}^{(m),+} \}\) has a commuting right action of the quotient group
such that if \(g \in (P^{(m),+}_{n,(i)}/Z(N_{n,(i)}^{(m)}))({\mathbb {A}}^\infty )\) maps to \(g_{{\text {lin}}}\in L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^\infty )\) then g acts by the coproduct of the maps
(See Sect. 4.2.)
If U is a neat open compact subgroup of \(P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )\) we define \(S_{n,(i),U}^{(m),+}\) to be the torus over \(X_{n,(i),U}^{(m),+}\) with cocharacter group constant over
(where \(\overline{{U}}\) denotes the image of U in \(L_{n,(i)}^{(m)}({\mathbb {A}}^\infty )\)) and identified with
In fact, we define
so that \(S_{n,(i),U}^{(m),+}\) is already defined over \(Y_{n,(i),U}^+\). The torus \(S_{n,(i),U}^{(m),+}\) has a left action of \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}})\) such that \(\delta \in L_{n,(i),{{\text {lin}}}}^{(m)}\) sends
via the morphism induced by the map on cocharacter groups
given by conjugation by \(\delta \). The system of tori \(\{ S_{n,(i),U}^{(m),+} \}\) also has a right action of \(P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )\) such that \(g \in P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )\) mapping to \(g_{{\text {lin}}}\in L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^\infty )\) sends
via the morphism induced by the map on cocharacter groups
by conjugation by \(hg_{{\text {lin}}}g^{-1}h^{-1}\). (See Sect. 4.3.)
If \((U,\varSigma ) \in {\mathcal {J}}^{(m),{{\text {tor}}}}_n\) then we define a partial fan (in the sense of Sect. 2.5) \(\varSigma (h)_0\) in \(X_*(S_{n,(i),hUh^{-1} \cap P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )}^{(m),+})\) as follows. Over
we take the cones
such that
and \(\sigma \cap {\mathfrak {C}}^{(m),>0}(V_{n,(i)})\) is non-empty. It is preserved by the action of \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}})\). (See Sect. 5.2.) This, in the sense of Sect. 4.4, is an ‘admissible cone decomposition’ of \(X_*(S_{n,(i),hUh^{-1} \cap P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )}^{(m),+})^{\succ 0}\). We write \(\widetilde{\varSigma (h)}_0\) for the fan consisting of all faces of elements of \(\varSigma (h)_0\).
If U is a neat open compact subgroup of \(P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )\) we define an \(S_{n,(i),U}^{(m),+}\)-torsor \(T_{n,(i),U}^{(m),+}/A_{n,(i),U}^{(m),+}\). It has an action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Q}})\). The system \(\{ T_{n,(i),U}^{(m),+} \}\) has a commuting action of \(P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )\). (See Sect. 4.3.) There is a torus embedding
over \(A_{n,(i),U}^{(m),+}\) corresponding to \(\widetilde{\varSigma (h)_0}\). We write
for the closed subset of the boundary corresponding to \(\varSigma (h)_0 \subset \widetilde{\varSigma (h)}_0\), and let
denote the completion of \(T_{n,(i),U,\widetilde{\varSigma (h)}_0}^{(m),+}\) along \(\partial _{\varSigma (h)_0} T_{n,(i),U,\widetilde{\varSigma (h)}_0}^{(m),+}\). (See Sects. 2.3, 2.5.) The \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Q}})\) action extends to \(T_{n,(i),U,\widetilde{\varSigma (h)}_0}^{(m),+}\) and \(\partial _{\varSigma (h)_0} T_{n,(i),U}^{(m),+}\) and \(T_{n,(i),U,\varSigma (h)_0}^{(m),+,\wedge }\). The quotients
and
make sense. Sometimes it will be convenient to take the quotient in two stages. Thus we set
and
These still carry an action of \(L_{n,(i)}({\mathbb {Q}})\). If \((U,\varSigma ) \ge (V,\varDelta )\), then there is also a natural \(L_{n,(i)}({\mathbb {Q}})\)-equivariant map
The inverse systems \(\{ T_{n,(i),U,\varSigma (h)_0}^{(m),\natural ,\wedge } \}\) and \(\{ \partial _{\varSigma (h)_0} T_{n,(i),U}^{(m),\natural } \}\) as U and \(\varSigma \) vary has an action of \(P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )\). (See Sect. 4.5.)
Then we have an identification
where the term indexed by h is exactly the open and closed subformal scheme whose underlying topological space is the pre-image of \(X^\natural _{n,(i),hUh^{-1} \cap P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )} \subset \partial _i^0 X_{n,U}^{{\text {min}}}\). These identifications are compatible with the action of \(G_n^{(m)}({\mathbb {A}}^\infty )\) and the maps
(See Sect. 5.3.)
There is a similar description for the formal completion of \({\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\varSigma }\) along \(\partial _i^0 {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\varSigma }\), which we will denote \({\mathcal {A}}^{(m),{{\text {ord}}},\wedge }_{n,U^p(N_1,N_2),\varSigma ,i}\).
We let \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}=L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) denote \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^{\infty ,p} \times {\mathbb {Z}}_p)\). We write
If \(U^p \subset L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^{\infty ,p})\) we write \(U^p(N)=U^p \times U_p(N)^{(m)}_{n,(i),{{\text {lin}}}}\). We also define
which has a left action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\). The inverse system \(\{ {\mathcal {Y}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+}\}\) has a commuting right action of \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\). We set
(See Sect. 4.1.)
We set
and
We set
and, if \(U^p \subset L_{n,(i)}^{(m)}({\mathbb {A}}^{\infty ,p})\), then we set \(U^p(N_1,N_2)=U^p \times U_p(N_1,N_2)^{(m)}_{n,(i)}\). We define
which has a left action of \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\). The inverse system \(\{ {\mathcal {X}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\}\) has a commuting right action of \(L_{n,(i)}^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\). We set
(See Sect. 4.1.)
Further we set \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) to be
and \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) to be
Moreover we define \(U_p(N_1,N_2)_{n,(i)}^{(m),+}\) to be the subgroup of \(P_{n,(i)}^{(m),+}({\mathbb {Z}}_p)\) consisting of matrices of the form
If \(U^p \subset P_{n,(i)}^{(m),+}({\mathbb {A}}^{\infty ,p})\) (resp. \((P^{(m),+}_{n,(i)}/Z(N_{n,(i)}^{(m)}))({\mathbb {A}}^{\infty ,p})\)) we define \(U^p(N_1,N_2)\) to be \(U^p \times U_p(N_1,N_2)_{n,(i)}^{(m),+}\) (resp. \(U^p \times U_p(N_1,N_2)_{n,(i)}^{(m),+}/(Z(N_{n,(i)}^{(m)})({\mathbb {Z}}_p))\)). If \(U^p\) is a neat open compact subgroup of \((P^{(m),+}_{n,(i)}/Z(N_{n,(i)}^{(m)}))({\mathbb {A}}^{\infty ,p})\) we define
which has a left action of \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\). The system of schemes \(\{ {\mathcal {A}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+} \}\) has a commuting right action of \(P^{(m),+}_{n,(i)}({\mathbb {A}}^\infty )^{{\text {ord}}}/(Z(N_{n,(i)}^{(m)})({\mathbb {A}}^{\infty ,p}\times {\mathbb {Z}}_p))\). (See Sect. 4.2.)
If \(U^p\) is a neat open compact subgroup of \(P_{n,(i)}^{(m),+}({\mathbb {A}}^{\infty ,p})\) we define \({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+}\) to be the torus over \({\mathcal {Y}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+}\) with cocharacter group over the \({{\text {Spec}}\,}{\mathbb {Z}}_{(p)}\) indexed by \(h \in L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }/U^p(N)\) identified with
The torus \({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+}\) has a left action of \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\). The system of tori \(\{ {\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+} \}\) also has a right action of \(P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )^{{\text {ord}}}\). (See Sect. 4.3.)
If \((U^p(N),\varSigma ) \in {\mathcal {J}}^{(m),{{\text {tor}}},{{\text {ord}}}}_n\) and \(h \in G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) then we define a partial fan \(\varSigma (h)_0\) in \(X_*({\mathcal {S}}_{n,(i),(hU^ph^{-1} \cap P_{n,(i)}^{(m),+}({\mathbb {A}}^{\infty ,p}))(N)}^{(m),{{\text {ord}}},+})\) as follows. Over the \({{\text {Spec}}\,}{\mathbb {Z}}_{(p)}\) corresponding to \(h' \in L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }/U^p(N)\) we take the cones
such that
and \(\sigma \cap {\mathfrak {C}}^{(m),>0}(V_{n,(i)})\) is non-empty. It is preserved by the action of \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\). (See Sect. 5.2.) This, in the sense of Sect. 4.4, is an ‘admissible cone decomposition’ of \(X_*({\mathcal {S}}_{n,(i),(hU^ph^{-1} \cap P_{n,(i)}^{(m),+}({\mathbb {A}}^{\infty ,p}))(N)}^{(m),+,{{\text {ord}}}})^{\succ 0}\). We write \(\widetilde{\varSigma (h)}_0\) for the fan consisting of all faces of elements of \(\varSigma (h)_0\).
If \(U^p\) is a neat open compact subgroup of \(P_{n,(i)}^{(m),+}({\mathbb {A}}^{\infty ,p})\) we define an \({\mathcal {S}}_{n,(i),U^p(N_1)}^{(m),{{\text {ord}}},+}\)-torsor \({\mathcal {T}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+}/{\mathcal {A}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+}\). It has an action of \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)})\). The system \(\{ {\mathcal {T}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+} \}\) has a commuting action of \(P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )^{{\text {ord}}}\). (See Sect. 4.3.) There is a torus embedding
over \({\mathcal {A}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+}\) corresponding to \(\widetilde{\varSigma (h)_0}\). We write
for the closed subset of the boundary corresponding to \(\varSigma (h)_0 \subset \widetilde{\varSigma (h)}_0\), and let
denote the completion of \({\mathcal {T}}_{n,(i),U^p(N_1,N_2),\widetilde{\varSigma (h)}_0}^{(m),{{\text {ord}}},+}\) along \(\partial _{\varSigma (h)_0} {\mathcal {T}}_{n,(i),U^p(N_1,N_2),\widetilde{\varSigma (h)}_0}^{(m),{{\text {ord}}},+}\). (See Sects. 2.3, 2.5.) The \(L_{n,(i),{{\text {lin}}}}^{(m)}({\mathbb {Z}}_{(p)})\) action extends to \({\mathcal {T}}_{n,(i),U^p(N_1,N_2),\widetilde{\varSigma (h)}_0}^{(m),{{\text {ord}}},+}\) and \(\partial _{\varSigma (h)_0} {\mathcal {T}}_{n,(i),U^p(N_1,N_2)}^{(m),+}\) and \({\mathcal {T}}_{n,(i),U^p(N_1,N_2),\varSigma (h)_0}^{(m),{{\text {ord}}},+,\wedge }\). The quotients
and
make sense. Sometimes it will be convenient to take the quotient in two stages. Thus we set
and
These still carry an action of \(L_{n,(i)}({\mathbb {Z}}_{(p)})\). The inverse systems \(\{ {\mathcal {T}}_{n,(i),U^p(N_1,N_2),\varSigma (h)_0}^{(m),{{\text {ord}}},\natural ,\wedge } \}\) and \(\{ \partial _{\varSigma (h)_0} {\mathcal {T}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},\natural } \}\) as \(U^p(N_1,N_2)\) and \(\varSigma \) vary have actions of \(P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )^{{\text {ord}}}\). (See Sect. 4.5.)
Then we have an identification
where the term indexed by h is exactly the open and closed subformal scheme whose underlying topological space is the pre-image of
or
These identifications are compatible with the action of \(G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) and the maps
(See Sect. 5.3.)
Appendix B. Comparison with [44]
This appendix is meant to be a guide to the notation system in [44] (and in the earlier works [41–43]). While we will not cover everything, we will highlight the key definitions and results used in this paper so that, it is hoped, the reader can understand the references more easily. We will adopt the notation and conventions introduced in [41, Notation and Conventions], which might be quite different from the ones introduced in this paper. (We will explain the differences when necessary.)
1.1 Shimura varieties as PEL moduli in characteristic zero
In [41, 44] the symbols \(\mathbb {Z}({1})\) are used to denote \(\ker (\exp : \mathbb {C}\rightarrow {\mathbb {C}}^\times )\). In those papers, unlike this paper, Tate twists just mean tensor products with powers of \(\mathbb {Z}({1})\) as a \(\mathbb {Z}\)-module, with no attached Galois actions at all. Let us fix the choice of a square-root \(\sqrt{-1}\) in \(\mathbb {C}\), which defines an isomorphism \(2\pi \sqrt{-1}: \mathbb {Z}\mathop {\rightarrow }\limits ^{\sim }\mathbb {Z}({1})= \ker (\exp : \mathbb {C}\rightarrow {\mathbb {C}}^\times )\).
Consider the integral PEL datum
in the sense of definition 1.1.1.1 of [44], where \(h_0\) is the \(\mathbb {R}\)-algebra homomorphism
with \(\varPsi _n\) being the \(n \times n\)-matrix with 1’s on the anti-diagonal and 0’s elsewhere (see Sect. 1.1). Note that the technical condition 1.4.3.10 in [41] or condition 1.2.1.1 in [44] are satisfied.
The group functor \(\mathrm {G}\) of definition 1.1.1.3 of [44] is just our \(G_n\), and \(\nu \) has the same meaning in this paper and in [44]. The reflex field \(F_0\) defined by \((L\otimes _{\mathbb {Z}} {\mathbb {R}}, \langle {\,\cdot \,,\,\cdot \,}\rangle , h_0)\) (see section 1.2.5 of [41]) is just \(\mathbb {Q}\), and \(\mathsf {S}_0 = {{\text {Spec}}\,}F_0= {{\text {Spec}}\,}\mathbb {Q}\).
For each open compact subgroup \({\mathcal {H}}\) of \(\mathrm {G}({\hat{\mathbb {Z}}})\), [44] defines a moduli problem
over \(\mathsf {S}_0 = {{\text {Spec}}\,}\mathbb {Q}\) (parameterizing isomorphism classes of abelian schemes with additional PEL structures) as in the beginning of section 1.1.2 of [44], which is representable by a smooth quasi-projective scheme over \({{\text {Spec}}\,}\mathbb {Q}\) when \({\mathcal {H}}\) is neat. As explained in remark 1.1.2.1 in [44], the definition of \(\mathsf {M}_{\mathcal {H}}\) only depends on the rational PEL datum \((\mathcal {O}\otimes _{\mathbb {Z}} {\mathbb {Q}}, \star , L\otimes _{\mathbb {Z}} {\mathbb {Q}}, \langle {\,\cdot \,,\,\cdot \,}\rangle \otimes _{\mathbb {Z}} {\mathbb {Q}}, h_0)\), and hence can be extended to the cases of all open compact subgroups \({\mathcal {H}}\) of \(\mathrm {G}({\mathbb {A}^{\infty }})\), up to replacing \(L\) with any lattice in \(L\otimes _{\mathbb {Z}} {\mathbb {Q}}\) stabilized by \({\mathcal {H}}\) (and replacing \(\langle {\,\cdot \,,\,\cdot \,}\rangle \) with a suitable \({\mathbb {Q}}^\times _{>0}\)-multiple of the induced pairing). By proposition 1.4.3.4 in [41], this moduli problem is canonically isomorphic to the
in this paper when \(U={\mathcal {H}}\) is contained in \(\mathrm {G}({\hat{\mathbb {Z}}})\) (see Sect. 3.1.1). (The condition on Lie algebras in this paper is equivalent to the determinantal condition in [39, 41].) The collection \(\{ \mathsf {M}_{\mathcal {H}}\}_{\mathcal {H}}\) indexed by neat open compact subgroups \({\mathcal {H}}\) of \(\mathrm {G}({\mathbb {A}^{\infty }})\) can be \(\mathrm {G}({\mathbb {A}^{\infty }})\)-equivariantly identified with \(\{ X_{n,U} \}_U\) (with \(U={\mathcal {H}}\)), because the (Hecke) actions of \(\mathrm {G}({\mathbb {A}^{\infty }})\) on them are defined by twisting level structures on quasi-isogeny classes in the same way (see remark 1.4.3.11 of [41]). (Later we will not repeat the explanation on such canonical isomorphisms between moduli of isomorphism classes and of quasi-isogeny classes. Also, we will tacitly assume that all Hecke actions to be introduced are compatible with previously introduced ones under canonical morphisms.)
1.2 Filtrations
Consider any filtration \({\texttt {V}}= \{ {\texttt {V}}_{-j} \}_j\) on \(L\otimes _{\mathbb {Z}} {\mathbb {Q}} \cong F^{2n}\) such that
which is symplectic in the sense of definition 1.2.6.8 in [41] (or definition 1.2.1.2(4) in [44]). By lemma A.4.3 in [45], up to the action of \(\mathrm {G}(\mathbb {Q})\), we may assume that there exists an integer \(0\le i\le n\) such that
which we call the i-th standard symplectic filtration, and write \({\texttt {V}}= {\texttt {V}}^{(i)}\). The stabilizer of \({\texttt {V}}^{(i)}\) is then the parabolic subgroup \(P_{n,(i)}^+(\mathbb {Q})\) of \(G_n(\mathbb {Q})\).
We will need to consider filtrations \({\texttt {Z}}= \{ {\texttt {Z}}_{-j} \}_j\) on \(L\otimes _{\mathbb {Z}} {{\hat{\mathbb {Z}}}}\) such that
which is fully symplectic admissible as in definition 5.2.7.1 in [41] or definition 1.2.1.4 in [44]. If \(0\le i\le n\) and \(g\in \mathrm {G}({\mathbb {A}^{\infty }})\), then we can define such a filtration \({{\texttt {Z}}}^{(i, g)}\) by
for each j. Since we are not in the so-called type \(\mathrm {D}\) case, by the proof of proposition A.5.9 of [45], any fully symplectic admissible filtration arises as \({{\texttt {Z}}}^{(i, g)}\) for some i and some g. In fact, using the Iwasawa decomposition, we may suppose that \(g \in \hbox {G}({\hat{\mathbb {Z}}})\). For each j, let \({\text {Gr}}^{\texttt {Z}}_{-j} := {\texttt {Z}}_{-j}/{\texttt {Z}}_{-j-1}\) as usual. Then \(\langle {\,\cdot \,,\,\cdot \,}\rangle \) canonically induces a perfect pairing
Let us temporarily fix such a \({\texttt {Z}}= {{\texttt {Z}}}^{(i, g)}\) with \(g \in \hbox {G}({\hat{\mathbb {Z}}})\) and fix an open compact subgroup \({\mathcal {H}}\subset \hbox {G}({\hat{\mathbb {Z}}})\). In [44], the following groups are defined for \({\hat{\mathbb {Z}}}\)-algebras R along with open compact subgroups (see definitions 1.2.1.9, 1.2.1.10, and 1.2.1.11 in [44]):
-
(1)
\(\mathrm {P}_{\texttt {Z}}(R) := \{ g\in \mathrm {G}(R) : g({\texttt {Z}}) = {\texttt {Z}}\} = g^{-1} P^+_{n,(i)}(R) g\). We set \({\mathcal {H}}_{\mathrm {P}_{\texttt {Z}}} := {\mathcal {H}}\cap \mathrm {P}_{\texttt {Z}}({\hat{\mathbb {Z}}})\).
-
(2)
\(\mathrm {G}_{l, {\texttt {Z}}}'(R)\) is set equal to the subgroup of \({\text {GL}}_\mathcal {O}({\text {Gr}}^{\texttt {Z}}_{-2}\otimes _{{\hat{\mathbb {Z}}}} {R}) \times {\text {GL}}_\mathcal {O}({\text {Gr}}^{\texttt {Z}}_0\otimes _{{\hat{\mathbb {Z}}}} {R})\) consisting of elements respecting the pairing \({\text {Gr}}^{\texttt {Z}}_{-2} \times {\text {Gr}}^{\texttt {Z}}_0 \rightarrow {\hat{\mathbb {Z}}}({1})\) induced by \(\langle {\,\cdot \,,\,\cdot \,}\rangle \). This group comes isomorphically from \(g^{-1} L_{n,(i),{{\text {lin}}}}(R) g\). There is a natural surjective map \((\nu ^{-1} {\text {Gr}}^{\texttt {Z}}_{-2}, {\text {Gr}}^{\texttt {Z}}_0): \mathrm {P}_{\texttt {Z}}(R) \twoheadrightarrow \mathrm {G}_{l, {\texttt {Z}}}'(R)\) corresponding to \(P^+_{n,(i)}(R) \twoheadrightarrow L_{n,(i),{{\text {lin}}}}(R)\). We denote the kernel by \(\mathrm {P}_{\texttt {Z}}'(R) = g^{-1} P_{n,(i)}(R) g\). We set \({\mathcal {H}}_{\mathrm {P}_{\texttt {Z}}'} := {\mathcal {H}}\cap \mathrm {P}_{\texttt {Z}}'({\hat{\mathbb {Z}}})\), \({\mathcal {H}}_{\mathrm {G}_{h, {\texttt {Z}}}'} := {\mathcal {H}}_{\mathrm {P}_{\texttt {Z}}'} / {\mathcal {H}}_{\mathrm {U}_{\texttt {Z}}}\), and \({\mathcal {H}}_{\mathrm {G}_{l, {\texttt {Z}}}'} := {\mathcal {H}}_{\mathrm {P}_{\texttt {Z}}} / {\mathcal {H}}_{\mathrm {P}_{\texttt {Z}}'}\).
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(3)
\(\mathrm {G}_{h, {\texttt {Z}}}(R) := \left\{ \begin{matrix} (g, r)\in {\text {GL}}_{\mathcal {O}\otimes _{\mathbb {Z}} {R}}({\text {Gr}}^{\texttt {Z}}_{-1}\otimes _{{\hat{\mathbb {Z}}}} {R}) \times \mathbf {G}_{\text {m}}(R): \\ \langle {gx, gy}\rangle _{-1} = r \langle {x, y}\rangle _{-1}, \; \forall x,y\in {\text {Gr}}^{\texttt {Z}}_{-1}\otimes _{{\hat{\mathbb {Z}}}} {R} \end{matrix} \right\} \), which comes isomorphically from \(g^{-1} L_{n,(i),{{\text {herm}}}}(R) g\). Then there is a natural surjective map \({\text {Gr}}^{\texttt {Z}}_{-1}: \mathrm {P}_{\texttt {Z}}(R) \twoheadrightarrow \mathrm {G}_{h, {\texttt {Z}}}(R)\) corresponding to \(P^+_{n,(i)}(R) \twoheadrightarrow L_{n,(i),{{\text {herm}}}}(R)\). We denote the kernel \(\mathrm {Z}_{\texttt {Z}}(R) = ( g^{-1} L_{n,(i),{{\text {lin}}}}(R) g ) \ltimes ( g^{-1} N_{n,(i)} (R) g )\). We set \({\mathcal {H}}_{\mathrm {Z}_{\texttt {Z}}} := {\mathcal {H}}\cap \mathrm {Z}_{\texttt {Z}}({\hat{\mathbb {Z}}})\) and \({\mathcal {H}}_{\mathrm {G}_{h, {\texttt {Z}}}} := {\mathcal {H}}_{\mathrm {P}_{\texttt {Z}}} / {\mathcal {H}}_{\mathrm {Z}_{\texttt {Z}}}\).
-
(4)
\(\mathrm {U}_{\texttt {Z}}(R)\) is the subgroup of \(\mathrm {P}_{\texttt {Z}}(R)\) consisting of elements g such that \({\text {Gr}}^{\texttt {Z}}(g) = {\text {Id}}_{{\text {Gr}}^{\texttt {Z}}\otimes _{{\hat{\mathbb {Z}}}} {R}}\). Thus \(\mathrm {U}_{\texttt {Z}}(R) = g^{-1} N_{n,(i)}(R) g\). We set \({\mathcal {H}}_{\mathrm {U}_{\texttt {Z}}} := {\mathcal {H}}\cap \mathrm {U}_{\texttt {Z}}({\hat{\mathbb {Z}}})\).
-
(5)
\(\mathrm {U}_{2, {\texttt {Z}}}(R)\) is the subgroup of \(\mathrm {P}_{\texttt {Z}}(R)\) consisting of elements g which induce \({\text {Id}}_{{\texttt {Z}}_{-1}\otimes _{{\hat{\mathbb {Z}}}} {R}}\) and \({\text {Id}}_{({\texttt {Z}}_0\otimes _{{\hat{\mathbb {Z}}}} {R})/({\texttt {Z}}_{-2}\otimes _{{\hat{\mathbb {Z}}}} {R})}\) on \({\texttt {Z}}_{-1}\otimes _{{\hat{\mathbb {Z}}}} {R}\) and \(({\texttt {Z}}_0\otimes _{{\hat{\mathbb {Z}}}} {R})/({\texttt {Z}}_{-2}\otimes _{{\hat{\mathbb {Z}}}} {R})\), respectively. Thus we have \(\mathrm {U}_{2, {\texttt {Z}}}(R) = g^{-1} Z(N_{n,(i)})(R) g \cong {{\text {Herm}}}^{(i)}(R) \hookrightarrow {\text {Hom}}_{\mathcal {O}\otimes _{\mathbb {Z}} {R}}({\text {Gr}}^{\texttt {Z}}_0\otimes _{{\hat{\mathbb {Z}}}} {R}, {\text {Gr}}^{\texttt {Z}}_{-2}\otimes _{{\hat{\mathbb {Z}}}} {R})\). We set \({\mathcal {H}}_{\mathrm {U}_{2, {\texttt {Z}}}} := {\mathcal {H}}\cap \mathrm {U}_{2, {\texttt {Z}}}({\hat{\mathbb {Z}}})\).
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(6)
\(\mathrm {U}_{1, {\texttt {Z}}}(R) := \mathrm {U}_{\texttt {Z}}(R) / \mathrm {U}_{2, {\texttt {Z}}}(R) = (g^{-1} N_{n,(i)}(R) g)/(g^{-1} Z(N_{n,(i)})(R) g)\), which is isomorphic to \({{\text {Hom}}}_{n-i}^{(i)}(R)\). We set \({\mathcal {H}}_{\mathrm {U}_{1, {\texttt {Z}}}} := {\mathcal {H}}_{\mathrm {U}_{\texttt {Z}}} / {\mathcal {H}}_{\mathrm {U}_{2, {\texttt {Z}}}}\).
-
(7)
\(\mathrm {G}_{l, {\texttt {Z}}}(R) := \mathrm {Z}_{\texttt {Z}}(R) / \mathrm {U}_{\texttt {Z}}(R)\). This maps isomorphically to \(\mathrm {G}_{l, {\texttt {Z}}}'(R)\), but [44] chooses to distinguish it as a subgroup rather than a quotient of \(\mathrm {P}_{\texttt {Z}}(R)/\mathrm {U}_{\texttt {Z}}(R)\). We set \({\mathcal {H}}_{\mathrm {G}_{l, {\texttt {Z}}}} := {\mathcal {H}}_{\mathrm {Z}_{\texttt {Z}}} / {\mathcal {H}}_{\mathrm {U}_{\texttt {Z}}}\), which may differ from \({\mathcal {H}}_{\mathrm {G}_{l, {\texttt {Z}}}'}\).
-
(8)
\(\mathrm {G}_{h, {\texttt {Z}}}'(R) := \mathrm {P}_{\texttt {Z}}'(R) / \mathrm {U}_{\texttt {Z}}(R)\). This maps isomorphically to \(\mathrm {G}_{h, {\texttt {Z}}}(R)\), but [44] chooses to distinguish it as a subgroup rather than a quotient of \(\mathrm {P}_{\texttt {Z}}(R)/\mathrm {U}_{\texttt {Z}}(R)\). We set \({\mathcal {H}}_{\mathrm {G}_{h, {\texttt {Z}}}'} := {\mathcal {H}}_{\mathrm {P}_{\texttt {Z}}'} / {\mathcal {H}}_{\mathrm {U}_{\texttt {Z}}}\), which may differ from \({\mathcal {H}}_{\mathrm {G}_{h, {\texttt {Z}}}}\).
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(9)
\(\mathrm {G}_{1,{\texttt {Z}}}(R):= \mathrm {P}_{\texttt {Z}}'(R) / \mathrm {U}_{2,{\texttt {Z}}}(R)\). It is isomorphic to \(G_{n-i}^{(i)}(R)\). We set \({\mathcal {H}}_{\mathrm {G}_{1, {\texttt {Z}}}'} := {\mathcal {H}}_{\mathrm {P}_{\texttt {Z}}'} / {\mathcal {H}}_{\mathrm {U}_{2,{\texttt {Z}}}}\).
For each open compact subgroup \({\mathcal {H}}=U\) of \(\mathrm {G}({\hat{\mathbb {Z}}})\), as g varies in \(\mathrm {G}({\mathbb {A}^{\infty }})\), the \({\mathcal {H}}\)-orbits \({{\texttt {Z}}}^{(i, g)}_{\mathcal {H}}\) of \({{\texttt {Z}}}^{(i, g)}\) are parameterized by the double coset space
(See, for example, Sect. 5.1.)
1.3 Cusp labels
Given any fully symplectic admissible filtration \({\texttt {Z}}\), we define a torus argument to be a rigidification \(\varPhi = (X, Y, \phi , \varphi _{-2}, \varphi _0)\) on the top and bottom graded pieces (see definition 5.4.1.3 in [41] or definition 1.2.1.5 in [44]):
-
(1)
\(X\) and \(Y\) are locally free \(\mathcal {O}\)-modules, and \(\phi : Y\rightarrow X\) is an isomorphism of \(\mathcal {O}\)-modules.
-
(2)
\(\varphi _{-2}: {\text {Gr}}^{\texttt {Z}}_{-2} \mathop {\rightarrow }\limits ^{\sim }{\text {Hom}}_{\mathbb {Z}}(X, {\hat{\mathbb {Z}}}({1}))\) and \(\varphi _0: {\text {Gr}}^{\texttt {Z}}_0 \mathop {\rightarrow }\limits ^{\sim }Y\otimes _{\mathbb {Z}} {{\hat{\mathbb {Z}}}}\) are isomorphisms of \(\mathcal {O}\otimes _{\mathbb {Z}} {{\hat{\mathbb {Z}}}}\)-modules such that the pairing \(\langle {\,\cdot \,, \,\cdot \,}\rangle _{20}: {\text {Gr}}^{\texttt {Z}}_{-2} \times {\text {Gr}}^{\texttt {Z}}_0 \rightarrow {\hat{\mathbb {Z}}}({1})\) induced by \(\langle {\,\cdot \,,\,\cdot \,}\rangle \) is the pull-back of the pairing
$$\begin{aligned} \langle {\,\cdot \,, \,\cdot \,}\rangle _\phi : {\text {Hom}}_{\hat{\mathbb {Z}}}(X\otimes _{\mathbb {Z}} {{\hat{\mathbb {Z}}}}, {\hat{\mathbb {Z}}}({1})) \times (Y\otimes _{\mathbb {Z}} {{\hat{\mathbb {Z}}}}) \rightarrow {\hat{\mathbb {Z}}}({1})\end{aligned}$$defined by \(\langle {x, y}\rangle _\phi = x(\phi (y))\).
(For a general integral PEL datum, the first condition should be weakened to \(\phi \) being injective with finite cokernel, but in our setting the second condition then forces it to be an isomorphism.) We say that two torus arguments \(\varPhi = (X, Y, \phi , \varphi _{-2}, \varphi _0)\) and \(\varPhi ' = (X', Y', \phi ', \varphi '_{-2}, \varphi '_0)\) are equivalent if there exists a pair of isomorphisms \((\gamma _X: X' \mathop {\rightarrow }\limits ^{\sim }X, \gamma _Y: Y\mathop {\rightarrow }\limits ^{\sim }Y')\) of \(\mathcal {O}\)-lattices matching all other data. For \(0\le i\le n\) we have a torus argument
for \({{\texttt {Z}}}^{(i, 1)}\), where \(\varPsi _i\) is (as before) the \(i \times i\)-matrix with 1’s on the anti-diagonal and 0’s elsewhere (see Sect. 1.1).
If \(\varPhi = (X, Y, \phi , \varphi _{-2}, \varphi _0)\) is a torus argument for \({\texttt {Z}}\) and \(g \in \mathrm {P}_{\texttt {Z}}({\mathbb {A}^{\infty }})\), then we obtain a torus argument \({\varPhi }^{(g)} = ({X}^{(g)}, {Y}^{(g)}, {\phi }^{(g)}, {\varphi }^{(g)}_{-2}, {\varphi }^{(g)}_0)\) for \({{\texttt {Z}}}^{(g)}\) induced by \(g: {\text {Gr}}^{g^{-1}{\texttt {Z}}} \rightarrow {\text {Gr}}^{\texttt {Z}}\) (see section 1.2.3 of [44] for more details). We shall write \({\varPhi }^{(i, g)} = ({\varPhi }^{(i)}){}^{(g)}\).
Consider triples of the form \(({\texttt {Z}}, \varPhi , \delta )\), where \({\texttt {Z}}\) is a filtration as in the previous section, where \(\varPhi \) is a torus argument as above, and where \(\delta : {\text {Gr}}^{\texttt {Z}}= {\text {Gr}}^{\texttt {Z}}_{-2} \oplus {\text {Gr}}^{\texttt {Z}}_{-1} \oplus {\text {Gr}}^{\texttt {Z}}_0 \mathop {\rightarrow }\limits ^{\sim }L\otimes _{\mathbb {Z}} {{\hat{\mathbb {Z}}}}\) is an \(\mathcal {O}\otimes _{\mathbb {Z}} {{\hat{\mathbb {Z}}}}\)-equivariant splitting. The group \(\mathrm {G}({\hat{\mathbb {Z}}})\) acts naturally on such triples, and hence we can consider their \({\mathcal {H}}\)-orbits of the form \(({\texttt {Z}}_{\mathcal {H}}, \varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})\), for each open compact subgroup \({\mathcal {H}}\) of \(\mathrm {G}({\hat{\mathbb {Z}}})\). We consider two such orbits \(({\texttt {Z}}_{\mathcal {H}}, \varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})\) and \(({\texttt {Z}}'_{\mathcal {H}}, \varPhi '_{\mathcal {H}}, \delta '_{\mathcal {H}})\) equivalent if \({\texttt {Z}}_{\mathcal {H}}= {\texttt {Z}}'_{\mathcal {H}}\) and if there exist representatives \(\varPhi \) and \(\varPhi '\) of \(\varPhi _{\mathcal {H}}\) and \(\varPhi '_{\mathcal {H}}\), respectively, which are equivalent. (Note that no compatibility condition is imposed on \(\delta _{\mathcal {H}}\) and \(\delta '_{\mathcal {H}}\).) We call the equivalence classes \([({\texttt {Z}}_{\mathcal {H}}, \varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]\), or simply \([(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]\), the cusp labels for \(\mathsf {M}_{\mathcal {H}}\).
For each \(({\texttt {Z}}, \varPhi , \delta )\) as above, and any \(\mathbb {Z}\)-algebra R, we define \(\mathrm {G}_{l, \varPhi }'(R) = {\text {GL}}_{\mathcal {O}\otimes _{\mathbb {Z}} {R}}(Y\otimes _{\mathbb {Z}} {R})\), which admits a canonical map to \(\mathrm {G}_{l, {\texttt {Z}}}'(R \otimes _{\mathbb {Z}} {{\hat{\mathbb {Z}}}})\) induced by \(\varPhi \). When R is a \({\hat{\mathbb {Z}}}\)-algebra, we have \(\mathrm {G}_{l, \varPhi }'(R) \cong \mathrm {G}_{l, {\texttt {Z}}}'(R)\). When \(R=\mathbb {Q}\), we have \(\mathrm {G}_{l, \varPhi }'(\mathbb {Q}) \hookrightarrow \mathrm {G}_{l, {\texttt {Z}}}'({\mathbb {A}^{\infty }})\), and we define
a congruence subgroup of \(\mathrm {G}_{l, \varPhi }'(\mathbb {Z}) = \mathrm {G}_{l, {\texttt {Z}}}'({\hat{\mathbb {Z}}}) \cap \mathrm {G}_{l, \varPhi }'(\mathbb {Q})\) depending only on \(\varPhi _{\mathcal {H}}\) (see definition 1.2.2.3 of [44] for an equivalent definition).
The map that sends \(g \in \mathrm {G}({\mathbb {A}}^\infty )\) to \([({{\texttt {Z}}}^{(i, g)}_{\mathcal {H}}, {\varPhi }^{(i, g)}_{\mathcal {H}}, \delta _{\mathcal {H}})]\) (for any \(\delta \)) sets up a bijection between the double coset space
and cusp labels \([({\texttt {Z}}_{\mathcal {H}}, \varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]\) such that \({\text {Gr}}^{\texttt {Z}}_0\) has rank i over \(\mathcal {O}\otimes _{\mathbb {Z}} {{\hat{\mathbb {Z}}}}\). This identifies the set of cusp labels of the form \([({\texttt {Z}}^{(i,g)}_{\mathcal {H}}, \varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]\) with
This last double coset space is the index set of
in this paper (see Sect. 4.1), with \(U={\mathcal {H}}\).
1.4 Minimal compactifications in characteristic zero
For each neat open compact subgroup \({\mathcal {H}}=U\), there is a normal projective scheme
containing \(\mathsf {M}_{\mathcal {H}}=X_{n,U}\) as an open dense subscheme (see theorem 1.3.1.5 of [44]).
The minimal compactification \({\mathsf {M}}^{\text {min}}_{\mathcal {H}}\) admits a stratification by locally closed subschemes
where \([(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]\) runs over cusp labels for \(\mathsf {M}_{\mathcal {H}}\) and g runs over the double quotient \(( \mathrm {G}_{l, {\varPhi }^{(i, 1)}}'(\mathbb {Q}) \ltimes \mathrm {P}_{{{\texttt {Z}}}^{(i, 1)}}'({\mathbb {A}^{\infty }}) ) \backslash \mathrm {G}({\mathbb {A}^{\infty }}) / {\mathcal {H}}\) (see theorem 1.3.1.5(4) of [44]). If \({\mathcal {H}}=U\) this induces
The decomposition
(see Sect. 5.1) corresponds to
Each \(\mathsf {Z}_{[(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]}\) is the quotient \(\mathsf {M}_{\mathcal {H}}^{{\texttt {Z}}_{\mathcal {H}}}\) of
by \(\varGamma _{\varPhi _{\mathcal {H}}}\), which is isomorphic to \(\mathsf {M}_{{\mathcal {H}}_{\mathrm {G}_{h, {\texttt {Z}}}, \varPhi }}\), where \({\mathcal {H}}_{\mathrm {G}_{h, {\texttt {Z}}}, \varPhi }\) is the image of \({\mathcal {H}}_{\mathrm {P}_{\texttt {Z}}} \cap ( \mathrm {G}_{l, \varPhi }'(\mathbb {Q}) \ltimes \mathrm {P}_{\texttt {Z}}'({\mathbb {A}^{\infty }}) )\) in \(\mathrm {G}_{h, {\texttt {Z}}}({\mathbb {A}^{\infty }})\) (see lemmas 1.3.2.1 and 1.3.2.5 of [44]). Here \(\mathsf {M}_{{\mathcal {H}}_{\mathrm {G}_{h, {\texttt {Z}}}'}}\) and \(\mathsf {M}_{{\mathcal {H}}_{\mathrm {G}_{h, {\texttt {Z}}}, \varPhi }}\) are moduli problems analogously defined by some integral PEL datum \((\mathcal {O}, \star , L^{\texttt {Z}}, \langle {\,\cdot \,,\,\cdot \,}\rangle ^{\texttt {Z}}, h_0^{\texttt {Z}})\) associated with a representative \({\texttt {Z}}\) of \({\texttt {Z}}_{\mathcal {H}}\) (see definition 1.2.1.15 and lemmas 1.3.2.1 and 1.3.2.5 of [44]). If we let \(U_g'\) (resp. \(U_{g,{{\text {lin}}}}'\)) denote the image of \(gUg^{-1}\cap P_{n,(i)}^+({\mathbb {A}}^\infty )\) in \(L_{n,(i)}({\mathbb {A}}^\infty )\) (resp. \(L_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )\)), then under the identifications
of Sect. 4.1,
-
the term \(X_{n,(i),U_g'\cap G_{n-i}({\mathbb {A}}^\infty )}\) indexed by h is identified with \(\mathsf {M}_{\mathcal {H}}^{{\varPhi }^{(i, hg)}_{\mathcal {H}}}\);
-
the group \(L_{n,(i),{{\text {lin}}}}({\mathbb {Q}}) \cap h U_{g,{{\text {lin}}}}' h^{-1}\) is identified with \(\varGamma _{{\varPhi }^{(i, hg)}_{\mathcal {H}}}\);
-
and the term
$$\begin{aligned} \Big (L_{n,(i),{{\text {lin}}}}({\mathbb {Q}}) \cap h U_{g,{{\text {lin}}}}' h^{-1}\Big ) \Big \backslash X_{n,(i),U_g'\cap G_{n-i}({\mathbb {A}}^\infty )} \end{aligned}$$is identified with \(\mathsf {M}_{\mathcal {H}}^{{{\texttt {Z}}}^{(i, hg)}_{\mathcal {H}}}\).
Similarly, under the identification
the term \(X_{n,(i),U_g'\cap G_{n-i}({\mathbb {A}}^\infty )}\) indexed by h is identified with \(\mathsf {M}_{\mathcal {H}}^{{\varPhi }^{(i, hg)}_{\mathcal {H}}}\). These identifications are all Hecke equivariant.
1.5 Toroidal compactifications of Shimura varieties in characteristic zero
For each representative \(({\texttt {Z}}_{\mathcal {H}}, \varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})\) of a cusp label \([({\texttt {Z}}_{\mathcal {H}}, \varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]\) for \(\mathsf {M}_{\mathcal {H}}\), there is a torsor
of an abelian scheme
(see lemma 1.3.2.7 and propositions 1.3.2.12 and 1.3.2.14 of [44]). The abelian scheme \(C_{\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}}^{\text {grp}}\) over \(\mathsf {M}_{\mathcal {H}}^{\varPhi _{\mathcal {H}}}\) is \({\mathbb {Q}}^\times \)-isogenous (i.e. quasi-isogenous) to \(\underline{{\text {Hom}}}_\mathcal {O}(X, B)\) (where \(B\) is the pull-back of the universal abelian scheme over \(\mathsf {M}_{\mathcal {H}}^{{\texttt {Z}}_{\mathcal {H}}}\)). We obtain an isomorphic abelian scheme torsor if we replace \(({\texttt {Z}}_{\mathcal {H}}, \varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})\) with another representative (but its universal property, which we have not described here, depends on this choice of representative).
If \(U_g'\) again denotes \(gUg^{-1}\cap P_{n,(i)}^+({\mathbb {A}}^\infty )\), then the map
in this paper (see Sects. 4.1, 4.2) with \(U={\mathcal {H}}\) is identified with
These maps are \(\mathrm {P}_{{{\texttt {Z}}}^{(i, g)}}({\mathbb {A}^{\infty }})/\mathrm {U}_{2, {{\texttt {Z}}}^{(i, g)}}({\mathbb {A}^{\infty }})\)-equivariant. (See proposition 1.3.2.24 of [44].) (Since our pairing \(\langle {\,\cdot \,,\,\cdot \,}\rangle \) is perfect, the universal property of \(C_{\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}}\) in proposition 1.3.2.14 of [44] can be simplified by suppressing the dual objects. This universal property then agrees with that of \(A_{n-i,hgUg^{-1}h^{-1}\cap P_{n,(i)}({\mathbb {A}}^\infty )}^{(i)}\) in Sect. 3.2 of this paper.)
For each representative \(({\texttt {Z}}_{\mathcal {H}}, \varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})\) of a cusp label \([({\texttt {Z}}_{\mathcal {H}}, \varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]\) for \(\mathsf {M}_{\mathcal {H}}\), there is a torsor
under the pull-back of a split torus
over \({{\text {Spec}}\,}\mathbb {Z}\) with character group
(see lemma 1.3.2.25 and proposition 1.3.2.31 of [44]). We obtain an isomorphic torus torsor if we replace \(({\texttt {Z}}_{\mathcal {H}}, \varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})\) with another representative, but its universal property depends on this choice of representative. For a fixed \(({\texttt {Z}}, \varPhi , \delta )\), the morphisms
are \(\mathrm {P}_{\texttt {Z}}'({\mathbb {A}^{\infty }})\)-equivariant. (See proposition 1.3.2.45 of [44].)
When \({\mathcal {H}}_{\mathrm {U}_{2, {\texttt {Z}}}} = \mathrm {G}({\hat{\mathbb {Z}}})_{\mathrm {U}_{2, {\texttt {Z}}}} = \mathrm {U}_{2, {\texttt {Z}}}({\hat{\mathbb {Z}}})\) and when \(\varPhi _{\mathcal {H}}\) is represented by some \(\varPhi = (X, Y, \phi , \varphi _{-2}, \varphi _0)\) (where \(\phi : Y\hookrightarrow X\) must be an isomorphism, as explained above), the group \(\mathbf {S}_{\varPhi _{\mathcal {H}}}\) is the group \(S(Y)^{{\text {TF}}}\) in the notation of this paper (see Sect. 1.1). For more general \({\mathcal {H}}\), we set \(\mathbf {S}_{\varPhi _1} = S(Y)^{{\text {TF}}}\) (see section 1.2.2 of [44]), and then \(\mathbf {S}_{\varPhi _{\mathcal {H}}}\) is the unique lattice in \(\mathbf {S}_{\varPhi _1} \otimes _{\mathbb {Z}} {\mathbb {Q}}\) such that \(\mathbf {S}_{\varPhi _{\mathcal {H}}} / \mathbf {S}_{\varPhi _1} \cong \mathbf {S}_{\varPhi _1}^\vee / \mathbf {S}_{\varPhi _{\mathcal {H}}}^\vee \cong \mathrm {U}_{2, {\texttt {Z}}}({\hat{\mathbb {Z}}}) / {\mathcal {H}}_{\mathrm {U}_{2, {\texttt {Z}}}}\) (see proposition 1.3.2.31 of [44]), where the superscript \(^\vee \) denotes the dual of \(\mathbb {Z}\)-modules. Then \((\mathbf {S}_{\varPhi _{\mathcal {H}}})_\mathbb {R}^\vee \) can be identified with the space of hermitian forms over \(Y\otimes _{\mathbb {Z}} {\mathbb {R}}\), and we define \(\mathbf {P}_{\varPhi _{\mathcal {H}}}\) (resp. \(\mathbf {P}_{\varPhi _{\mathcal {H}}}^+\)) to be the subset of \((\mathbf {S}_{\varPhi _{\mathcal {H}}})_\mathbb {R}^\vee \) corresponding to positive semi-definite hermitian forms with rational radicals (resp. positive definite hermitian forms). When \({\texttt {Z}}_{\mathcal {H}}= {{\texttt {Z}}}^{(i, g)}_{\mathcal {H}}\) for some \(g\in \mathrm {G}({\mathbb {A}^{\infty }})\), we have \(Y\otimes _{\mathbb {Z}} {\mathbb {Q}} \cong F^i\); and the sets \(\mathbf {S}_{\varPhi _{\mathcal {H}}} \otimes _{\mathbb {Z}} {\mathbb {Q}}\) and \(\mathbf {P}_{\varPhi _{\mathcal {H}}}^+\) are the sets \(S(F^i)\) and \({{\text {Herm}}}_{F^i}^{>0}\), respectively, in this paper (see Sects. 1.1, 1.4).
The torus
in this paper (see Sect. 4.3) with \(U={\mathcal {H}}\) is identified with
Moreover, the sheaves \(X^*(S_{n,(i),U_g'}^+)\), \(X_*(S_{n,(i),U_g'}^+)_{\mathbb {R}}^{\succ 0}\), and \(X_*(S_{n,(i),U_g'}^+)_{\mathbb {R}}^{>0}\) are identified with
respectively. The \(S_{n,(i),U_g'}^+\)-torsor
is identified with
These maps are \(\mathrm {P}_{{{\texttt {Z}}}^{(i, g)}}({\mathbb {A}^{\infty }})\)-equivariant.
Consider any compatible collection
of admissible projective smooth rational polyhedral cone decompositions for \(\mathsf {M}_{\mathcal {H}}\), where each \(\varSigma _{\varPhi _{\mathcal {H}}}\) is a \(\varGamma _{\varPhi _{\mathcal {H}}}\)-admissible projective smooth rational polyhedral cone decomposition of \(\mathbf {P}_{\varPhi _{\mathcal {H}}}\), as in definitions 1.2.2.13 and 1.2.2.14 of [44]. (Note that in [44] rational polyhedral cones are open cones, whereas in this paper they are closed cones.) Each such \(\varSigma \) considered in [44] induces a pair
in \({\mathcal {J}}_n^{{\text {tor}}}\) (with \(U={\mathcal {H}}\)) in Sect. 5.2 of this paper, because, in order to define \((U,\varDelta )\) as in Sect. 5.2, it suffices to define the admissible cone decomposition \(\varDelta (g)_0\) for \(X_*(S_{n,(i),gUg^{-1}\cap P_{n,(i)}^+({\mathbb {A}}^\infty )}^+)_{\mathbb {R}}^{\succ 0}\), for each \(g\in \mathrm {G}({\mathbb {A}^{\infty }})\), which can be taken to be the pull-back of the subcollection \(\{ \varSigma _{\varPhi _{\mathcal {H}}} \}_{[(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]}\) of \(\varSigma \) indexed by the cusp labels \([(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]\) with underlying \({\texttt {Z}}_{\mathcal {H}}\) equal to \({{\texttt {Z}}}^{(i, g)}_{\mathcal {H}}\). In fact, \({\mathcal {J}}_n^{{\text {tor}}}\) is exactly the set of such induced pairs (as \(U={\mathcal {H}}\) varies).
Each \(\varSigma _{\varPhi _{\mathcal {H}}}\) defines an affine toroidal embedding
over \(C_{\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}}\). Rather confusingly \(\varXi _{\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}}(\sigma )\) (in the notation of [44]) is what in this paper we would have denoted \(\varXi _{\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}, \sigma }\); and \(\varXi _{\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}, \sigma }\) is what in this paper we would have denoted \(\partial _\sigma {\varXi }_{\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}, \varSigma _{\varPhi _{\mathcal {H}}}}\). The formal completion of \(\overline{\varXi }_{\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}, \varSigma _{\varPhi _{\mathcal {H}}}}\) along the union of the \(\sigma \)-strata \(\varXi _{\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}, \sigma }\) for all \(\sigma \in \varSigma _{\varPhi _{\mathcal {H}}}\) such that \(\sigma \cap \mathbf {P}_{\varPhi _{\mathcal {H}}}^+ \ne \emptyset \) is denoted
(See (1.3.2.34), (1.3.2.35), (1.3.2.36), and lemma 1.3.2.41 of [44].) The schemes
of this paper (see Sect. 4.4) are identified with
Moreover, \(T_{n,(i),U_g',\varDelta (g)_0}^{+,\wedge }\) is identified with
and \(T_{n,(i),U_g',\varDelta (g)_0}^{\natural ,\wedge }\) is identified with
where the second disjoint union runs over all cusp labels \([(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]\) with underlying \({\texttt {Z}}_{\mathcal {H}}\) equal to \({{\texttt {Z}}}^{(i, g)}_{\mathcal {H}}\). (Again \(U={\mathcal {H}}\).)
For each pair \(({\mathcal {H}},\varSigma )\) as above, we have a smooth projective scheme
containing \(\mathsf {M}_{\mathcal {H}}\) as an open dense subscheme, called a toroidal compactification of \(\mathsf {M}_{\mathcal {H}}\) (see theorems 1.3.1.3 and 1.3.1.10 of [44]). This is the
in this paper (see Sect. 5.3), with \(U={\mathcal {H}}\) and with \((U,\varDelta ) \in {\mathcal {J}}_n^{{\text {tor}}}\) induced by \(\varSigma \) as above. This identification is compatible with the actions of \(\mathrm {G}({\mathbb {A}^{\infty }})\) on the collections \(\{ {\mathsf {M}}^{\text {tor}}_{{\mathcal {H}}, \varSigma } \}_{({\mathcal {H}}, \varSigma )}\) (see proposition 1.3.1.15 of [44]) and \(\{ X_{n,U,\varDelta } \}_{(U,\varDelta )}\).
The toroidal compactification \({\mathsf {M}}^{\text {tor}}_{{\mathcal {H}}, \varSigma }\) admits a stratification by locally closed subschemes
(see theorem 1.3.1.3(2) of [44]) indexed by equivalence classes \([(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}, \sigma )]\) as in definition 1.2.2.10 of [44]. Without repeating the definition in detail, let us just note that the equivalence classes \([(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}, \sigma )]\) with the same underlying cusp label \([(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]\) can be identified with the \(\varGamma _{\varPhi _{\mathcal {H}}}\)-orbits of the cones \(\sigma \in \varSigma _{\varPhi _{\mathcal {H}}}\) such that \(\sigma \cap \mathbf {P}_{\varPhi _{\mathcal {H}}}^+ \ne \emptyset \). Each stratum \(\mathsf {Z}_{[(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}, \sigma )]}\) is canonically isomorphic to \(\varXi _{\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}, \sigma }\), and the formal completion of \({\mathsf {M}}^{\text {tor}}_{{\mathcal {H}}, \varSigma }\) along the union of the strata \(\mathsf {Z}_{[(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}, \sigma )]}\) labelled by equivalence classes \([(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}, \sigma )]\) with the same underlying cusp label \([(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]\) is canonically isomorphic to \(\mathfrak {X}_{\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}} / \varGamma _{\varPhi _{\mathcal {H}}}\).
When \(U={\mathcal {H}}\), and when \((U,\varDelta ) \in {\mathcal {J}}_n^{{\text {tor}}}\) is induced by \(\varSigma \) as above, the formal completion of \({\mathsf {M}}^{\text {tor}}_{{\mathcal {H}}, \varSigma }\) along the union of all strata \(\mathsf {Z}_{[(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}, \sigma )]}\) labelled by equivalence classes \([(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}, \sigma )]\) with underlying \({\texttt {Z}}_{\mathcal {H}}\) equal to \({{\texttt {Z}}}^{(i, g)}_{\mathcal {H}}\) for some \(g \in \mathrm {G}({\mathbb {A}^{\infty }})\) is
where the indices g and h run over \(P_{n,(i)}^+({\mathbb {A}}^\infty ) \backslash G_n({\mathbb {A}}^\infty ) / U\) and
respectively, in this paper (see Sect. 5.3).
These identifications are all Hecke equivariant (see proposition 1.3.2.45 of [44]).
1.6 Kuga families in characteristic zero
Consider any \(\mathcal {O}\)-lattice \(Q\). Define \(\widehat{\mathrm {G}}\) to be the subgroup of automorphisms of \(L \oplus Q\) which preserve L, act trivially on the quotient \((L \oplus Q)/L\) and preserve, up to scalar multiples, the pairing \(\langle {\,\cdot \,,\,\cdot \,}\rangle \) on L. Restriction to L gives a surjective homomorphism \(\widehat{\mathrm {G}} \rightarrow \mathrm {G}\), and we denote the kernel \(\widehat{\mathrm {U}}\). This homomorphism is naturally split. (Compare with definition 1.2.4.3 of [44].) If \(Q={\mathcal {O}}_F^m\), then \(\widehat{\mathrm {G}}=G_n^{(m)}\) and \(\widehat{\mathrm {U}}={{\text {Hom}}}_n^{(m)}\). If \(\widehat{{\mathcal {H}}}\) is any open compact subgroup of \(\widehat{\mathrm {G}}({\hat{\mathbb {Z}}})\), then \(\widehat{{\mathcal {H}}}_{\widehat{\mathrm {U}}}\) denotes \(\widehat{{\mathcal {H}}} \cap \widehat{\mathrm {U}}({\hat{\mathbb {Z}}})\), and \(\widehat{{\mathcal {H}}}_{\mathrm {G}}\) denotes \(\widehat{{\mathcal {H}}}/\widehat{{\mathcal {H}}}_{\widehat{\mathrm {U}}}\).
To each neat open compact subgroup \(\widehat{{\mathcal {H}}}\) of \(\widehat{\mathrm {G}}({\hat{\mathbb {Z}}})\) with image \({\mathcal {H}}= \widehat{{\mathcal {H}}}_\mathrm {G}\) in \(\mathrm {G}({\hat{\mathbb {Z}}})\), [44] attached a generalized Kuga family \(\mathsf {N}\rightarrow \mathsf {M}_{\mathcal {H}}\). (See definition 1.3.3.4 of [44].) If \(Q={\mathcal {O}}_F^m\) and \(U=\widehat{{\mathcal {H}}}\), then it is the scheme denoted \(A^{(m)}_{n,U}\) in this paper. The generalized Kuga family attached to \(\widehat{{\mathcal {H}}}_{\mathrm {G}} \ltimes \widehat{{\mathcal {H}}}_{\widehat{\mathrm {U}}}\) is denoted \(\mathsf {N}^{\text {grp}}\rightarrow \mathsf {M}_{\mathcal {H}}\), and is a Kuga family as in definition 1.3.3.3 of [44]. It is an abelian scheme \({\mathbb {Q}}^\times \)-isogenous to the m-fold fibre product of the universal abelian scheme over \(\mathsf {M}_{\mathcal {H}}\) when \(Q\cong {\mathcal {O}}_F^m\). The generalized Kuga family \(\mathsf {N}\rightarrow \mathsf {M}_{\mathcal {H}}\) is a torsor for \(\mathsf {N}^{\text {grp}}\rightarrow \mathsf {M}_{\mathcal {H}}\).
To study these schemes and their compactifications [44] realizes them, in a non-canonical way, inside the boundary of a larger Shimura variety. Concretely, as in section 1.2.4 of [44], to define such generalized Kuga families, we start with an \(\mathcal {O}\)-lattice \(Q\); consider
(where \({\text {Diff}}_{\mathcal {O}/\mathbb {Z}}^{-1}\) denotes the inverse different), with the natural perfect pairing
induced by the trace pairing; and set
with the (self-dual) pairing
defined by
for \(x_{-2}, y_{-2} \in Q_{-2}\), \(x_{-1}, y_{-1} \in L\), and \(x_0, y_0 \in Q_0\). We shall fix the choice of
in what follows, since this is all we need in this paper. Then the above \(\widetilde{L}\) and \(\langle {\,\cdot \,,\,\cdot \,}\rangle \widetilde{~}\) can be identified with the \(\varLambda _{m+n}\) and \(\langle \,\,\, ,\,\,\, \rangle _{m+n}: \varLambda _{m+n} \times \varLambda _{m+n} \rightarrow {\mathbb {Z}}\) in this paper (see Sects. 1.1, B.1), up to reversing the ordering of the coordinates of \(Q_{-2} = ({\text {Diff}}_{\mathcal {O}_F/\mathbb {Z}}^{-1}({1}))^m\) and dividing by \(2 \pi \sqrt{-1}\). Then there is a natural choice of \(\widetilde{h}_0: \mathbb {C}\rightarrow {\text {End}}_{\mathcal {O}\otimes _{\mathbb {Z}} {\mathbb {R}}}(\widetilde{L}\otimes _{\mathbb {Z}} {\mathbb {R}})\) extending \(h_0: \mathbb {C}\rightarrow {\text {End}}_{\mathcal {O}\otimes _{\mathbb {Z}} {\mathbb {R}}}(L\otimes _{\mathbb {Z}} {\mathbb {R}})\) (see section 1.2.4 of [44]) which makes
an integral PEL datum as in definition 1.1.1.1 of [44], which defines a group functor \(\widetilde{\mathrm {G}}\) and a PEL moduli problem \(\widetilde{\mathsf {M}}_{\widetilde{{\mathcal {H}}}}\) for each open compact subgroup \(\widetilde{{\mathcal {H}}}\) of \(\widetilde{\mathrm {G}}({\hat{\mathbb {Z}}})\) as in the case of \(\mathrm {G}\) and \(\mathsf {M}_{\mathcal {H}}\) in Sect. B.1. We shall always denote the analogues of objects for \((\mathcal {O}, \star , \widetilde{L}, \langle {\,\cdot \,,\,\cdot \,}\rangle \widetilde{~}, \widetilde{h}_0)\) with a tilde \(\widetilde{~}\), without explicitly introducing them.
By the definition of \((\widetilde{L}, \langle {\,\cdot \,,\,\cdot \,}\rangle \widetilde{~})\), there is a fully symplectic admissible filtration \(\widetilde{{\texttt {Z}}}\) on \(\widetilde{L}\otimes _{\mathbb {Z}} {{\hat{\mathbb {Z}}}}\) induced by
Let \(\widetilde{X} := {\text {Hom}}_\mathcal {O}(Q_{-2}, {\text {Diff}}_{\mathcal {O}/\mathbb {Z}}^{-1}({1}))\) and \(\widetilde{Y} := Q_0\). The pairing \(\langle {\,\cdot \,, \,\cdot \,}\rangle _Q: Q_{-2} \times Q_0 \rightarrow \mathbb {Z}({1})\) induces a canonical isomorphism \(\widetilde{\phi }: \widetilde{Y} \mathop {\rightarrow }\limits ^{\sim }\widetilde{X}\), and there are canonical isomorphisms \(\widetilde{\varphi }_{-2}: {\text {Gr}}^{\widetilde{{\texttt {Z}}}}_{-2} \mathop {\rightarrow }\limits ^{\sim }{\text {Hom}}_{\hat{\mathbb {Z}}}(\widetilde{X}\otimes _{\mathbb {Z}} {{\hat{\mathbb {Z}}}}, {\hat{\mathbb {Z}}}({1}))\) and \(\widetilde{\varphi }_0: {\text {Gr}}^{\widetilde{{\texttt {Z}}}}_0 \mathop {\rightarrow }\limits ^{\sim }\widetilde{Y}\otimes _{\mathbb {Z}} {{\hat{\mathbb {Z}}}}\) (of \(\mathcal {O}\otimes _{\mathbb {Z}} {{\hat{\mathbb {Z}}}}\)-modules). These data define a torus argument
for \(\widetilde{{\texttt {Z}}}\) (see definition 5.4.1.3 in [41] or definition 1.2.1.5 in [44]). Let \(\widetilde{\delta }\) be the obvious splitting of \(\widetilde{{\texttt {Z}}}\) induced by the equality \(Q_{-2} \oplus L\oplus Q_0 = \widetilde{L}\).
Then we can define algebraic groups \(\widetilde{\mathrm {P}}_{\widetilde{{\texttt {Z}}}}\), \(\widetilde{\mathrm {U}}_{\widetilde{{\texttt {Z}}}}\), \(\widetilde{\mathrm {U}}_{2, \widetilde{{\texttt {Z}}}}\), \(\widetilde{\mathrm {U}}_{1, \widetilde{{\texttt {Z}}}}\), \(\widetilde{\mathrm {G}}_{l, \widetilde{{\texttt {Z}}}}\), \(\widetilde{\mathrm {G}}_{l, \widetilde{{\texttt {Z}}}}'\), \(\widetilde{\mathrm {P}}_{\widetilde{{\texttt {Z}}}}'\), \(\widetilde{\mathrm {G}}_{1, \widetilde{{\texttt {Z}}}}\), \(\widetilde{\mathrm {G}}_{h, \widetilde{{\texttt {Z}}}}\) and \(\widetilde{\mathrm {G}}_{h, \widetilde{{\texttt {Z}}}}'\). By definition, we have canonical isomorphisms
and
For each open compact subgroup \(\widetilde{{\mathcal {H}}}\) of \(\widetilde{\mathrm {G}}({\hat{\mathbb {Z}}})\), we define
-
(1)
\(\widehat{{\mathcal {H}}} := \widetilde{{\mathcal {H}}}_{\widehat{\mathrm {G}}} := \widetilde{{\mathcal {H}}}_{\widetilde{\mathrm {G}}_{1, \widetilde{{\texttt {Z}}}}}\).
-
(2)
\(\widehat{{\mathcal {H}}}_{\widehat{\mathrm {U}}} := \widetilde{{\mathcal {H}}}_{\widehat{\mathrm {U}}} := \widetilde{{\mathcal {H}}}_{\widetilde{\mathrm {U}}_{1, \widetilde{{\texttt {Z}}}}}\).
-
(3)
\(\widehat{{\mathcal {H}}}_\mathrm {G}:= \widehat{{\mathcal {H}}} / \widehat{{\mathcal {H}}}_{\widehat{\mathrm {U}}} \cong \widetilde{{\mathcal {H}}}_{\widetilde{\mathrm {G}}_{h, \widetilde{{\texttt {Z}}}}'}\).
(See definition 1.2.4.4 of [44].)
Given any neat open compact \(\widehat{{\mathcal {H}}} \subset \widehat{\mathrm {G}}({\hat{\mathbb {Z}}})\), we can always find some neat \(\widetilde{{\mathcal {H}}} \subset \widetilde{\mathrm {G}}({\hat{\mathbb {Z}}})\) such that \(\widehat{{\mathcal {H}}} = \widetilde{{\mathcal {H}}}_{\widehat{\mathrm {G}}}\) as above and
(cf. condition 1.2.4.7 of [44]). In this case the abelian scheme torsor
depends only on \(\widehat{{\mathcal {H}}}\) (but not on the auxiliary choice of \(\widetilde{{\mathcal {H}}}\)) and equals the above generalized Kuga family \(\mathsf {N}\rightarrow \mathsf {M}_{\mathcal {H}}\) attached to \(\widehat{{\mathcal {H}}}\).
We can explicitly compare some of the above groups with the related groups defined in our paper (see Sects. 1.1, 1.2), as follows:
-
(1)
\(\widetilde{\mathrm {G}}_{h, \widetilde{{\texttt {Z}}}}'(R) \cong \widetilde{\mathrm {G}}_{h, \widetilde{{\texttt {Z}}}}(R) = \mathrm {G}(R) = G_n(R)\).
-
(2)
\(\widetilde{\mathrm {G}}_{l,\widetilde{{\texttt {Z}}}}(R) \cong \widetilde{\mathrm {G}}_{l,\widetilde{{\texttt {Z}}}}'(R) \cong {\text {GL}}_m(\mathcal {O}_F\otimes _{\mathbb {Z}} {R})\).
-
(3)
\(\widetilde{\mathrm {P}}_{\widetilde{{\texttt {Z}}}}(R) \cong {\text {GL}}_m(\mathcal {O}_F\otimes _{\mathbb {Z}} {R}) \ltimes \widetilde{{G}}_n^{(m)}(R) \cong ({\text {GL}}_m(\mathcal {O}_F\otimes _{\mathbb {Z}} {R}) \times G_n(R)) \ltimes N_n^{(m)}(R)\).
-
(4)
\(\widetilde{\mathrm {U}}_{\widetilde{{\texttt {Z}}}}(R) = N_n^{(m)}(R)\).
-
(5)
\(\widetilde{\mathrm {U}}_{2, \widetilde{{\texttt {Z}}}}(R) = Z(N_n^{(m)})(R) = {{\text {Herm}}}^{(m)}(R)\).
-
(6)
\(\widehat{\mathrm {U}}(R) = \widetilde{\mathrm {U}}_{1, \widetilde{{\texttt {Z}}}}(R) = {{\text {Hom}}}_n^{(m)}(R)\).
-
(7)
\(\widetilde{\mathrm {P}}_{\widetilde{{\texttt {Z}}}}'(R) = \widetilde{{G}}_n^{(m)}(R) = G_n(R) \ltimes N_n^{(m)}(R)\).
-
(8)
\(\widehat{\mathrm {G}}(R) = \widetilde{\mathrm {G}}_{1, \widetilde{{\texttt {Z}}}}(R) \cong G_n^{(m)}(R) = G_n(R) \ltimes {{\text {Hom}}}_n^{(m)}(R)\).
-
(9)
\(\widetilde{\mathrm {P}}_{\widetilde{{\texttt {Z}}}}(R) / \widetilde{\mathrm {U}}_{\widetilde{{\texttt {Z}}}}(R) \cong {\text {GL}}_m(\mathcal {O}_F\otimes _{\mathbb {Z}} {R}) \times G_n(R)\).
1.7 Toroidal compactifications of Kuga families in characteristic zero
Let us fix the choice of some \((\widetilde{{\texttt {Z}}}, \widetilde{\varPhi }, \widetilde{\delta })\) as above. Then there is a bijection between the fully symplectic admissible filtrations \(\breve{{\texttt {Z}}}\) of \(\widetilde{L}\otimes _{\mathbb {Z}} {{\hat{\mathbb {Z}}}}\) such that
and the fully symplectic admissible filtrations \({\texttt {Z}}\) of \(L\otimes _{\mathbb {Z}} {{\hat{\mathbb {Z}}}}\) such that \({\texttt {Z}}_{-2} = \breve{{\texttt {Z}}}_{-2}/\widetilde{{\texttt {Z}}}_{-2}\). (The notation \(\breve{~}\) will always mean some objects related to such a filtration \(\breve{{\texttt {Z}}}\).) When \({\texttt {Z}}= {{\texttt {Z}}}^{(i, 1)}\), we have \(\breve{{\texttt {Z}}} = {\widetilde{{\texttt {Z}}}}^{(i+m, 1)}\).
For each \(\breve{{\texttt {Z}}}\) as above, and for each \({\hat{\mathbb {Z}}}\)-algebra R, we define the following quotients of subgroups of \(\widehat{\mathrm {G}}(R)\) (see definition 1.2.4.53 of [44]):
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(1)
\(\widehat{\mathrm {P}}_{\breve{{\texttt {Z}}}}(R) := (\widetilde{\mathrm {P}}_{\breve{{\texttt {Z}}}}(R) \cap \widetilde{\mathrm {P}}_{\widetilde{{\texttt {Z}}}}'(R)) / \widetilde{\mathrm {U}}_{2, \widetilde{{\texttt {Z}}}}(R)\), so that \(\widehat{\mathrm {P}}_{{\widetilde{{\texttt {Z}}}}^{(i+m, 1)}}(R) = P_{n,(i)}^{(m),+}(R)\).
-
(2)
\(\widehat{\mathrm {P}}_{\breve{{\texttt {Z}}}}'(R) := \widetilde{\mathrm {P}}_{\breve{{\texttt {Z}}}}'(R) / \widetilde{\mathrm {U}}_{2, \widetilde{{\texttt {Z}}}}(R)\), so that \(\widehat{\mathrm {P}}_{{\widetilde{{\texttt {Z}}}}^{(i+m, 1)}}'(R) = P_{n,(i)}^{(m)}(R)\).
-
(3)
\(\widehat{\mathrm {U}}_{\breve{{\texttt {Z}}}}(R) := \widetilde{\mathrm {U}}_{\breve{{\texttt {Z}}}}(R) / \widetilde{\mathrm {U}}_{2, \widetilde{{\texttt {Z}}}}(R)\), so that \(\widehat{\mathrm {U}}_{{\widetilde{{\texttt {Z}}}}^{(i+m, 1)}}(R)=N^{(m)}_{n,(i)}(R)\).
-
(4)
\(\widehat{\mathrm {U}}_{2, \breve{{\texttt {Z}}}}(R) := \widetilde{\mathrm {U}}_{2, \breve{{\texttt {Z}}}}(R) / \widetilde{\mathrm {U}}_{2, \widetilde{{\texttt {Z}}}}(R)\), so that
$$\begin{aligned} \widehat{\mathrm {U}}_{2, {\widetilde{{\texttt {Z}}}}^{(i+m, 1)}}(R)=Z(N_{n,(i)}^{(m)})(R)\cong {{\text {Herm}}}^{(i+m)}(R)/{{\text {Herm}}}^{(m)}(R). \end{aligned}$$ -
(5)
\(\widehat{\mathrm {U}}_{1, \breve{{\texttt {Z}}}}(R) := \widehat{\mathrm {U}}_{\breve{{\texttt {Z}}}}(R) / \widehat{\mathrm {U}}_{2, \breve{{\texttt {Z}}}}(R) \cong \widetilde{\mathrm {U}}_{1, \breve{{\texttt {Z}}}}(R)\), so that
$$\begin{aligned} \widehat{\mathrm {U}}_{1, {\widetilde{{\texttt {Z}}}}^{(i+m, 1)}}(R) = N_{n,(i)}^{(m)}(R)\big /Z(N_{n,(i)}^{(m)})(R) \cong {{\text {Hom}}}_{n-i}^{(i+m)}(R). \end{aligned}$$ -
(6)
\(\widehat{\mathrm {G}}_{h, \breve{{\texttt {Z}}}}(R) := \widetilde{\mathrm {G}}_{h, \breve{{\texttt {Z}}}}(R)\) and \(\widehat{\mathrm {G}}_{h, \breve{{\texttt {Z}}}}'(R) := \widetilde{\mathrm {G}}_{h, \breve{{\texttt {Z}}}}'(R)\), so that \(\widehat{\mathrm {G}}_{h, \breve{{\texttt {Z}}}}(R) \cong \widehat{\mathrm {G}}_{h, \breve{{\texttt {Z}}}}'(R) \cong \mathrm {G}_{h, {{\texttt {Z}}}}(R)\), and \(\widehat{\mathrm {G}}_{h, {\widetilde{{\texttt {Z}}}}^{(i+m, 1)}}(R) = L_{n,(i),{{\text {herm}}}}(R)\).
-
(7)
\(\widehat{\mathrm {G}}_{l, \breve{{\texttt {Z}}}}(R) := ( \widetilde{\mathrm {Z}}_{\breve{{\texttt {Z}}}}(R) \cap \widetilde{\mathrm {P}}_{\widetilde{{\texttt {Z}}}}'(R) ) / \widetilde{\mathrm {U}}_{\breve{{\texttt {Z}}}}(R)\) and \(\widehat{\mathrm {G}}_{l, \breve{{\texttt {Z}}}}'(R) := \widehat{\mathrm {P}}_{\breve{{\texttt {Z}}}}(R) / \widehat{\mathrm {P}}_{\breve{{\texttt {Z}}}}'(R)\), so that \(\widehat{\mathrm {G}}_{l, \breve{{\texttt {Z}}}}(R) \mathop {\rightarrow }\limits ^{\sim }\widehat{\mathrm {G}}_{l, \breve{{\texttt {Z}}}}'(R)\) and \(\widehat{\mathrm {G}}_{l, {\widetilde{{\texttt {Z}}}}^{(i+m, 1)}}'(R) \cong L_{n,(i),{{\text {lin}}}}^{(m)}(R)\).
-
(8)
\(\widehat{\mathrm {G}}_{1, \breve{{\texttt {Z}}}}(R) := \widehat{\mathrm {P}}_{\breve{{\texttt {Z}}}}'(R) / \widehat{\mathrm {U}}_{2, \breve{{\texttt {Z}}}}(R) \cong \widetilde{\mathrm {G}}_{1, \breve{{\texttt {Z}}}}(R)\), so that \(\widehat{\mathrm {G}}_{1, {\widetilde{{\texttt {Z}}}}^{(i+m, 1)}}(R) \cong G_{n-i}^{(i+m)}(R)\).
Hence it makes sense to define \(\widehat{{\mathcal {H}}}_{\widehat{\mathrm {P}}_{\breve{{\texttt {Z}}}}} := ( \widetilde{{\mathcal {H}}}_{\widetilde{\mathrm {P}}_{\breve{{\texttt {Z}}}}} \cap \widetilde{{\mathcal {H}}}_{\widetilde{\mathrm {P}}_{\widetilde{{\texttt {Z}}}}'} ) / \widetilde{{\mathcal {H}}}_{\widetilde{\mathrm {U}}_{2, \widetilde{{\texttt {Z}}}}}\), etc., when \(\widehat{{\mathcal {H}}} = \widetilde{{\mathcal {H}}}_{\widehat{\mathrm {G}}}\), so that we have \(\widehat{{\mathcal {H}}}_{\widehat{\mathrm {P}}_{\breve{{\texttt {Z}}}}} \rightarrow {\mathcal {H}}_{\mathrm {P}_{\texttt {Z}}}\), etc.
If \(\widetilde{g} \in \widetilde{\mathrm {P}}_{\widetilde{{\texttt {Z}}}}'({\mathbb {A}^{\infty }})\), then \({\widetilde{{\texttt {Z}}}}^{(i+m, \widetilde{g})}\) depends only on the image \(\widehat{g}\) of \(\widetilde{g}\) in \(\widehat{\mathrm {G}}({\hat{\mathbb {Z}}})\), so we will denote it \({\breve{{\texttt {Z}}}}^{(i, \widehat{g})}\). This sets up a bijection between the \(\widehat{{\mathcal {H}}}\)-orbits \({\widetilde{{\texttt {Z}}}}^{(i+m, \widetilde{g})}_{\widehat{{\mathcal {H}}}} = {\breve{{\texttt {Z}}}}^{(i, \widehat{g})}_{\widehat{{\mathcal {H}}}}\) of \({\widetilde{{\texttt {Z}}}}^{(i+m, \widetilde{g})}={\breve{{\texttt {Z}}}}^{(i, \widehat{g})}\) and the double coset space
which equals
in this paper (see, for example, Sect. 1.4), with \(U=\widehat{{\mathcal {H}}}\).
By taking graded pieces with respect to filtrations induced by \(0 \subset \widetilde{{\texttt {Z}}}_{-2} \subset \breve{{\texttt {Z}}}_{-2} \subset \breve{{\texttt {Z}}}_{-1} \subset \widetilde{{\texttt {Z}}}_{-1} \subset \widetilde{L}\otimes _{\mathbb {Z}} {{\hat{\mathbb {Z}}}}\), there is also a bijection between the torus arguments \(\breve{\varPhi } = (\breve{X}, \breve{Y}, \breve{\phi }, \breve{\varphi }_{-2}, \breve{\varphi }_0)\) for \(\breve{{\texttt {Z}}}\) which induce \(\widetilde{\varPhi }\) and the torus arguments \(\varPhi = (X, Y, \phi , \varphi _{-2}, \varphi _0)\) for the filtration \({\texttt {Z}}\) corresponding to \(\breve{{\texttt {Z}}}\). If we set \({\mathcal {H}}= \widehat{{\mathcal {H}}}_\mathrm {G}\), then this bijection is compatible with the formation of the \(\widehat{{\mathcal {H}}}\)-orbits of \(\breve{\varPhi }\) and the \({\mathcal {H}}\)-orbits of the corresponding \(\varPhi \), which induces bijections among the following three sets (see lemmas 1.2.4.15 and 1.2.4.16 of [44]):
-
(1)
The cusp labels \([(\breve{{\texttt {Z}}}_{\widetilde{{\mathcal {H}}}}, \breve{\varPhi }_{\widetilde{{\mathcal {H}}}}, \breve{\delta }_{\widetilde{{\mathcal {H}}}})]\) for \(\widetilde{\mathsf {M}}_{\widetilde{{\mathcal {H}}}}\) such that the stratum \(\widetilde{\mathsf {Z}}_{[(\breve{\varPhi }_{\widetilde{{\mathcal {H}}}}, \breve{\delta }_{\widetilde{{\mathcal {H}}}})]}\) of \({\widetilde{\mathsf {M}}}^{\text {min}}_{\widetilde{{\mathcal {H}}}}\) is contained in the closure of \(\widetilde{\mathsf {Z}}_{[(\widetilde{\varPhi }_{\widetilde{{\mathcal {H}}}}, \widetilde{\delta }_{\widetilde{{\mathcal {H}}}})]}\).
-
(2)
The \(\widehat{{\mathcal {H}}}\)-orbits of equivalence classes of \((\breve{{\texttt {Z}}}, \breve{\varPhi }, \breve{\delta })\), where \(\breve{{\texttt {Z}}}\) and \(\breve{\varPhi }\) are compatible with \(\widetilde{{\texttt {Z}}}\) and \(\widetilde{\varPhi }\) in the sense that \(0 \subset \widetilde{{\texttt {Z}}}_{-2} \subset \breve{{\texttt {Z}}}_{-2} \subset \breve{{\texttt {Z}}}_{-1} \subset \widetilde{{\texttt {Z}}}_{-1} \subset \widetilde{L}\otimes _{\mathbb {Z}} {{\hat{\mathbb {Z}}}}\) and that \(\breve{\varPhi }\) induces \(\widetilde{\varPhi }\) as above, which we denote by \([(\breve{{\texttt {Z}}}_{\widehat{{\mathcal {H}}}}, \breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}})]\) and call it a cusp label (at level \(\widehat{{\mathcal {H}}}\)) for \((\widetilde{L}, \langle {\,\cdot \,,\,\cdot \,}\rangle \widetilde{~}, \widetilde{h}_0, \widetilde{{\texttt {Z}}})\) (see definition 1.2.4.17 of [44]).
-
(3)
The cusp labels \([({\texttt {Z}}_{\mathcal {H}}, \varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]\) for \(\mathsf {M}_{\mathcal {H}}\).
The stratum \(\widetilde{\mathsf {Z}}_{[(\breve{\varPhi }_{\widetilde{{\mathcal {H}}}}, \breve{\delta }_{\widetilde{{\mathcal {H}}}})]} \cong \widetilde{\mathsf {M}}_{\widetilde{{\mathcal {H}}}}^{\widetilde{{\texttt {Z}}}_{\widetilde{{\mathcal {H}}}}}\), the finite étale cover \(\widetilde{\mathsf {M}}_{\widetilde{{\mathcal {H}}}}^{\widetilde{\varPhi }_{\widetilde{{\mathcal {H}}}}} \rightarrow \widetilde{\mathsf {M}}_{\widetilde{{\mathcal {H}}}}^{\widetilde{{\texttt {Z}}}_{\widetilde{{\mathcal {H}}}}}\), the abelian scheme torsor \(\widetilde{C}_{\widetilde{\varPhi }_{\widetilde{{\mathcal {H}}}}, \widetilde{\delta }_{\widetilde{{\mathcal {H}}}}} \rightarrow \widetilde{\mathsf {M}}_{\widetilde{{\mathcal {H}}}}^{\widetilde{\varPhi }_{\widetilde{{\mathcal {H}}}}}\) and the abelian scheme \(\widetilde{C}_{\widetilde{\varPhi }_{\widetilde{{\mathcal {H}}}}, \widetilde{\delta }_{\widetilde{{\mathcal {H}}}}}^{\text {grp}}\rightarrow \widetilde{\mathsf {M}}_{\widetilde{{\mathcal {H}}}}^{\widetilde{\varPhi }_{\widetilde{{\mathcal {H}}}}}\) depend (up to canonical isomorphism) only on the \(\widehat{{\mathcal {H}}}\)-orbit of \((\breve{{\texttt {Z}}}, \breve{\varPhi }, \breve{\delta })\) (see lemma 1.3.2.50 of [44]), and hence we shall denote them by
respectively. For a fixed \((\breve{{\texttt {Z}}}, \breve{\varPhi }, \breve{\delta })\), the morphisms
are equivariant with
(see propositions 1.3.2.24 and 1.3.2.55 of [44]).
For each \([(\breve{{\texttt {Z}}}, \breve{\varPhi }, \breve{\delta })]\) as above, and any \(\mathbb {Z}\)-algebra R, we define \(\widehat{\mathrm {G}}_{l, \breve{\varPhi }}'(R)\) to be the subgroup of \({\text {GL}}_{\mathcal {O}\otimes _{\mathbb {Z}} {R}}(\breve{Y} \otimes _{\mathbb {Z}} {R})\) of elements stabilizing the kernel of \(\breve{Y} \twoheadrightarrow \widetilde{Y}\) and inducing \({\text {Id}}_{\widetilde{Y}}\), which admits a canonical map to \(\widehat{\mathrm {G}}_{l, \breve{{\texttt {Z}}}}'(R \otimes _{\mathbb {Z}} {{\hat{\mathbb {Z}}}})\) induced by \(\breve{\varPhi }\). When R is a \({\hat{\mathbb {Z}}}\)-algebra, we have \(\widehat{\mathrm {G}}_{l, \breve{\varPhi }}'(R) \cong \widehat{\mathrm {G}}_{l, \breve{{\texttt {Z}}}}'(R)\). When \(R=\mathbb {Q}\), we have \(\widehat{\mathrm {G}}_{l, \breve{\varPhi }}'(\mathbb {Q}) \hookrightarrow \widehat{\mathrm {G}}_{l, \breve{{\texttt {Z}}}}'({\mathbb {A}^{\infty }})\), and we define
a congruence subgroup of \(\widehat{\mathrm {G}}_{l, \breve{\varPhi }}'(\mathbb {Z}) = \widehat{\mathrm {G}}_{l, \breve{{\texttt {Z}}}}'({\hat{\mathbb {Z}}}) \cap \widehat{\mathrm {G}}_{l, \breve{\varPhi }}'(\mathbb {Q})\) depending only on \(\breve{\varPhi }_{\widehat{{\mathcal {H}}}}\) (see definition 1.2.4.21 of [44] for an equivalent definition).
If \(\widetilde{g} \in \widetilde{\mathrm {P}}_{\widetilde{{\texttt {Z}}}}'({\mathbb {A}^{\infty }})\), then the equivalence class of \(({\widetilde{{\texttt {Z}}}}^{(i+m, \widetilde{g})}, {\widetilde{\varPhi }}^{(i+m, \widetilde{g})}, \breve{\delta })\) depends only on the image \(\widehat{g}\) of \(\widetilde{g}\) in \(\widehat{\mathrm {G}}({\mathbb {A}^{\infty }})\). We will denote it \(({\breve{{\texttt {Z}}}}^{(i, \widehat{g})}, {\breve{\varPhi }}^{(i, \widehat{g})}, \breve{\delta })\). The map
sets up a bijection between the double coset space
and the set of cusp labels \([(\breve{{\texttt {Z}}}_{\widehat{{\mathcal {H}}}}, \breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}})]\) such that \(\breve{{\texttt {Z}}}_{\widehat{{\mathcal {H}}}} = {\breve{{\texttt {Z}}}}^{(i, \widehat{g})}_{\widehat{{\mathcal {H}}}}\) for some \(\widehat{g} \in \widehat{\mathrm {G}}({\mathbb {A}^{\infty }})\). The forgetful map sending \([(\breve{{\texttt {Z}}}_{\widehat{{\mathcal {H}}}}, \breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}})]\) to \(\breve{{\texttt {Z}}}_{\widehat{{\mathcal {H}}}}\) can be identified with the canonical map
whose fibre above the double coset of \(\widehat{g} \in \widehat{\mathrm {G}}({\mathbb {A}^{\infty }})\) can be identified with
This last double coset space is the index set of
in this paper (see Sect. 4.1), with \(g=\widehat{g}\) and \(U=\widehat{{\mathcal {H}}}\).
If \(U=\widehat{{\mathcal {H}}}\) is any neat open compact subgroup of \(G_n^{(m)}({\mathbb {A}}^\infty )=\widehat{\mathrm {G}}({\mathbb {A}^{\infty }})\), if \(g=\widehat{g} \in G_n^{(m)}({\mathbb {A}}^\infty )=\widehat{\mathrm {G}}({\mathbb {A}^{\infty }})\) and if \(U_g'=gUg^{-1} \cap P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )\), then the maps
in this paper (see Sects. 4.1, 4.2) are identified with
For each representative \((\breve{{\texttt {Z}}}_{\widehat{{\mathcal {H}}}}, \breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}})\) of a cusp label \([(\breve{{\texttt {Z}}}_{\widehat{{\mathcal {H}}}}, \breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}})]\) as above, there is a torsor
under the pull-back of a split torus
over \({{\text {Spec}}\,}\mathbb {Z}\) with character group
(which only depends on \(\widehat{{\mathcal {H}}}_{\widehat{\mathrm {P}}_{\breve{{\texttt {Z}}}}}\); see definition 1.2.4.29 and proposition 1.3.2.56 of [44]). We obtain an isomorphic torus torsor if we replace \((\breve{{\texttt {Z}}}_{\widehat{{\mathcal {H}}}}, \breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}})\) with another representative, but its universal property depends on this choice of representative.
For a fixed \((\breve{{\texttt {Z}}}, \breve{\varPhi }, \breve{\delta })\), the collection \(\{ \widehat{\varXi }_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}}} \}_{\widehat{{\mathcal {H}}}_{\widehat{\mathrm {P}}_{\breve{{\texttt {Z}}}}'}}\) admits an action of \(\widehat{\mathrm {P}}_{\breve{{\texttt {Z}}}}'({\mathbb {A}^{\infty }})\) such that the morphisms
are equivariant with
(see proposition 1.3.2.67 of [44]).
When \({\mathcal {H}}_{\widehat{\mathrm {U}}_{2, \breve{{\texttt {Z}}}}} = \widehat{\mathrm {G}}({\hat{\mathbb {Z}}})_{\widehat{\mathrm {U}}_{2, \breve{{\texttt {Z}}}}} = \widehat{\mathrm {U}}_{2, \breve{{\texttt {Z}}}}({\hat{\mathbb {Z}}})\) and when \(\breve{\varPhi }_{\widehat{{\mathcal {H}}}}\) is represented by some \(\breve{\varPhi } = (\breve{X}, \breve{Y}, \breve{\phi }, \breve{\varphi }_{-2}, \breve{\varphi }_0)\) (where \(\breve{\phi }: \breve{Y} \hookrightarrow \breve{X}\) must be an isomorphism, as explained above), which admits a surjection \(\breve{Y} \twoheadrightarrow \widetilde{Y}\) with kernel \(Y\) an \(\mathcal {O}\)-lattice, the group \(\widehat{\mathbf {S}}_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}\) is the group \(\ker ( S(\breve{Y})^{{\text {TF}}}\rightarrow S(\widetilde{Y})^{{\text {TF}}})\) in the notation of this paper (see Sect. 1.1). For more general \(\widehat{{\mathcal {H}}}\), we write \(\widehat{\mathbf {S}}_{\breve{\varPhi }_1} = \ker ( S(\breve{Y})^{{\text {TF}}}\rightarrow S(\widetilde{Y})^{{\text {TF}}})\), and \(\widehat{\mathbf {S}}_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}\) is the unique lattice in \(\widehat{\mathbf {S}}_{\breve{\varPhi }_1} \otimes _{\mathbb {Z}} {\mathbb {Q}}\) such that \(\widehat{\mathbf {S}}_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}} / \widehat{\mathbf {S}}_{\breve{\varPhi }_1} \cong \widehat{\mathbf {S}}_{\breve{\varPhi }_1}^\vee / \widehat{\mathbf {S}}_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}^\vee \cong \widehat{\mathrm {U}}_{2, \breve{{\texttt {Z}}}}({\hat{\mathbb {Z}}}) / \widehat{{\mathcal {H}}}_{\widehat{\mathrm {U}}_{2, \breve{{\texttt {Z}}}}}\) (see proposition 1.3.2.56 of [44]). Then \((\widehat{\mathbf {S}}_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}})_\mathbb {R}^\vee \) is a quotient of the space of hermitian forms over \(\breve{Y}\otimes _{\mathbb {Z}} {\mathbb {R}}\), which also admits a projection to the space of hermitian forms over \(Y\otimes _{\mathbb {Z}} {\mathbb {R}}\). We define \(\widehat{\mathbf {P}}_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}\) (resp. \(\widehat{\mathbf {P}}_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}^+\)) to be the subset of \((\widehat{\mathbf {S}}_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}})_\mathbb {R}^\vee \) consisting of images of positive semi-definite hermitian forms with rational radicals (resp. positive definite hermitian forms) over \(\breve{Y}\otimes _{\mathbb {Z}} {\mathbb {R}}\) (see (1.2.4.33) and (1.2.4.34) of [44]), which can be identified with the subset consisting of pre-images of positive semi-definite hermitian forms with rational radicals (resp. positive definite hermitian forms) over \(Y\otimes _{\mathbb {Z}} {\mathbb {R}}\).
The torus
in this paper (see Sect. 4.3) with \(U={\mathcal {H}}\) is identified with
Moreover, the sheaves \(X^*(S^{(m),+}_{n,(i),U_g'})\), \(X_*(S^{(m),+}_{n,(i),U_g'})_{\mathbb {R}}^{\succ 0}\), and \(X_*(S^{(m),+}_{n,(i),U_g'})_{\mathbb {R}}^{>0}\) are identified with
respectively. The \(S^{(m),+}_{n,(i),U_g'}\)-torsor
is identified with
These maps are \(\widehat{\mathrm {P}}_{{\breve{{\texttt {Z}}}}^{(i, g)}}({\mathbb {A}^{\infty }})\)-equivariant (see proposition 1.3.2.67 of [44]).
Consider any compatible collection
of admissible projective smooth rational polyhedral cone decompositions, where each \(\widehat{\varSigma }_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}\) is a \(\varGamma _{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}\)-admissible projective smooth rational polyhedral cone decomposition of \(\widehat{\mathbf {P}}_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}\), as in lemma 1.2.4.42 of [44]. (We caution the reader that the definition there is rather ad hoc.) The set of pairs \(\kappa = (\widehat{{\mathcal {H}}}, \widehat{\varSigma })\) with \(\widehat{{\mathcal {H}}}_\mathrm {G}\subset {\mathcal {H}}\) is denoted \(\mathbf {K}_{Q, {\mathcal {H}}}^{++}\); the subset of \(\mathbf {K}_{Q, {\mathcal {H}}}^{++}\) consisting of \(\kappa = (\widehat{{\mathcal {H}}}, \widehat{\varSigma })\) with \(\widehat{{\mathcal {H}}}_\mathrm {G}= {\mathcal {H}}\) is denoted \(\mathbf {K}_{Q, {\mathcal {H}}}^+\); and the subset of \(\mathbf {K}_{Q, {\mathcal {H}}}^+\) consisting of \(\kappa = (\widehat{{\mathcal {H}}}, \widehat{\varSigma })\) with \(\widehat{{\mathcal {H}}}_\mathrm {G}= {\mathcal {H}}\) and \(\widehat{{\mathcal {H}}} = {\mathcal {H}}\ltimes \widehat{{\mathcal {H}}}_{\widehat{\mathrm {U}}}\) is denoted \(\mathbf {K}_{Q, {\mathcal {H}}}\) (see definitions 1.2.4.11 and 1.2.4.44 of [44]). For any given compatible collection \(\varSigma \) of admissible projective smooth rational polyhedral cone decompositions for \(\mathsf {M}_{\mathcal {H}}\), and for \(? = \emptyset \), \(+\) or \(++\), the subset of \(\mathbf {K}_{Q, {\mathcal {H}}}^?\) consisting of \(\kappa = (\widehat{{\mathcal {H}}}, \widehat{\varSigma })\) such that \(\widehat{\varSigma }\) is compatible with \(\varSigma \) in the sense that each \(\widehat{\rho } \in \widehat{\varSigma }_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}\) is mapped into some \(\sigma \in \varSigma _{\varPhi _{\mathcal {H}}}\) (see condition 1.2.4.49 and definition 1.2.4.50 of [44]) is denoted \(\mathbf {K}_{Q, {\mathcal {H}}, \varSigma }^?\). This notion of compatibility agrees with the one in this paper (see Sect. 5.2).
Each such \(\kappa = (\widehat{{\mathcal {H}}}, \widehat{\varSigma })\) in \(\mathbf {K}_{Q, {\mathcal {H}}}^{++}\) induces a pair
in \({\mathcal {J}}_n^{(m),{{\text {tor}}}}\) (with \(U=\widehat{{\mathcal {H}}}\)) in Sect. 5.2 of this paper, because, in order to define \((U,\varSigma )\) as in Sect. 5.2, it suffices to define the admissible cone decomposition \(\varSigma (g)_0\) for \(X_*(S_{n,(i),gUg^{-1}\cap P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )}^{(m),+})_{\mathbb {R}}^{\succ 0}\), for each \(g\in G_n^{(m)}({\mathbb {A}}^\infty )\), which can be taken to be the pull-back of the subcollection \(\{ \widehat{\varSigma }_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}} \}_{[(\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}})]}\) of \(\widehat{\varSigma }\) indexed by the cusp labels \([(\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}})]\) with underlying \(\breve{{\texttt {Z}}}_{\widehat{{\mathcal {H}}}}\) equal to \({\breve{{\texttt {Z}}}}^{(i, \widehat{g})}_{\widehat{{\mathcal {H}}}}\) for some \(\widehat{g}\) in \(\widehat{\mathrm {G}}({\mathbb {A}^{\infty }})\) corresponding to \(g \in G_n^{(m)}({\mathbb {A}}^\infty ) \cong \widehat{\mathrm {G}}({\mathbb {A}^{\infty }})\). In fact, \({\mathcal {J}}_n^{(m),{{\text {tor}}}}\) is exactly the set of such induced pairs (as \(U=\widehat{{\mathcal {H}}}\) varies). (It is hard to explicitly describe the set \({\mathcal {J}}_n^{(m),{{\text {tor}}}}\) when \(m>0\), because they are induced by auxiliary choices of compatible collections \(\widetilde{\varSigma }\) for \(\widetilde{\mathsf {M}}_{\widetilde{{\mathcal {H}}}}\). Nevertheless, this is unnecessary for our purposes.)
Each \(\widehat{\varSigma }_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}\) defines an affine toroidal embedding
over \(\widehat{C}_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}}}\). The formal completion of \(\overline{\widehat{\varXi }}_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}}, \widehat{\varSigma }_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}}\) along the union of the \(\widehat{\rho }\)-strata \(\widehat{\varXi }_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}}, \widehat{\rho }}\) for all \(\widehat{\rho } \in \widehat{\varSigma }_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}\) such that \(\widehat{\rho } \cap \widehat{\mathbf {P}}_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}^+ \ne \emptyset \) is denoted
(see (1.3.2.62), (1.3.2.63), (1.3.2.64) and (1.3.2.66) of [44]). The schemes
of this paper (see Sect. 4.4) are identified with
Moreover, \(T^{(m),+,\wedge }_{n,(i),U_g',\varSigma (g)_0}\) is identified with
and \(T^{(m),\natural ,\wedge }_{n,(i),U_g',\varSigma (g)_0}\) is identified with
where the second disjoint union is over cusp labels with underlying \(\breve{{\texttt {Z}}}_{\widehat{{\mathcal {H}}}}\) equal to \({\breve{{\texttt {Z}}}}^{(i, \widehat{g})}_{\widehat{{\mathcal {H}}}}\) for a fixed \(\widehat{g}\) in \(\widehat{\mathrm {G}}({\mathbb {A}^{\infty }})\) corresponding to \(g \in G_n^{(m)}({\mathbb {A}}^\infty ) \cong \widehat{\mathrm {G}}({\mathbb {A}^{\infty }})\). (Again \(\widehat{{\mathcal {H}}}=U\).)
For each \(\kappa = (\widehat{{\mathcal {H}}}, \widehat{\varSigma }) \in \mathbf {K}_{Q, {\mathcal {H}}}^{++}\), we have a smooth projective scheme
containing \(\mathsf {N}\) (of Sect. B.6, which we will henceforth write as \(\mathsf {N}_\kappa \) to emphasize the dependence on \(\widehat{{\mathcal {H}}}\)) as an open dense subscheme, called a toroidal compactification of \(\mathsf {N}_\kappa \) (see theorem 1.3.3.15 of [44]). When \(U=\widehat{{\mathcal {H}}}\), and when \((U,\varSigma ) \in {\mathcal {J}}_n^{(m),{{\text {tor}}}}\) is induced by \(\kappa = (\widehat{{\mathcal {H}}}, \widehat{\varSigma })\) as above, the toroidal compactification
is the
in this paper (see Sect. 5.3). Such toroidal compactifications and the identifications between them are compatible with the actions of \(\widehat{\mathrm {G}}({\mathbb {A}^{\infty }}) = G_n^{(m)}({\mathbb {A}}^\infty )\) (see theorem 1.3.3.15(4) of [44]).
The toroidal compactification \({\mathsf {N}}^{\text {tor}}_\kappa \) admits a stratification by locally closed subschemes
(see theorem 1.3.3.15(1) of [44]) indexed by equivalence classes \([(\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}}, \widehat{\rho })]\) as in lemma 1.2.4.42 of [44]. Without repeating the definition in detail, let us just note that the equivalence classes \([(\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}}, \widehat{\rho })]\) with the same underlying cusp label \([(\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}})]\) can be identified with the \(\varGamma _{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}\)-orbits of the cones \(\widehat{\rho } \in \widehat{\varSigma }_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}\) such that \(\widehat{\rho } \cap \widehat{\mathbf {P}}_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}^+ \ne \emptyset \). Each stratum \(\widehat{\mathsf {Z}}_{[(\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}}, \widehat{\rho })]}\) is canonically isomorphic to \(\widehat{\varXi }_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}}, \widehat{\rho }}\), and the formal completion of \({\mathsf {N}}^{\text {tor}}_\kappa \) along the union of the strata \(\widehat{\mathsf {Z}}_{[(\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}}, \widehat{\rho })]}\) labelled by equivalence classes \([(\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}}, \widehat{\rho })]\) with the same underlying cusp label \([(\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}})]\) is canonically isomorphic to \(\widehat{\mathfrak {X}}_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}}} / \varGamma _{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}\) (see theorem 1.3.3.15(1) of [44]).
When \(U=\widehat{{\mathcal {H}}}\), and when \((U,\varSigma ) \in {\mathcal {J}}_n^{(m),{{\text {tor}}}}\) is induced by \(\kappa = (\widehat{{\mathcal {H}}}, \widehat{\varSigma })\) as above, the formal completion of \({\mathsf {N}}^{\text {tor}}_\kappa \) along the union of all strata \(\widehat{\mathsf {Z}}_{[(\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}}, \widehat{\rho })]}\) labelled by equivalence classes \([(\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}}, \widehat{\rho })]\) with underlying \(\breve{{\texttt {Z}}}_{\widehat{{\mathcal {H}}}}\) equal to \({\breve{{\texttt {Z}}}}^{(i, \widehat{g})}_{\widehat{{\mathcal {H}}}}\) for some \(\widehat{g} \in \widehat{\mathrm {G}}({\mathbb {A}^{\infty }})\) is
where the indices g and h run over \(P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty ) \backslash G_n^{(m)}({\mathbb {A}}^\infty ) / U\) and
respectively, in this paper (see Sect. 5.3).
These identifications are all Hecke equivariant (see theorem 1.3.3.15(4) of [44]).
If \(\kappa \in \mathbf {K}_{Q, {\mathcal {H}}, \varSigma }^{++}\), then the canonical morphism
extends to a canonical log smooth morphism
(see theorem 1.3.3.15(2) of [44]). When \(U=\widehat{{\mathcal {H}}}\) and \(U'={\mathcal {H}}\), and when \((U,\varSigma ) \in {\mathcal {J}}_n^{(m),{{\text {tor}}}}\) and \((U',\varDelta ) \in {\mathcal {J}}_n^{{\text {tor}}}\) are induced by \(\kappa = (\widehat{{\mathcal {H}}}, \widehat{\varSigma })\) and \(({\mathcal {H}}, \varSigma )\), respectively, we have
(see Sect. 5.2 in this paper) and the above morphism is the log smooth morphism
in this paper (see Sect. 5.3). These morphisms and identifications are equivariant with \(\widehat{\mathrm {G}}({\mathbb {A}^{\infty }}) \rightarrow \mathrm {G}({\mathbb {A}^{\infty }})\) (see proposition 1.3.3.15(4) of [44]).
1.8 Automorphic bundles in characteristic zero
Since the maximal isotropic submodule \(V_0\) of \(L\otimes _{\mathbb {Z}} {\mathbb {C}}\) on which \(h_0(z)\) acts by \(1 \otimes z\) is isomorphic to \((F\otimes _{\mathbb {Q}} {\mathbb {C}})^n\) as an \(F\otimes _{\mathbb {Q}} {\mathbb {C}}\)-module, we can take \(F_0' = F_0= \mathbb {Q}\) and \(L_0 = F^n\) in the beginning of section 1.4.1 of [44], and define
for each \(\mathbb {Q}\)-algebra R. (See definition 6.2 of [42], definition 1.4.1.1 of [44]. We will not need the other groups \(\mathrm {G}_0(R)\) and \(\mathrm {P}_0(R)\) in this paper.) This can be canonically identified with the group \(L_{n,(n)}(R)\) in this paper (see Sect. 1.2), by matching \({\text {GL}}_{\mathcal {O}\otimes _{\mathbb {Z}} {R}}({L}^{\vee }_0({1})\otimes _{\mathbb {Q}} {R})\) with \(L_{n,(n),{{\text {lin}}}}(R)\), and by matching \(\mathbf {G}_{\text {m}}(R)\) with \(L_{n,(n),{{\text {herm}}}}(R)\).
Let \({\mathcal {H}}\) be any neat open compact subgroup of \(\mathrm {G}({\hat{\mathbb {Z}}})\), so that \(\mathsf {M}_{\mathcal {H}}\) is defined over \({{\text {Spec}}\,}\mathbb {Q}\) as in Sect. B.1. Then the tautological abelian scheme \(A\) over \(\mathsf {M}_{\mathcal {H}}\) defines a locally free sheaf
(where \(e_A\) denotes the identity section), which is the
in this paper (see Sect. 3.4.1), with \(U={\mathcal {H}}\). We can similarly define \({\underline{{\text {Lie}}}}^{\vee }_{{A}^{\vee }/\mathsf {M}_{\mathcal {H}}}\). The action of \(\mathrm {G}({\mathbb {A}^{\infty }})\) on \(\{ \mathsf {M}_{\mathcal {H}}\}_{\mathcal {H}}\) is defined by respecting their tautological abelian schemes up to canonical \({\mathbb {Q}}^\times \)-isogenies, which induces actions of \(\mathrm {G}({\mathbb {A}^{\infty }})\) on \(\{ {\underline{{\text {Lie}}}}^{\vee }_{A/\mathsf {M}_{\mathcal {H}}} \}_{\mathcal {H}}\) and \(\{ {\underline{{\text {Lie}}}}^{\vee }_{{A}^{\vee }/\mathsf {M}_{\mathcal {H}}} \}_{\mathcal {H}}\) covering the one on \(\{ \mathsf {M}_{\mathcal {H}}\}_{\mathcal {H}}\), which are compatible with the isomorphisms
induced by the tautological polarizations \(\lambda : A\rightarrow {A}^{\vee }\). Here the formal Tate twist is induced by the one on de Rham homology, realized by tensor products with \(\mathbb {Z}({1})\) over \(\mathbb {Z}\). Therefore, the corresponding Hecke action must be twisted by the similitude character \(\nu \), which corresponds to the tensor product with
in this paper (see Sect. 3.4.1).
Let \(\varSigma \) be a compatible collection of admissible projective smooth rational polyhedral cone decompositions for \(\mathsf {M}_{\mathcal {H}}\), so that \({\mathsf {M}}^{\text {tor}}_{{\mathcal {H}}, \varSigma }\) is defined over \({{\text {Spec}}\,}\mathbb {Q}\) as in Sect. B.5. Then the tautological semi-abelian scheme \(G\) over \({\mathsf {M}}^{\text {tor}}_{{\mathcal {H}}, \varSigma }\) defines a locally free sheaf
(where \(e_G\) denotes the identity section), which is the
in this paper (see Sect. 5.4), with \(U={\mathcal {H}}\), and with \(\varDelta \) induced by \(\varSigma \) as in Sect. B.5. We can similarly define \({\underline{{\text {Lie}}}}^{\vee }_{{G}^{\vee }/{\mathsf {M}}^{\text {tor}}_{{\mathcal {H}}, \varSigma }}\), where \({G}^{\vee }\) denotes the tautological ‘dual semi-abelian scheme’ over \({\mathsf {M}}^{\text {tor}}_{{\mathcal {H}}, \varSigma }\) extending \({A}^{\vee }\). (Note that dual semi-abelian schemes only make sense as such extensions.) The action of \(\mathrm {G}({\mathbb {A}^{\infty }})\) on \(\{ {\mathsf {M}}^{\text {tor}}_{{\mathcal {H}}, \varSigma } \}_{({\mathcal {H}}, \varSigma )}\) is defined by respecting their tautological semi-abelian schemes up to canonical \({\mathbb {Q}}^\times \)-isogenies, which induces actions of \(\mathrm {G}({\mathbb {A}^{\infty }})\) on \(\{ {\underline{{\text {Lie}}}}^{\vee }_{G/{\mathsf {M}}^{\text {tor}}_{{\mathcal {H}}, \varSigma }} \}_{({\mathcal {H}}, \varSigma )}\) and \(\{ {\underline{{\text {Lie}}}}^{\vee }_{{G}^{\vee }/{\mathsf {M}}^{\text {tor}}_{{\mathcal {H}}, \varSigma }} \}_{({\mathcal {H}}, \varSigma )}\) covering the one on \(\{ {\mathsf {M}}^{\text {tor}}_{{\mathcal {H}}, \varSigma } \}_{({\mathcal {H}}, \varSigma )}\), which are compatible with the isomorphisms
induced by the tautological polarizations \(\lambda : G\rightarrow {G}^{\vee }\). Here the formal Tate twist requires (as before) the Hecke action to be twisted by the similitude character \(\nu \), which corresponds to the tensor product with
in this paper (see Sect. 5.4).
Then we have the principal \(\mathrm {M}_0\)-bundle
which is an \(\mathrm {M}_0\)-torsor over \(\mathsf {M}_{\mathcal {H}}\) (see definition 1.4.1.5 and lemma 1.4.1.7 of [44]), which canonically extends (as an \(\mathrm {M}_0\)-torsor) to a principal \(\mathrm {M}_0\)-bundle
over \({\mathsf {M}}^{\text {tor}}_{{\mathcal {H}}, \varSigma }\) (see (1.4.2.7) and lemma 1.4.2.8 of [44]). These are the
in this paper (see Sects. 3.4.1, 5.4), with \(U={\mathcal {H}}\), and with \(\varDelta \) induced by \(\varSigma \) as in Sect. B.5.
For each \(\mathbb {Q}\)-algebra \(R\), we denote by \(\mathrm {Rep}_R(\mathrm {M}_0)\) the category of \(R\)-modules with algebraic actions of \(\mathrm {M}_0 \otimes _{\mathbb {Q}} {R}\) (see definition 1.4.1.8 of [44]). Then we also define, for each \(W\in \mathrm {Rep}_R(\mathrm {M}_0)\) that is locally free of finite rank as an \(R\)-module, the automorphic bundle
over \(\mathsf {M}_{\mathcal {H}}\otimes _{\mathbb {Q}} {R}\) (see definition 1.4.1.9 of [44]), which extends to the canonical extension
and the subcanonical extension
over \({\mathsf {M}}^{\text {tor}}_{\mathcal {H}}\otimes _{\mathbb {Q}} {R}\) (see definition 1.4.2.9 of [44]), where \(\mathscr {I}_{\mathsf {D}_{\infty , {\mathcal {H}}, \varSigma }}\) is the \(\mathscr {O}_{{\mathsf {M}}^{\text {tor}}_{{\mathcal {H}}, \varSigma }}\)-ideal defining the boundary divisor \(\mathsf {D}_{\infty , {\mathcal {H}}, \varSigma } := {\mathsf {M}}^{\text {tor}}_{{\mathcal {H}}, \varSigma } - \mathsf {M}_{\mathcal {H}}\) (with its canonical reduced subscheme structure). These are the vector bundles
in this paper (see Sects. 3.4.1, 5.4), with \(U={\mathcal {H}}\) and \(W_\rho =W\). The bundles \(\mathcal {E}_{\mathrm {M}_0, R}(W)\) and \(\mathcal {E}_{\mathrm {M}_0, R}^{\mathrm {can}}(W)\) admit compatible actions of \(\mathrm {G}({\mathbb {A}^{\infty }})\) (see proposition 1.4.3.1 of [44]), which are compatible with the compatible actions of \(G_n({\mathbb {A}}^\infty )\) on \({\mathcal {E}}_{U,\rho }\) and \({\mathcal {E}}_{U,\varDelta ,\rho }^{{\text {can}}}\), covering the ones on their respective base schemes.
1.9 Total objects in mixed characteristics
For each open compact subgroup \({\mathcal {H}}\) of \(\mathrm {G}({\hat{\mathbb {Z}}})\) whose image \({\mathcal {H}}^p\) under the canonical homomorphism \(\mathrm {G}({\hat{\mathbb {Z}}}) \rightarrow \mathrm {G}({\hat{\mathbb {Z}}^{p}})\) is neat, which implies, a fortiori, that \({\mathcal {H}}\) is also neat, we have a normal scheme
which is quasi-projective and flat over \(\vec {\mathsf {S}}_0 = {{\text {Spec}}\,}{\mathcal {O}_{F_0, (p)}}= {{\text {Spec}}\,}{\mathbb {Z}_{(p)}}\) and satisfies \(\vec {\mathsf {M}}_{\mathcal {H}}\otimes _{\mathbb {Z}} {\mathbb {Q}} \cong \mathsf {M}_{\mathcal {H}}\) (see proposition 2.2.1.1 in [44]). This is simply the normalization of \(\mathsf {M}_{\mathcal {H}}\) over the auxiliary (Siegel) moduli \(\mathsf {M}_{{\mathcal {H}}_{\text {aux}}}\) over \({{\text {Spec}}\,}{\mathcal {O}_{F_{0, {\text {aux}}}, (p)}}= {{\text {Spec}}\,}{\mathbb {Z}_{(p)}}\) defined by the auxiliary integral PEL datum
forgetting the actions of \(\mathcal {O}= \mathcal {O}_F\), for any neat open compact subgroup \({\mathcal {H}}_{\text {aux}}\) of \(\mathrm {G}_{\text {aux}}({\hat{\mathbb {Z}}})\) (defined by the above auxiliary integral PEL datum) containing the image of \({\mathcal {H}}\) under the canonical homomorphism \(\mathrm {G}({\hat{\mathbb {Z}}}) \rightarrow \mathrm {G}_{\text {aux}}({\hat{\mathbb {Z}}})\) (see lemma 2.1.1.18 of [44] for the existence of \({\mathcal {H}}_{\text {aux}}\)).
Similarly, we have a normal scheme
which is projective and flat over \({{\text {Spec}}\,}{\mathbb {Z}_{(p)}}\) and satisfies \({\vec {\mathsf {M}}}^{\text {min}}_{\mathcal {H}}\otimes _{\mathbb {Z}} {\mathbb {Q}} \cong {\mathsf {M}}^{\text {min}}_{\mathcal {H}}\), and contains \(\vec {\mathsf {M}}_{\mathcal {H}}\) as an open fibrewise dense subscheme (see propositions 2.2.1.2 and 2.2.1.7, and corollary 2.2.1.15, in [44]). The scheme \({\vec {\mathsf {M}}}^{\text {min}}_{\mathcal {H}}\) is the
in this paper with \(U={\mathcal {H}}\), and the special fibre
is the
in this paper (see Sect. 5.1).
The projective scheme \({\vec {\mathsf {M}}}^{\text {min}}_{\mathcal {H}}\) is equipped with an ample invertible sheaf
(see proposition 2.2.1.2 in [44]). (Since \(\langle {\,\cdot \,,\,\cdot \,}\rangle \) is self-dual, we can take \(a_1=1\), \(a_2=0\), \(a_0=1\) and \(a=1\) in lemmas 2.1.1.1 and 2.1.2.35 of [44].) This is the
in this paper (see Sect. 5.1). We have a section
(see corollary 6.3.1.7 of [44]), which is the
in this paper (see Sect. 5.1), whose vanishing and non-vanishing loci are
(see definition 6.3.2.1 of [44]), which are the
respectively, in this paper (see Sect. 5.1).
The collections \(\{ \vec {\mathsf {M}}_{\mathcal {H}}\}_{\mathcal {H}}\), \(\{ {\vec {\mathsf {M}}}^{\text {min}}_{\mathcal {H}}\}_{\mathcal {H}}\), \(\{ \omega _{{\vec {\mathsf {M}}}^{\text {min}}_{\mathcal {H}}} \}_{\mathcal {H}}\) and \(\{ \text {Hasse}_{\mathcal {H}}\}_{\mathcal {H}}\) admit compatible actions of \(\mathrm {G}({\mathbb {A}^{\infty , p}}) \times \mathrm {G}(\mathbb {Z}_p)\) (see proposition 2.2.3.1 and corollary 6.3.1.8 of [44]).
While the reader might be interested in knowing more about \(\vec {\mathsf {M}}_{\mathcal {H}}\) and \({\vec {\mathsf {M}}}^{\text {min}}_{\mathcal {H}}\), we emphasize that we need to know almost nothing about them in this paper. What we really need to know in detail are their ordinary loci (or more precisely just their multiplicative-type ordinary loci, rather than the whole \(({\vec {\mathsf {M}}}^{\text {min}}_{\mathcal {H}}\otimes _{\mathbb {Z}} {\mathbb {F}_p})^{\text {full-ord}}\) as above), which we will explain below.
1.10 Ordinary loci of Shimura varieties
To define the ordinary loci \(\vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\) in [44], which will be compared with the \({\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}\) for suitable choices of \({\mathcal {H}}\) and \(U^p(N_1,N_2)\), we consider the maximal totally isotropic filtration
(see lemma 3.2.2.1 of [44]) given by
Since \(\langle {\,\cdot \,,\,\cdot \,}\rangle \) is self-dual, the dual filtration \({\texttt {D}}^\#\) in lemma 3.2.2.4 of [44] can be identified with \({\texttt {D}}\), and the induced inclusions \(\phi _{{\texttt {D}}}^0: {\text {Gr}}_{\texttt {D}}^0\hookrightarrow {\text {Gr}}_{{\texttt {D}}^\#}^0\) and \(\phi _{{\texttt {D}}}^{-1}: {\text {Gr}}_{\texttt {D}}^{-1}\hookrightarrow {\text {Gr}}_{{\texttt {D}}^\#}^{-1}\) (again, see lemma 3.2.2.4 of [44]) are isomorphisms. Moreover, the group \(\vec {\mathbf {S}}_{\texttt {D}}\) in theorem 3.4.1.9 of [44] is torsion free because it can be identified with \(S({\mathcal {O}}_{F,p}^n)\) in this paper (see Sect. 1.1), and hence the invariant \({r_{\texttt {D}}}\) in definition 3.4.2.1 of [44] is just zero under the assumptions of this paper.
Then we define the following groups, for each \(\mathbb {Z}_p\)-algebra R (see definition 3.2.2.7 of [44]):
and
Then \(\mathrm {G}({\mathbb {A}^{\infty , p}}) \times \mathrm {P}^{\text {ord}}_{\texttt {D}}(\mathbb {Z}_p) = G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\), in the notation of this paper.
For all integers \(0\le r\) and \(0 \le {r_1}\le {r_0}\), we set (see definition 3.2.2.8 of [44]):
-
(1)
\(\mathcal {U}_{p, 0}(p^r) := ( \mathrm {G}(\mathbb {Z}_p) \rightarrow \mathrm {G}(\mathbb {Z}/p^r\mathbb {Z}) )^{-1} ( \mathrm {P}^{\text {ord}}_{\texttt {D}}(\mathbb {Z}/p^r\mathbb {Z}) ) = U_p(0,r)\).
-
(2)
\(\mathcal {U}_{p, 1}(p^r) := ( \mathrm {G}(\mathbb {Z}_p) \rightarrow \mathrm {G}(\mathbb {Z}/p^r\mathbb {Z}) )^{-1} ( \mathrm {U}^{{\text {ord}}, {-1}}_{\texttt {D}}(\mathbb {Z}/p^r\mathbb {Z}) ) = U_p(r,r)\).
-
(3)
\(\mathcal {U}_{p, 1}^{\text {bal}}(p^r) := ( \mathrm {G}(\mathbb {Z}_p) \rightarrow \mathrm {G}(\mathbb {Z}/p^r\mathbb {Z}) )^{-1} ( \mathrm {U}^{\text {ord}}_{\texttt {D}}(\mathbb {Z}/p^r\mathbb {Z}) )\).
-
(4)
\(\mathcal {U}_{p, 1, 0}(p^{r_1}, p^{r_0}) := \mathcal {U}_{p, 1}(p^{r_1}) \cap \mathcal {U}_{p, 0}(p^{r_0}) = U_p(r_1,r_0)\).
-
(5)
\(\mathcal {U}^{\text {ord}}(p^r) := \ker (\mathrm {M}^{\text {ord}}_{\texttt {D}}(\mathbb {Z}_p) \rightarrow \mathrm {M}^{\text {ord}}_{\texttt {D}}(\mathbb {Z}/p^r\mathbb {Z})) = U_p(r)_{n,(n)} \times {(1+p^r{\mathbb {Z}}_p)}^\times \).
An open compact subgroup \({\mathcal {H}}_p\) of \(\mathrm {G}({\mathbb {Q}}_p)\) is said to be of standard form and of depth \(r\) in the sense of definition 3.2.2.9 of [44] if
In this case we define \({\mathcal {H}}_p^{\text {ord}}\) to be the unique open compact subgroup of \(\mathrm {M}^{\text {ord}}_{\texttt {D}}(\mathbb {Z}_p)\) such that \({\mathcal {H}}_p^{\text {ord}}/ \mathcal {U}^{\text {ord}}(p^r) \cong {\mathcal {H}}_p/ \mathcal {U}_{p, 1}^{\text {bal}}(p^r)\) (see definition 3.3.3.4 of [44]).
The theory in [44] is developed for open compact subgroups \({\mathcal {H}}\) of \(\mathrm {G}({\hat{\mathbb {Z}}})\) of the form \({\mathcal {H}}= {\mathcal {H}}^p{\mathcal {H}}_p\), where \({\mathcal {H}}^p\) is a neat open compact subgroup of \({\mathrm {G}}({\hat{\mathbb {Z}}^{p}})\), and where \({\mathcal {H}}_p\) is an open compact subgroup of \({\mathrm {G}}(\mathbb {Z}_p)\) of standard form. In this paper, we will only need \({\mathcal {H}}\) of the form \(U^p(N_1,N_2)\), which satisfies the above requirement with \({\mathcal {H}}^p= U^p\) and \({\mathcal {H}}_p= \mathcal {U}_{p, 1, 0}(p^{N_1}, p^{N_2})\). In this case, since \(\nu ({\mathcal {H}}_p) = {\mathbb {Z}}^\times _p\) and \({r_{\texttt {D}}}=0\), the invariant \({r_{\mathcal {H}}}\) in definition 3.4.2.1 of [44] is just zero. Then we have a smooth quasi-projective scheme
over \(\vec {\mathsf {S}}_0 = {{\text {Spec}}\,}{\mathcal {O}_{F_0, (p)}}= {{\text {Spec}}\,}{\mathbb {Z}_{(p)}}\), which satisfies \(\vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\otimes _{\mathbb {Z}} {\mathbb {Q}} \cong \mathsf {M}_{\mathcal {H}}\) and can be canonically embedded as an open subscheme of \(\vec {\mathsf {M}}_{\mathcal {H}}\) (see theorem 3.4.2.5 and proposition 3.4.6.3 of [44]). Note that, in theorem 3.4.2.5 of [44], \(\vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\) is defined as a normalization of the base change from \({{\text {Spec}}\,}{\mathbb {Z}_{(p)}}\) to \({{\text {Spec}}\,}({\mathcal {O}_{F_0, (p)}}[\zeta _{p^{r_{\mathcal {H}}}}])\) of a naive moduli \(\dddot{\mathsf {M}}_{\mathcal {H}}^{\text {ord}}\) over \({{\text {Spec}}\,}{\mathbb {Z}_{(p)}}\) (see definition 3.4.1.1 and theorem 3.4.1.9 of [44]), but since \(F_0=\mathbb {Q}\), \({r_{\texttt {D}}}=0\), and \({r_{\mathcal {H}}}=0\), the base change has no effect, and the normalization merely singles out the correct components satisfying the condition on Lie algebra. Hence, \(\vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\) coincides with the moduli problem
in this paper (see Sect. 3.1.1), with \(U^p(N_1,N_2)={\mathcal {H}}\) (that is, with \(U^p={\mathcal {H}}^p\), \(N_1={r_1}\), and \(N_2={r_0}\)). (See remark 3.4.2.8 of [44] for the comparison between the definition using isomorphism classes of abelian schemes with additional structures in [44], and the definition of \({\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}\) using prime-to-p quasi-isogeny classes in this paper, and for the extension of the definition of \(\vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\) to allow \({\mathcal {H}}= {\mathcal {H}}^p{\mathcal {H}}_p\) for all open compact subgroups \({\mathcal {H}}^p\) of \(\mathrm {G}({\hat{\mathbb {Z}}^{p}})\). Again, since the pairing \(\langle {\,\cdot \,,\,\cdot \,}\rangle \) is self-dual, the consideration of dual objects in [44] can be suppressed, although they were clarifying when developing the general theory.)
The formal completion of \(\vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\cong {\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}\) (with \(U^p(N_1,N_2)={\mathcal {H}}\)) along
is denoted
(see definition 3.4.4.2 of [44]). Their independence of \(N_2={r_0}\) is explained in corollary 3.4.4.4 of [44].
The collection \(\{ \vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\}_{\mathcal {H}}\) indexed by neat open compact subgroups \({\mathcal {H}}\) of \(\mathrm {G}({\mathbb {A}^{\infty }})\) of the form considered above admits compatible actions (see proposition 3.4.4.1 of [44]) of \(\mathrm {G}({\mathbb {A}^{\infty , p}}) \times \mathrm {P}^{\text {ord}}_{\texttt {D}}(\mathbb {Z}_p)\) and of the element of \(\mathrm {P}^{\text {ord}}_{\texttt {D}}(\mathbb {Q}_p) \subset \mathrm {G}(\mathbb {Q}_p)\) corresponding to \(\varsigma _p \in L_{n,(n),{{\text {herm}}}}({\mathbb {Q}}_p)\) in this paper (see Sect. 1.2) under the identification between \(\mathrm {G}(\mathbb {Q}_p)\) and \(G_n({\mathbb {Q}}_p)\), which are compatible with the action of \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\) on \(\{ {\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}\}_{U^p(N_1,N_2)}\). (In fact, proposition 3.4.4.1 of [44] gives the actions of many more elements, but we omit them because they are not needed in this paper.) While these actions are given by quasi-finite morphisms which are often not finite, the induced morphisms on the \(p\)-adic completions are always finite (see corollary 3.4.4.3 of [44]). The element of \(\mathrm {P}^{\text {ord}}_{\texttt {D}}(\mathbb {Q}_p)\) corresponding to \(\varsigma _p\) is a typical example of an element of \({\text {U}}_p\) type in definition 3.3.6.1 of [44], which induces the composite of absolute Frobenius and forgetful morphisms in characteristic \(p\) (see corollary 3.4.4.6 of [44]).
1.11 Ordinary cusp labels
Let \({\mathcal {H}}\) be any open compact subgroup of \(\mathrm {G}({\hat{\mathbb {Z}}})\) as above. We say that a cusp label \(({\texttt {Z}}_{\mathcal {H}}, \varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})\) is ordinary (see definition 3.2.3.8 of [44]) if \({\texttt {Z}}_{\mathcal {H}}\) contains an element \({\texttt {Z}}\) that is compatible with the filtration \({\texttt {D}}\) in the sense that
(see definition 3.2.3.1 of [44]). Then we have an induced filtration \({\texttt {D}}_{-1}\) on \({\text {Gr}}^{\texttt {Z}}_{-1} \otimes _{{\hat{\mathbb {Z}}}} {\mathbb {Z}_p}\) given by
For any such \({\texttt {Z}}\), and for each \(\mathbb {Z}_p\)-algebra R, we define the following quotients of subgroups of \(\mathrm {P}_{\texttt {Z}}(R)\) (see definition 3.2.3.9 of [44]):
-
(1)
\(\mathrm {P}^{\text {ord}}_{{\texttt {Z}}, {\texttt {D}}}(R) := \mathrm {P}_{\texttt {Z}}(R) \cap \mathrm {P}^{\text {ord}}_{\texttt {D}}(R)\).
-
(2)
\(\mathrm {P}^{{\text {ord}}, \prime }_{{\texttt {Z}}, {\texttt {D}}}(R) := \mathrm {P}_{\texttt {Z}}'(R) \cap \mathrm {P}^{\text {ord}}_{\texttt {D}}(R)\).
-
(3)
\(\mathrm {P}^{\text {ord}}_{1, {\texttt {Z}}, {\texttt {D}}}(R) := \mathrm {P}^{{\text {ord}}, \prime }_{{\texttt {Z}}, {\texttt {D}}}(R) / \mathrm {U}_{2, {\texttt {Z}}}(R)\).
-
(4)
\(\mathrm {P}^{\text {ord}}_{h, {\texttt {Z}}, {\texttt {D}}}(R)\) is the subgroup of elements of \(\mathrm {G}_{h, {\texttt {Z}}}(R)\) preserving the filtration \({\texttt {D}}_{-1}\) induced by \({\texttt {D}}\) on \({\text {Gr}}^{\texttt {Z}}_{-1}\otimes _{{\hat{\mathbb {Z}}}} {\mathbb {Z}_p}\).
-
(5)
\(\mathrm {P}^{{\text {ord}}, \prime }_{h, {\texttt {Z}}, {\texttt {D}}}(R) := \mathrm {P}^{{\text {ord}}, \prime }_{{\texttt {Z}}, {\texttt {D}}}(R) / \mathrm {U}_{\texttt {Z}}(R) \mathop {\rightarrow }\limits ^{\sim }\mathrm {P}^{\text {ord}}_{h, {\texttt {Z}}, {\texttt {D}}}(R)\).
We have, for example,
in the notation of this paper (see Sect. 1.2). (In this paper, all intersections with \(\mathrm {P}^{\text {ord}}_{\texttt {D}}(\mathbb {Z}_p) = P_{n,(n)}^+({\mathbb {Z}}_p)\) at the factors at \(p\) are denoted by the superscript \(^{{{\text {ord}}},\times }\).) By definition, as g varies in \(\mathrm {G}({\mathbb {A}^{\infty }})\), the filtration \({{\texttt {Z}}}^{(i, g)}\) is compatible with \({\texttt {D}}\) if and only if \(g \in \mathrm {G}({\mathbb {A}^{\infty , p}}) \times \mathrm {P}_{{{\texttt {Z}}}^{(i, 1)}}(\mathbb {Q}_p) \mathrm {P}^{\text {ord}}_{{\texttt {D}}}(\mathbb {Q}_p)\).
Now suppose \({\mathcal {H}}= {\mathcal {H}}^p{\mathcal {H}}_p\), where \({\mathcal {H}}^p\) is an open compact subgroup of \(\mathrm {G}({\mathbb {A}^{\infty , p}})\) and \({\mathcal {H}}_p= \mathcal {U}_{p, 1, 0}(p^{r_1}, p^{r_0})\) for some integers \(0 \le {r_1}\le {r_0}\). The \({\mathcal {H}}\)-orbits \({{\texttt {Z}}}^{(i, g)}_{\mathcal {H}}\) of \({{\texttt {Z}}}^{(i, g)}\) that contains a filtration compatible with \({\texttt {D}}\) are parameterized by the image of \(\mathrm {G}({\mathbb {A}^{\infty , p}}) \times \mathrm {P}_{{{\texttt {Z}}}^{(i, 1)}}(\mathbb {Q}_p) \mathrm {P}^{\text {ord}}_{{\texttt {D}}}(\mathbb {Q}_p)\) in \(\mathrm {P}_{{{\texttt {Z}}}^{(i, 1)}}({\mathbb {A}^{\infty }}) \backslash \mathrm {G}({\mathbb {A}^{\infty }}) / {\mathcal {H}}\). As in Sect. 5.1, this image is in bijection with the double coset space
which is the double coset space
in this paper (see, e.g., Sect. 5.1), with \(U^p(N_1,N_2)={\mathcal {H}}\) and with \(U^p(N_1)\) denoting the intersection of \(U^p(N_1,N_2)\) with \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\). The ordinary cusp labels \([({\texttt {Z}}_{\mathcal {H}}, \varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]\) for \(\mathsf {M}_{\mathcal {H}}\) with underlying \({\texttt {Z}}_{\mathcal {H}}\) equal to \({{\texttt {Z}}}^{(i, g)}_{\mathcal {H}}\) for some \(g \in \mathrm {G}({\mathbb {A}^{\infty , p}}) \times \mathrm {P}^{\text {ord}}_{\texttt {D}}(\mathbb {Z}_p)\) are parameterized by the double coset space
and the forgetful map sending \([({\texttt {Z}}_{\mathcal {H}}, \varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]\) to \({\texttt {Z}}_{\mathcal {H}}\) can be identified with the canonical map from this double coset space to
whose fibre above the double coset of \(g \in \mathrm {G}({\mathbb {A}^{\infty , p}}) \times \mathrm {P}^{\text {ord}}_{\texttt {D}}(\mathbb {Z}_p)\) can be identified with
This last double coset space is the index set of
in this paper (see Sect. 4.1), with \(U^p(N_1,N_2)={\mathcal {H}}\).
1.12 Partial minimal compactifications of ordinary PEL moduli
For each \({\mathcal {H}}= {\mathcal {H}}^p{\mathcal {H}}_p\), where \({\mathcal {H}}^p\) is a neat open compact subgroup of \(\mathrm {G}({\mathbb {A}^{\infty , p}})\), and where \({\mathcal {H}}_p= \mathcal {U}_{p, 1, 0}(p^{r_1}, p^{r_0})\) for some integers \(0 \le {r_1}\le {r_0}\), there is a normal scheme
quasi-projective and flat over \({{\text {Spec}}\,}{\mathbb {Z}_{(p)}}\), with geometrically normal fibres, which contains \(\vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\) as an open fibrewise dense subscheme and can be canonically embedded as an open subscheme of \({\vec {\mathsf {M}}}^{\text {min}}_{\mathcal {H}}\), called the partial minimal compactification of \(\vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\), whose characteristic zero pull-back \({\vec {\mathsf {M}}}^{{\text {ord}}, \text {min}}_{\mathcal {H}}\otimes _{\mathbb {Z}} {\mathbb {Q}}\) is an open subscheme of \({\mathsf {M}}^{\text {min}}_{\mathcal {H}}\) which can be identified with the union of the strata \(\mathsf {Z}_{[(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]}\) indexed by ordinary cusp labels (see theorem 6.2.1.1 and proposition 6.2.1.6 of [44]). The union
(by gluing along their common open subscheme described above) is the
in this paper (see Sect. 5.1), with \(U^p(N_1,N_2)={\mathcal {H}}\). The collection \(\{ {\vec {\mathsf {M}}}^{{\text {ord}}, \text {min}}_{\mathcal {H}}\}_{\mathcal {H}}\) admits compatible actions of \(\mathrm {G}({\mathbb {A}^{\infty , p}}) \times \mathrm {P}^{\text {ord}}_{\texttt {D}}(\mathbb {Z}_p)\) and of the element of \(\mathrm {P}^{\text {ord}}_{\texttt {D}}(\mathbb {Q}_p)\) corresponding to \(\varsigma _p\) (see proposition 6.2.2.1 of [44]), which are compatible with the action of \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\) on \(\{ {\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {min}}}\}_{U^p(N_1,N_2)}\).
The partial minimal compactification \({\vec {\mathsf {M}}}^{{\text {ord}}, \text {min}}_{\mathcal {H}}\) admits a stratification by locally closed subschemes
where \([(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]\) runs over ordinary cusp labels for \(\mathsf {M}_{\mathcal {H}}\) and g runs over
(see theorem 6.2.1.1(4) of [44]). If \({\mathcal {H}}=U^p(N_1,N_2)\), this matches
(where the indices g are as above) with the subscheme
of \(\partial _i^0 {\mathcal {X}}^{{{\text {ord}}},\min }_{n,U^p(N_1,N_2)}\) (see Sect. 5.1), where
Each \(\vec {\mathsf {Z}}^{\text {ord}}_{[(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]}\) is the quotient \(\vec {\mathsf {M}}_{\mathcal {H}}^{{\text {ord}}, {\texttt {Z}}_{\mathcal {H}}}\) of
by \(\varGamma _{\varPhi _{\mathcal {H}}}\), which is isomorphic to \(\vec {\mathsf {M}}^{\text {ord}}_{{\mathcal {H}}_{\mathrm {G}_{h, {\texttt {Z}}}, \varPhi }}\), where \({\mathcal {H}}_{\mathrm {G}_{h, {\texttt {Z}}}, \varPhi }\) is as in Sect. B.4 (see lemmas 5.2.4.1 and 5.2.4.5 of [44]). Here \(\vec {\mathsf {M}}^{\text {ord}}_{{\mathcal {H}}_{\mathrm {G}_{h, {\texttt {Z}}}'}}\) and \(\vec {\mathsf {M}}^{\text {ord}}_{{\mathcal {H}}_{\mathrm {G}_{h, {\texttt {Z}}}, \varPhi }}\) are analogues of \(\vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\) defined by an integral PEL datum \((\mathcal {O}, \star , L^{\texttt {Z}}, \langle {\,\cdot \,,\,\cdot \,}\rangle ^{\texttt {Z}}, h_0^{\texttt {Z}})\) defining \(\mathsf {M}_{{\mathcal {H}}_{\mathrm {G}_{h, {\texttt {Z}}}'}}\) and \(\mathsf {M}_{{\mathcal {H}}_{\mathrm {G}_{h, {\texttt {Z}}}, \varPhi }}\), which is associated with a representative \({\texttt {Z}}\) of \({\texttt {Z}}_{\mathcal {H}}\) that is compatible with \({\texttt {D}}\), and by the filtration \({\texttt {D}}_{-1}\) on \(L^{\texttt {Z}}\otimes _{\mathbb {Z}} {\mathbb {Z}_p} \cong {\text {Gr}}^{\texttt {Z}}_{-1} \otimes _{{\hat{\mathbb {Z}}}} {\mathbb {Z}_p}\) determined by \({\texttt {D}}\) as above (see definition 1.2.1.15 and lemma 5.2.4.1 of [44]). If we let \(U_g^{p,\prime }\) (resp. \(U^{p,\prime }_{g,{{\text {lin}}}}\)) denote the image of \(gU^pg^{-1}\cap P_{n,(i)}^+({\mathbb {A}}^{\infty ,p})\) in \(L_{n,(i)}({\mathbb {A}}^{\infty ,p})\) (resp. \(L_{n,(i),{{\text {lin}}}}({\mathbb {A}}^{\infty ,p})\)), then under the identifications
of Sect. 4.1,
-
the term \({\mathcal {X}}^{{\text {ord}}}_{n,(i),(U_g^{p,\prime }\cap G_{n-i}({\mathbb {A}}^{\infty ,p}))(N_1,N_2)}\) indexed by h is identified with \(\vec {\mathsf {M}}_{\mathcal {H}}^{{\text {ord}}, {\varPhi }^{(i, hg)}_{\mathcal {H}}}\);
-
the group \(L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)}) \cap (h U_{g,{{\text {lin}}}}^{p,\prime } h^{-1})(N_1)\) is identified with \(\varGamma _{{\varPhi }^{(i, hg)}_{\mathcal {H}}}\);
-
and the term
$$\begin{aligned} \left( L_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)}) \cap \left( h U_{g,{{\text {lin}}}}^{p,\prime } h^{-1}\right) (N_1)\right) \Big \backslash {\mathcal {X}}^{{\text {ord}}}_{n,(i),\left( U_g^{p,\prime }\cap G_{n-i}({\mathbb {A}}^{\infty ,p})\right) (N_1,N_2)} \end{aligned}$$is identified with \(\vec {\mathsf {M}}_{\mathcal {H}}^{{\text {ord}}, {{\texttt {Z}}}^{(i, hg)}_{\mathcal {H}}}\).
Similarly, under the identification
the term \({\mathcal {X}}^{{\text {ord}}}_{n,(i),(U_g^{p,\prime }\cap G_{n-i}({\mathbb {A}}^{\infty ,p}))(N_1,N_2)}\) indexed by h is identified with \(\vec {\mathsf {M}}_{\mathcal {H}}^{{\text {ord}}, {\varPhi }^{(i, hg)}_{\mathcal {H}}}\). These identifications are Hecke equivariant.
If \({\mathcal {H}}=U^p(N_1,N_2)\), then the formal completion of \({\vec {\mathsf {M}}}^{{\text {ord}}, \text {min}}_{\mathcal {H}}\cong {\mathcal {X}}^{{{\text {ord}}},{{\text {min}}}}_{n,U^p(N_1,N_2)}\) along
is denoted
(see definition 3.4.4.2 of [44]). Their independence of \(N_2={r_0}\) is explained in corollary 6.2.2.8 and example 3.4.4.5 of [44]. For the Hecke actions on these formal schemes, see corollaries 6.2.2.7, 6.2.2.8 and 6.2.2.9 of [44].
1.13 Partial toroidal compactifications of ordinary PEL moduli
For each representative \(({\texttt {Z}}_{\mathcal {H}}, \varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})\) of an ordinary cusp label \([({\texttt {Z}}_{\mathcal {H}}, \varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]\) for \(\mathsf {M}_{\mathcal {H}}\), there is a torsor
of an abelian scheme
this abelian scheme is \({\mathbb {Q}}^\times \)-isogenous to \(\underline{{\text {Hom}}}_\mathcal {O}(X, B)\), and in fact its \({{\mathbb {Z}}^\times _{(p)}}\)-isogeny class (i.e. prime-to-p quasi-isogeny class) can be described explicitly (see lemma 5.2.4.7 and propositions 5.2.4.11 and 5.2.4.13 of [44]). We obtain an isomorphic abelian scheme torsor if we replace \(({\texttt {Z}}_{\mathcal {H}}, \varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})\) with another representative, but its universal property depends on this choice of representative.
If \(U_g^{p,\prime }\) again denotes \(gU^pg^{-1}\cap P_{n,(i)}^+({\mathbb {A}}^{\infty ,p})\) then the map
in this paper (see Sects. 4.1, 4.2) with \(U^p(N_1,N_2)={\mathcal {H}}\) is identified with
These maps are equivariant with compatible actions of \(\mathrm {P}_{{{\texttt {Z}}}^{(i, g)}}({\mathbb {A}^{\infty , p}}) \times \mathrm {P}^{\text {ord}}_{{{\texttt {Z}}}^{(i, g)}, {\texttt {D}}}(\mathbb {Z}_p)\) and of the element of \(\mathrm {P}^{\text {ord}}_{{{\texttt {Z}}}^{(i, g)}, {\texttt {D}}}(\mathbb {Q}_p)\) corresponding to \(\varsigma _p\). (See proposition 5.2.4.25 of [44].) (Since our pairing \(\langle {\,\cdot \,,\,\cdot \,}\rangle \) is perfect, the universal property of \(\vec {C}^{\text {ord}}_{\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}}\) in proposition 5.2.4.13 of [44] can be simplified by suppressing the dual objects. This universal property then agrees with that of \({\mathcal {A}}_{n-i,(hgU^pg^{-1}h^{-1}\cap P_{n,(i)}({\mathbb {A}}^{\infty ,p}))(N_1,N_2)}^{(i),{{\text {ord}}}}\) in Sect. 3.2 of this paper.)
For each representative \(({\texttt {Z}}_{\mathcal {H}}, \varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})\) of an ordinary cusp label \([({\texttt {Z}}_{\mathcal {H}}, \varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]\) for \(\mathsf {M}_{\mathcal {H}}\), there is a torsor
under the pull-back of the same split torus \(E_{\varPhi _{\mathcal {H}}}\) over \({{\text {Spec}}\,}\mathbb {Z}\) as before (see Sect. B.5; and see lemma 5.2.4.26 and proposition 5.2.4.30 of [44]). We obtain an isomorphic torus torsor if we replace \(({\texttt {Z}}_{\mathcal {H}}, \varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})\) with another representative, but its universal property depends on this choice of representative. For a fixed \(({\texttt {Z}}, \varPhi , \delta )\), the collection \(\{ \vec {\varXi }^{\text {ord}}_{\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}} \}_{{\mathcal {H}}_{\mathrm {P}_{\texttt {Z}}'}}\) admits compatible actions of \(\mathrm {P}_{\texttt {Z}}'({\mathbb {A}^{\infty , p}}) \times \mathrm {P}^{{\text {ord}}, \prime }_{{\texttt {Z}}, {\texttt {D}}}(\mathbb {Z}_p)\) and of the element of \(\mathrm {P}^{{\text {ord}}, \prime }_{{\texttt {Z}}, {\texttt {D}}}(\mathbb {Q}_p)\) corresponding to \(\varsigma _p\) such that the morphisms
are equivariant with
and with the compatible actions of the elements corresponding to \(\varsigma _p\) (see proposition 5.2.4.41 of [44]).
The torus
in this paper (see Sect. 4.3) with \(U^p(N_1,N_2)={\mathcal {H}}\) is identified with
Moreover, the sheaves \(X^*({\mathcal {S}}_{n,(i),U_g^{p,\prime }(N_1)}^{{{\text {ord}}},+})\), \(X_*({\mathcal {S}}_{n,(i),U_g^{p,\prime }(N_1)}^{{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\), and \(X_*({\mathcal {S}}_{n,(i),U_g^{p,\prime }}^{{{\text {ord}}},+})_{\mathbb {R}}^{>0}\) are identified with
and
respectively. The \({\mathcal {S}}_{n,(i),U_g^{p,\prime }(N_1)}^{{{\text {ord}}},+}\)-torsor
is identified with
These maps are equivariant with the compatible actions of \(\mathrm {P}_{{{\texttt {Z}}}^{(i, g)}}({\mathbb {A}^{\infty , p}}) \times \mathrm {P}^{{\text {ord}}, \prime }_{{{\texttt {Z}}}^{(i, g)}, {\texttt {D}}}(\mathbb {Z}_p)\) and of the elements corresponding to \(\varsigma _p\). (See proposition 5.2.4.41 of [44].)
Consider any compatible collection
of admissible projective smooth rational polyhedral cone decompositions for \(\vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\), where each \(\varSigma _{\varPhi _{\mathcal {H}}}\) is a \(\varGamma _{\varPhi _{\mathcal {H}}}\)-admissible projective smooth rational polyhedral cone decomposition of \(\mathbf {P}_{\varPhi _{\mathcal {H}}}\), and where the indices \([(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]\) are ordinary cusp labels for \(\mathsf {M}_{\mathcal {H}}\), as in definitions 5.1.3.1 and 5.1.3.3 of [44]. Any compatible collection \(\varSigma \) for \(\mathsf {M}_{\mathcal {H}}\) induces a compatible collection \(\varSigma ^{\text {ord}}\) for \(\vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\) by restricting to the indices given by ordinary cusp labels, and conversely any compatible collection \(\varSigma ^{\text {ord}}\) for \(\vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\) extends to a compatible collection \(\varSigma \) for \(\mathsf {M}_{\mathcal {H}}\) (see proposition 5.1.3.4 of [44]). Each such \(\varSigma ^{\text {ord}}\) considered in [44] induces a pair
in \({\mathcal {J}}_n^{{{\text {tor}}},{{\text {ord}}}}\) (with \(U^p(N_1,N_2)={\mathcal {H}}\)) in Sect. 5.2 of this paper, because, in order to define \((U^p(N_1),\varDelta )\) as in Sect. 5.2, it suffices to define the admissible cone decomposition \(\varDelta (g)_0\) for \(X_*({\mathcal {S}}_{n,(i),(gU^pg^{-1}\cap P_{n,(i)}^+({\mathbb {A}}^{\infty ,p}))(N_1)}^{{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\), for each \(g\in G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times } = \mathrm {G}({\mathbb {A}^{\infty , p}}) \times \mathrm {P}^{\text {ord}}_{\texttt {D}}(\mathbb {Z}_p)\), which can be taken to be the pull-back of the subcollection \(\{ \varSigma _{\varPhi _{\mathcal {H}}} \}_{[(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]}\) of \(\varSigma ^{\text {ord}}\) indexed by the cusp labels \([(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]\) with underlying \({\texttt {Z}}_{\mathcal {H}}\) equal to \({{\texttt {Z}}}^{(i, g)}_{\mathcal {H}}\). In fact, \({\mathcal {J}}_n^{{{\text {tor}}},{{\text {ord}}}}\) is exactly the set of such induced pairs (as \(U^p(N_1,N_2)={\mathcal {H}}\) varies).
Each \(\varSigma _{\varPhi _{\mathcal {H}}}\) defines an affine toroidal embedding
over \(\vec {C}^{\text {ord}}_{\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}}\). Rather confusingly \(\vec {\varXi }^{\text {ord}}_{\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}}(\sigma )\) (in the notation of [44]) is what in this paper we would have denoted \(\vec {\varXi }^{\text {ord}}_{\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}, \sigma }\); and \(\vec {\varXi }^{\text {ord}}_{\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}, \sigma }\) is what in this paper we would have denoted \(\partial _\sigma \vec {\varXi }^{\text {ord}}_{\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}, \varSigma _{\varPhi _{\mathcal {H}}}}\). The formal completion of \(\overline{\vec {\varXi }}_{\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}, \varSigma _{\varPhi _{\mathcal {H}}}}^{\text {ord}}\) along the union of the \(\sigma \)-strata \(\vec {\varXi }^{\text {ord}}_{\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}, \sigma }\) for all \(\sigma \in \varSigma _{\varPhi _{\mathcal {H}}}\) such that \(\sigma \cap \mathbf {P}_{\varPhi _{\mathcal {H}}}^+ \ne \emptyset \) is denoted
(see (4.2.2.4), (5.2.4.32), (5.2.4.33), and lemma 5.2.4.38 of [44]).
For a fixed \(g \in \mathrm {G}({\mathbb {A}^{\infty , p}}) \times \mathrm {P}^{\text {ord}}_{\texttt {D}}(\mathbb {Z}_p)\), the schemes
of this paper (see Sect. 4.4) are identified with
Moreover, \({\mathcal {T}}_{n,(i),U_g^{p,\prime }(N_1,N_2),\varDelta (g)_0}^{{{\text {ord}}},+,\wedge }\) is identified with
and \({\mathcal {T}}_{n,(i),U_g^{p,\prime }(N_1,N_2),\varDelta (g)_0}^{{{\text {ord}}},\natural ,\wedge }\) is identified with
where the second disjoint union runs over all ordinary cusp labels \([(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]\) with underlying \({\texttt {Z}}_{\mathcal {H}}\) equal to \({{\texttt {Z}}}^{(i, g)}_{\mathcal {H}}\). (Again \(U={\mathcal {H}}\).)
For each \(\varSigma ^{\text {ord}}\) as above (and each \({\mathcal {H}}\) as above; see the beginning of Sect. B.12), we have a smooth quasi-projective scheme
over \({{\text {Spec}}\,}{\mathbb {Z}_{(p)}}\), which contains \(\vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\) as an open fibrewise dense subscheme, and is called a partial toroidal compactification of \(\vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\). Its characteristic zero pull-back \({\vec {\mathsf {M}}}^{{\text {ord}}, \text {tor}}_{{\mathcal {H}}, \varSigma ^{\text {ord}}} \otimes _{\mathbb {Z}} {\mathbb {Q}}\) is an open subscheme of \({\mathsf {M}}^{\text {tor}}_{{\mathcal {H}}, \varSigma }\), for any \(\varSigma \) extending \(\varSigma ^{\text {ord}}\), which can be identified with the union of the strata \(\mathsf {Z}_{[(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}, \sigma )]}\) indexed by equivalence classes whose underlying cusp labels \([(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]\) are ordinary (see theorems 5.2.1.1 and 6.2.3.1 and remark 5.2.1.5 of [44]). The union
(by gluing along their common open subscheme described above) is the
in this paper (see Sect. 5.3), with \(U^p(N_1,N_2)={\mathcal {H}}\) and with \((U^p(N_1,N_2),\varDelta ) \in {\mathcal {J}}_n^{{\text {tor}}}\) induced by \(\varSigma \) as in Sect. B.5. In this case, the \((U^p(N_1),\varDelta ^{{\text {ord}}}) \in {\mathcal {J}}_n^{{{\text {tor}}},{{\text {ord}}}}\) induced by \((U^p(N_1,N_2),\varDelta )\) is induced by the \(\varSigma ^{\text {ord}}\) induced by \(\varSigma \). The collection \(\{ {\vec {\mathsf {M}}}^{{\text {ord}}, \text {tor}}_{{\mathcal {H}}, \varSigma ^{\text {ord}}} \}_{({\mathcal {H}}, \varSigma ^{\text {ord}})}\) admits compatible actions of \(\mathrm {G}({\mathbb {A}^{\infty , p}}) \times \mathrm {P}^{\text {ord}}_{\texttt {D}}(\mathbb {Z}_p)\) and of the element of \(\mathrm {P}^{\text {ord}}_{\texttt {D}}(\mathbb {Q}_p)\) corresponding to \(\varsigma _p\) (see proposition 5.2.2.2 of [44]), which are compatible with the action of \(G_n({\mathbb {A}}^\infty )^{{\text {ord}}}\) on \(\{ {\mathcal {X}}_{n,U^p(N_1,N_2),\varDelta } \}_{(U^p(N_1,N_2),\varDelta )}\).
The partial toroidal compactification \({\vec {\mathsf {M}}}^{{\text {ord}}, \text {tor}}_{{\mathcal {H}}, \varSigma ^{\text {ord}}}\) admits a stratification by locally closed subschemes
(see theorem 5.2.1.1(2) of [44]) indexed by equivalence classes \([(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}, \sigma )]\) as in definition 1.2.2.10 of [44] whose underlying cusp labels \([(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]\) are ordinary. Each stratum \(\vec {\mathsf {Z}}^{\text {ord}}_{[(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}, \sigma )]}\) is canonically isomorphic to \(\vec {\varXi }^{\text {ord}}_{\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}, \sigma }\), and the formal completion of \({\vec {\mathsf {M}}}^{{\text {ord}}, \text {tor}}_{{\mathcal {H}}, \varSigma ^{\text {ord}}}\) along the union of the strata \(\vec {\mathsf {Z}}^{\text {ord}}_{[(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}, \sigma )]}\) labelled by equivalence classes \([(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}, \sigma )]\) with the same underlying ordinary cusp label \([(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]\) is canonically isomorphic to \(\vec {\mathfrak {X}}^{\text {ord}}_{\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}} / \varGamma _{\varPhi _{\mathcal {H}}}\) (see lemma 5.2.4.38 of [44]).
When \(U^p(N_1,N_2)={\mathcal {H}}\), and when \((U^p(N_1),\varDelta ^{{\text {ord}}}) \in {\mathcal {J}}_n^{{{\text {tor}}},{{\text {ord}}}}\) is induced by \(\varSigma ^{\text {ord}}\) as above, the formal completion of \({\vec {\mathsf {M}}}^{{\text {ord}}, \text {tor}}_{{\mathcal {H}}, \varSigma ^{\text {ord}}}\) along the union of all strata \(\vec {\mathsf {Z}}^{\text {ord}}_{[(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}, \sigma )]}\) labelled by equivalence classes \([(\varPhi _{\mathcal {H}}, \delta _{\mathcal {H}}, \sigma )]\) with underlying \({\texttt {Z}}_{\mathcal {H}}\) equal to \({{\texttt {Z}}}^{(i, g)}_{\mathcal {H}}\) for some \(g \in \mathrm {G}({\mathbb {A}^{\infty , p}}) \times \mathrm {P}^{\text {ord}}_{\texttt {D}}(\mathbb {Z}_p)\) is
in this paper, whose union with \(X_{n,U^p(N_1,N_2),\varDelta ,i}^\wedge \) is
(see Sect. 5.3). It can be identified with
where g runs over \(P_{n,(i)}^+({\mathbb {A}}^\infty )^{{{\text {ord}}},\times } \backslash G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times } / U^p(N_1)\) and h runs over
These identifications are all Hecke equivariant (see proposition 5.2.4.41 of [44]).
The formal completion of \({\vec {\mathsf {M}}}^{{\text {ord}}, \text {tor}}_{{\mathcal {H}}, \varSigma ^{\text {ord}}}\) along
is denoted
in [44] (see definition 3.4.4.2). When \(U^p(N_1,N_2)={\mathcal {H}}\), and when \((U^p(N_1),\varDelta ) \in {\mathcal {J}}_n^{{{\text {tor}}},{{\text {ord}}}}\) induced by \(\varSigma ^{\text {ord}}\) as above, these are denoted \(\overline{{X}}_{n,U^p(N_1),\varDelta }^{{\text {ord}}}\) and \({\mathfrak {X}}_{n,U^p(N_1),\varDelta }^{{\text {ord}}}\), respectively, in this paper (see Sect. 5.3). (Their independence of \(N_2={r_0}\) is explained in corollary 5.2.2.4 of [44].) For the Hecke actions on these formal schemes, see corollaries 5.2.2.3, 5.2.2.4 and 5.2.2.5 of [44].
1.14 Ordinary loci of Kuga families
Recall the choice of \(Q= {\mathcal {O}}_F^m\) in Sect. B.6, and the associated algebraic groups. For a \({\mathbb {Z}}_p\)-algebra R, we set
to be the pre-image of \(\mathrm {P}^{\text {ord}}_{\texttt {D}}(R)\) under the homomorphism \(\widehat{\mathrm {G}}(R) \twoheadrightarrow \mathrm {G}(R)\). This is \(P_{n,(n)}^{(m),+}(R)\) in the notation of this paper. Thus,
For all integers \(0\le r\) and \(0 \le {r_1}\le {r_0}\), we set:
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\(\widehat{\mathcal {U}}_{p, 0}(p^r) := ( \widehat{\mathrm {G}}(\mathbb {Z}_p) \rightarrow \widehat{\mathrm {G}}(\mathbb {Z}/p^r\mathbb {Z}) )^{-1} ( \widehat{\mathrm {P}}^{\text {ord}}_{\texttt {D}}(\mathbb {Z}/p^r\mathbb {Z}) ) = U_p(0,r)_n^{(m)}\).
-
(2)
\(\widehat{\mathcal {U}}_{p, 1}(p^r) := U_p(r,r)_n^{(m)}\).
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(3)
\(\widehat{\mathcal {U}}_{p, 1}^{\text {bal}}(p^r) := \ker (\nu : \widehat{\mathcal {U}}_{p, 1}(p^r) \rightarrow ({\mathbb {Z}}/p^r{\mathbb {Z}})^\times )\).
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\(\widehat{\mathcal {U}}_{p, 1, 0}(p^{r_1}, p^{r_0}) := \widehat{\mathcal {U}}_{p, 1}(p^{r_1}) \cap \widehat{\mathcal {U}}_{p, 0}(p^{r_0}) = U_p(r_1,r_0)_n^{(m)}\).
(The first and third of these definitions are consistent with definition 7.1.1.2 of [44].) An open compact subgroup \(\widehat{{\mathcal {H}}}_p\subset \widehat{G}({\mathbb {Q}}_p)\) is said to be of standard form and of depth \(r\) if
(see definition 7.1.1.2 of [44]).
The theory in [44] is developed for open compact subgroups \(\widehat{{\mathcal {H}}}\) of \(\widehat{\mathrm {G}}({\hat{\mathbb {Z}}})\) of the form \(\widehat{{\mathcal {H}}} = \widehat{{\mathcal {H}}}^p\widehat{{\mathcal {H}}}_p\), where \(\widehat{{\mathcal {H}}}^p\) is a neat open compact subgroup of \(\widehat{\mathrm {G}}({\hat{\mathbb {Z}}^{p}})\), and where \(\widehat{{\mathcal {H}}}_p\) is an open compact subgroup of \(\widehat{\mathrm {G}}(\mathbb {Z}_p)\) of standard form. In this paper, we will only need \(\widehat{{\mathcal {H}}}\) of the form \(U^p(N_1,N_2)\), which satisfies the above requirement with \(\widehat{{\mathcal {H}}}^p= U^p\) and \(\widehat{{\mathcal {H}}}_p= \widehat{\mathcal {U}}_{p, 1, 0}(p^{N_1}, p^{N_2})\). Then \(r_{\widehat{{\mathcal {H}}}_\mathrm {G}}=0\) and \(\widehat{{\mathcal {H}}}\) satisfies conditions 7.1.1.4 and 7.1.1.5 of [44]. To each \(\widehat{{\mathcal {H}}}\) as above with image \({\mathcal {H}}= \widehat{{\mathcal {H}}}_\mathrm {G}\) in \(\mathrm {G}({\hat{\mathbb {Z}}})\), [44] attached a generalized ordinary Kuga family
(see definition 7.1.3.2 of [44]). Its characteristic zero pull-back is a generalized Kuga family \(\mathsf {N}\rightarrow \mathsf {M}_{\mathcal {H}}\) as in Sect. B.6. Since \(Q={\mathcal {O}}_F^m\), it is the scheme denoted \({\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)}\) in this paper. The generalized ordinary Kuga family attached to \(\widehat{{\mathcal {H}}}_{\widehat{\mathrm {G}}} \ltimes \widehat{{\mathcal {H}}}_{\widehat{\mathrm {U}}}\) is denoted
and is a Kuga family as in definition 7.1.3.1 of [44]. It is an abelian scheme and, since \(Q={\mathcal {O}}_F^m\), it is \({\mathbb {Q}}^\times \)-isogenous to the m-fold fibre product of the universal abelian scheme over \(\vec {\mathsf {M}}^{\text {ord}}_{\widehat{{\mathcal {H}}}_\mathrm {G}}\). Its characteristic zero pull-back is a Kuga family \(\mathsf {N}^{\text {grp}}\rightarrow \mathsf {M}_{\mathcal {H}}\) as in Sect. B.6. The generalized ordinary Kuga family \(\vec {\mathsf {N}}^{\text {ord}} \rightarrow \vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\) is a torsor for \(\vec {\mathsf {N}}^{{\text {ord}},{\text {grp}}} \rightarrow \vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\).
To study these schemes and their compactification [44] again realizes them in a non-canonical way inside the boundary of a larger Shimura variety. We will continue to use the notation of Sect. B.6. We consider the maximal totally isotropic filtration
of \(\widetilde{L}\otimes _{\mathbb {Z}} {\mathbb {Z}_p}\) (see the beginning of section 7.1.1 of [44]). We have
We define (see definition 7.1.1.22 of [44]):
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\(\widetilde{\mathrm {P}}^{\text {ord}}_{\widetilde{{\texttt {Z}}}, \widetilde{{\texttt {D}}}}(R) := \widetilde{\mathrm {P}}_{\widetilde{{\texttt {Z}}}}(R) \cap \widetilde{\mathrm {P}}^{\text {ord}}_{\widetilde{{\texttt {D}}}}(R)\).
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(2)
\(\widetilde{\mathrm {P}}^{{\text {ord}}, \prime }_{\widetilde{{\texttt {Z}}}, \widetilde{{\texttt {D}}}}(R) := \widetilde{\mathrm {P}}_{\widetilde{{\texttt {Z}}}}'(R) \cap \widetilde{\mathrm {P}}^{\text {ord}}_{\widetilde{{\texttt {D}}}}(R)=\widetilde{{P}}^{(m),+}_{n,(n)}\).
For any such \(\widehat{{\mathcal {H}}}\), we can always find some
with \(\widetilde{{\mathcal {H}}}^p\) neat such that \(\widehat{{\mathcal {H}}} = \widetilde{{\mathcal {H}}}_{\widehat{\mathrm {G}}}\) (see Sect. B.6) and
(Then \({r_{\widetilde{{\mathcal {H}}}}}=0\).) In this case, the abelian scheme torsor
depends only on \(\widehat{{\mathcal {H}}}\) (but not on the auxiliary choice of \(\widetilde{{\mathcal {H}}}\)) and equals the above generalized ordinary Kuga family \(\vec {\mathsf {N}}^{\text {ord}} \rightarrow \vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\) attached to \(\widehat{{\mathcal {H}}}\).
1.15 Partial toroidal compactifications of ordinary loci of Kuga families
Let us fix the choice of some \((\widetilde{{\texttt {Z}}}, \widetilde{\varPhi }, \widetilde{\delta })\) as in Sect. B.6. Then the bijection described at the start of Sect. B.7 restricts to a bijection between the following two sets:
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The fully symplectic admissible filtrations \(\breve{{\texttt {Z}}}\) of \(\widetilde{L}\otimes _{\mathbb {Z}} {{\hat{\mathbb {Z}}}}\) compatible with \(\widetilde{{\texttt {D}}}\) such that \(0 \subset \widetilde{{\texttt {Z}}}_{-2} \subset \breve{{\texttt {Z}}}_{-2} \subset \breve{{\texttt {Z}}}_{-1} \subset \widetilde{{\texttt {Z}}}_{-1} \subset \widetilde{L}\otimes _{\mathbb {Z}} {{\hat{\mathbb {Z}}}}\).
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(2)
The fully symplectic admissible filtrations \({\texttt {Z}}\) of \(L\otimes _{\mathbb {Z}} {{\hat{\mathbb {Z}}}}\) compatible with \({\texttt {D}}\) such that \({\texttt {Z}}_{-2} = \breve{{\texttt {Z}}}_{-2}/\widetilde{{\texttt {Z}}}_{-2}\).
(Recall that the notation \(\breve{~}\) will always mean objects related to such a filtration \(\breve{{\texttt {Z}}}\).)
For each \(\breve{{\texttt {Z}}}\) as above, and for each \(\mathbb {Z}_p\)-algebra R, we define the following quotients of subgroups of \(\widehat{\mathrm {P}}_{\breve{{\texttt {Z}}}}(R)\) (see definition 7.1.1.27 of [44]):
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\(\widehat{\mathrm {P}}^{\text {ord}}_{\breve{{\texttt {Z}}}, {\texttt {D}}}(R) := (\widetilde{\mathrm {P}}^{\text {ord}}_{\breve{{\texttt {Z}}}, \widetilde{{\texttt {D}}}}(R) \cap \widetilde{\mathrm {P}}_{\widetilde{{\texttt {Z}}}}'(R)) / \widetilde{\mathrm {U}}_{2, \widetilde{{\texttt {Z}}}}(R)\).
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(2)
\(\widehat{\mathrm {P}}^{{\text {ord}}, \prime }_{\breve{{\texttt {Z}}}, {\texttt {D}}}(R) := \widetilde{\mathrm {P}}^{{\text {ord}}, \prime }_{\breve{{\texttt {Z}}}, \widetilde{{\texttt {D}}}}(R) / \widetilde{\mathrm {U}}_{2, \widetilde{{\texttt {Z}}}}(R)\).
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(3)
\(\widehat{\mathrm {P}}^{\text {ord}}_{1, \breve{{\texttt {Z}}}, {\texttt {D}}}(R) := \widehat{\mathrm {P}}^{{\text {ord}}, \prime }_{\breve{{\texttt {Z}}}, {\texttt {D}}}(R) / \widehat{\mathrm {U}}_{2, \breve{{\texttt {Z}}}}(R)\).
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\(\widehat{\mathrm {P}}^{{\text {ord}}, \prime }_{h, \breve{{\texttt {Z}}}, {\texttt {D}}}(R) := \widehat{\mathrm {P}}^{\text {ord}}_{1, \breve{{\texttt {Z}}}, {\texttt {D}}}(R) / \widehat{\mathrm {U}}_{1, \breve{{\texttt {Z}}}}(R) \cong \widehat{\mathrm {P}}^{\text {ord}}_{h, \breve{{\texttt {Z}}}, {\texttt {D}}}(R)\).
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(5)
\(\widehat{\mathrm {P}}^{\text {ord}}_{h, \breve{{\texttt {Z}}}, {\texttt {D}}}(R)\) denotes the subgroup of elements of \(\widehat{\mathrm {G}}_{h, \breve{{\texttt {Z}}}}(R) \cong \mathrm {G}_{h, {\texttt {Z}}}(R)\) which preserve \(\widetilde{{\texttt {D}}}_{-1} = {\texttt {D}}\subset {\text {Gr}}^{\breve{{\texttt {Z}}}}_{-1}\otimes _{{\hat{\mathbb {Z}}}} {\mathbb {Z}_p} \cong {\text {Gr}}^{\texttt {Z}}_{-1}\otimes _{{\hat{\mathbb {Z}}}} {\mathbb {Z}_p}\). Thus \(\widehat{\mathrm {P}}^{{\text {ord}}, \prime }_{h, \breve{{\texttt {Z}}}, {\texttt {D}}}(R) \mathop {\rightarrow }\limits ^{\sim }\widehat{\mathrm {P}}^{\text {ord}}_{h, \breve{{\texttt {Z}}}, {\texttt {D}}}(R)\).
When \({\texttt {Z}}= {{\texttt {Z}}}^{(i, 1)}\), in which case \(\breve{{\texttt {Z}}} = {\widetilde{{\texttt {Z}}}}^{(i+m, 1)} = {\breve{{\texttt {Z}}}}^{(i, 1)}\), we have, for example,
in the notation of this paper (see Sect. 1.2).
The filtration \({\breve{{\texttt {Z}}}}^{(i, \widehat{g})}\) is compatible with \(\widetilde{{\texttt {D}}}\) if and only if
where \(\widehat{\mathrm {P}}^{\text {ord}}_{\texttt {D}}(\mathbb {Q}_p) = \widetilde{\mathrm {P}}^{\text {ord}}_{1, \widetilde{{\texttt {Z}}}, \widetilde{{\texttt {D}}}}(\mathbb {Q}_p)\).
Now suppose \(\widehat{{\mathcal {H}}}\) is an open compact subgroup \(\widehat{{\mathcal {H}}}\) of \(\widehat{\mathrm {G}}({\hat{\mathbb {Z}}})\) of the form \(\widehat{{\mathcal {H}}} = \widehat{{\mathcal {H}}}^p\widehat{{\mathcal {H}}}_p\), where \(\widehat{{\mathcal {H}}}^p\) is a neat open compact subgroup of \(\widehat{\mathrm {G}}({\hat{\mathbb {Z}}^{p}})\), and where \(\widehat{{\mathcal {H}}}_p= \widehat{\mathcal {U}}_{p, 1, 0}(p^{r_1}, p^{r_0})\) for some integers \(0\le {r_1}\le {r_0}\). The \(\widehat{{\mathcal {H}}}\)-orbits \({\breve{{\texttt {Z}}}}^{(i, \widehat{g})}_{\widehat{{\mathcal {H}}}}\) containing a filtration compatible with \(\widetilde{{\texttt {D}}}\) are parameterized by the image of \(\widehat{\mathrm {G}}({\mathbb {A}^{\infty , p}}) \times \widehat{\mathrm {P}}^{\text {ord}}_{{\breve{{\texttt {Z}}}}^{(i, 1)}}(\mathbb {Q}_p) \widehat{\mathrm {P}}^{\text {ord}}_{\texttt {D}}(\mathbb {Q}_p)\) in \(\widehat{\mathrm {P}}^{\text {ord}}_{{\breve{{\texttt {Z}}}}^{(i, 1)}}({\mathbb {A}^{\infty }}) \backslash \widehat{\mathrm {G}}({\mathbb {A}^{\infty }}) / \widehat{{\mathcal {H}}}\). As in Sect. 5.1, this set is in bijection with the double coset space
which is the double coset space
in this paper (see, e.g., Sect. 1.4), with \(U^p(N_1,N_2)=\widehat{{\mathcal {H}}}\). It maps isomorphically to
The correspondence of cusp labels from Sect. B.7 sets up bijections between the following sets (see lemmas 1.2.4.15 and 1.2.4.16, definitions 1.2.4.17 and 3.2.3.8, and lemma 7.1.1.8 of [44]):
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The ordinary cusp labels \([(\breve{{\texttt {Z}}}_{\widetilde{{\mathcal {H}}}}, \breve{\varPhi }_{\widetilde{{\mathcal {H}}}}, \breve{\delta }_{\widetilde{{\mathcal {H}}}})]\) for \(\widetilde{\mathsf {M}}_{\widetilde{{\mathcal {H}}}}\) such that the stratum \(\widetilde{\mathsf {Z}}_{[(\breve{\varPhi }_{\widetilde{{\mathcal {H}}}}, \breve{\delta }_{\widetilde{{\mathcal {H}}}})]}\) of \({\widetilde{\mathsf {M}}}^{\text {min}}_{\widetilde{{\mathcal {H}}}}\) is contained in the closure of \(\widetilde{\mathsf {Z}}_{[(\widetilde{\varPhi }_{\widetilde{{\mathcal {H}}}}, \widetilde{\delta }_{\widetilde{{\mathcal {H}}}})]}\).
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(2)
The \(\widehat{{\mathcal {H}}}\)-orbits, \([(\breve{{\texttt {Z}}}_{\widehat{{\mathcal {H}}}}, \breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}})]\), of equivalence classes of \((\breve{{\texttt {Z}}}, \breve{\varPhi }, \breve{\delta })\), where \(\breve{{\texttt {Z}}}\) and \(\breve{\varPhi }\) are compatible with \(\widetilde{{\texttt {Z}}}\) and \(\widetilde{\varPhi }\), as described in Sect. B.7, and with \(\widetilde{{\texttt {D}}}\). We call such an orbit an ordinary cusp label for \((\widetilde{L}, \langle {\,\cdot \,,\,\cdot \,}\rangle \widetilde{~}, \widetilde{h}_0, \widetilde{{\texttt {Z}}}, \widetilde{{\texttt {D}}})\).
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(3)
The ordinary cusp labels \([({\texttt {Z}}_{\mathcal {H}}, \varPhi _{\mathcal {H}}, \delta _{\mathcal {H}})]\) for \(\mathsf {M}_{\mathcal {H}}\).
The stratum \(\vec {\widetilde{\mathsf {Z}}}^{\text {ord}}_{[(\breve{\varPhi }_{\widetilde{{\mathcal {H}}}}, \breve{\delta }_{\widetilde{{\mathcal {H}}}})]} \cong \vec {\widetilde{\mathsf {M}}}_{\widetilde{{\mathcal {H}}}}^{{\text {ord}}, \widetilde{{\texttt {Z}}}_{\widetilde{{\mathcal {H}}}}}\), the finite étale cover \(\vec {\widetilde{\mathsf {M}}}_{\widetilde{{\mathcal {H}}}}^{{\text {ord}}, \widetilde{\varPhi }_{\widetilde{{\mathcal {H}}}}} \rightarrow \vec {\widetilde{\mathsf {M}}}_{\widetilde{{\mathcal {H}}}}^{{\text {ord}}, \widetilde{{\texttt {Z}}}_{\widetilde{{\mathcal {H}}}}}\), the abelian scheme torsor \(\vec {\widetilde{C}}^{\text {ord}}_{\widetilde{\varPhi }_{\widetilde{{\mathcal {H}}}}, \widetilde{\delta }_{\widetilde{{\mathcal {H}}}}} \rightarrow \vec {\widetilde{\mathsf {M}}}_{\widetilde{{\mathcal {H}}}}^{{\text {ord}}, \widetilde{\varPhi }_{\widetilde{{\mathcal {H}}}}}\), and the abelian scheme \(\vec {\widetilde{C}}_{\widetilde{\varPhi }_{\widetilde{{\mathcal {H}}}}, \widetilde{\delta }_{\widetilde{{\mathcal {H}}}}}^{{\text {ord}}, {\text {grp}}} \rightarrow \vec {\widetilde{\mathsf {M}}}_{\widetilde{{\mathcal {H}}}}^{{\text {ord}}, \widetilde{\varPhi }_{\widetilde{{\mathcal {H}}}}}\) depend (up to canonical isomorphism) only on the \(\widehat{{\mathcal {H}}}\)-orbit of \((\breve{{\texttt {Z}}}, \breve{\varPhi }, \breve{\delta })\) (see lemma 7.1.2.1 of [44]), and hence we shall denote them by
respectively. For a fixed \((\breve{{\texttt {Z}}}, \breve{\varPhi }, \breve{\delta })\), the morphisms
are equivariant with
and with the compatible actions of the elements corresponding to \(\varsigma _p\) (see propositions 5.2.4.25 and 7.1.2.5 of [44]).
The cusp labels \([(\breve{{\texttt {Z}}}_{\widehat{{\mathcal {H}}}}, \breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}})]\) such that \(\breve{{\texttt {Z}}}_{\widehat{{\mathcal {H}}}} = {\breve{{\texttt {Z}}}}^{(i, \widehat{g})}_{\widehat{{\mathcal {H}}}}\) for some \(\widehat{g} \in \widehat{\mathrm {G}}({\mathbb {A}^{\infty , p}}) \times \widehat{\mathrm {P}}^{\text {ord}}_{\texttt {D}}(\mathbb {Z}_p)\) are parameterized by the double coset space
and the forgetful map sending \([(\breve{{\texttt {Z}}}_{\widehat{{\mathcal {H}}}}, \breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}})]\) to \(\breve{{\texttt {Z}}}_{\widehat{{\mathcal {H}}}}\) can be identified with the canonical map from this double coset space to
whose fibre above the double coset of \(\widehat{g} \in \widehat{\mathrm {G}}({\mathbb {A}^{\infty , p}}) \times \widehat{\mathrm {P}}^{\text {ord}}_{\texttt {D}}(\mathbb {Z}_p)\) can be identified with
(See Sect. B.7 for the definition of \(\widehat{\mathrm {G}}_{l, {\breve{\varPhi }}^{(i, 1)}}'\).) This last double coset space is the index set of
in this paper (see Sect. 4.1), with \(g=\widehat{g}\) and \(U^p(N_1,N_2)=\widehat{{\mathcal {H}}}\).
If \(U^p(N_1,N_2)=\widehat{{\mathcal {H}}}\) is a neat open compact subgroup of \(G_n^{(m)}({\mathbb {A}}^\infty )=\widehat{\mathrm {G}}({\mathbb {A}^{\infty }})\) as above, if \(g=\widehat{g} \in G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times } = \widehat{\mathrm {G}}({\mathbb {A}^{\infty , p}}) \times \widehat{\mathrm {P}}^{\text {ord}}_{\texttt {D}}(\mathbb {Z}_p)\) and if \(U_g^{p,\prime }=gU^pg^{-1} \cap P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )\), then the maps
in this paper (see Sects. 4.1, 4.2) are identified with
For each representative \((\breve{{\texttt {Z}}}_{\widehat{{\mathcal {H}}}}, \breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}})\) of an ordinary cusp label \([(\breve{{\texttt {Z}}}_{\widehat{{\mathcal {H}}}}, \breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}})]\) as above, there is a torsor
under the pull-back of the same split torus \(\widehat{E}_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}\) over \({{\text {Spec}}\,}\mathbb {Z}\) with character group \(\widehat{\mathbf {S}}_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}\) as before (see Sect. B.7; and see proposition 7.1.2.6 of [44]). We obtain an isomorphic torus torsor if we replace \((\breve{{\texttt {Z}}}_{\widehat{{\mathcal {H}}}}, \breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}})\) with another representative, but its universal property depends on this choice of representative.
The torus
in this paper (see Sect. 4.3) with \(U^p(N_1,N_2)={\mathcal {H}}\) is identified with
Moreover, the sheaves \(X^*({\mathcal {S}}^{(m),{{\text {ord}}},+}_{n,(i),U_g^{p,\prime }(N_1)})\), \(X_*({\mathcal {S}}^{(m),{{\text {ord}}},+}_{n,(i),U_g^{p,\prime }(N_1)})_{\mathbb {R}}^{\succ 0}\), and \(X_*({\mathcal {S}}^{(m),{{\text {ord}}},+}_{n,(i),U_g^{p,\prime }(N_1)})_{\mathbb {R}}^{>0}\) are identified with
and
respectively. The \({\mathcal {S}}^{(m),{{\text {ord}}},+}_{n,(i),U_g^{p,\prime }(N_1)}\)-torsor
is identified with
These maps are equivariant with \(\widehat{\mathrm {P}}_{{\breve{{\texttt {Z}}}}^{(i, g)}}({\mathbb {A}^{\infty , p}}) \times \widehat{\mathrm {P}}^{\text {ord}}_{{\breve{{\texttt {Z}}}}^{(i, g)}, {\texttt {D}}}(\mathbb {Z}_p)\) and with the compatible actions of the elements corresponding to \(\varsigma _p\) (see proposition 1.3.2.67 of [44]).
In lemma 7.1.1.9 of [44], we have introduced an ad hoc definition of a compatible collection
of admissible projective smooth rational polyhedral cone decompositions, where each \(\widehat{\varSigma }_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}\) is a \(\varGamma _{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}\)-admissible projective smooth rational polyhedral cone decomposition of \(\widehat{\mathbf {P}}_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}\), and where the indices are ordinary cusp labels \([(\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}})]\). In [44], we allow \(\widehat{{\mathcal {H}}}_p\) to be all subgroups of \(\widehat{\mathrm {G}}(\mathbb {Z}_p)\) of standard form (which is more general than just of the form \(\widehat{\mathcal {U}}_{p, 1, 0}(p^{r_1}, p^{r_0})\) for some integers \(0\le {r_1}\le {r_0}\)), and the set of pairs \(\kappa = (\widehat{{\mathcal {H}}}, \widehat{\varSigma }^{\text {ord}})\) with \(\widehat{{\mathcal {H}}}_\mathrm {G}\subset {\mathcal {H}}\) is denoted \(\mathbf {K}_{Q, {\mathcal {H}}}^{{\text {ord}}, ++}\); the subset of \(\mathbf {K}_{Q, {\mathcal {H}}}^{{\text {ord}}, ++}\) consisting of \(\kappa = (\widehat{{\mathcal {H}}}, \widehat{\varSigma }^{\text {ord}})\) with \(\widehat{{\mathcal {H}}}_\mathrm {G}= {\mathcal {H}}\) is denoted \(\mathbf {K}_{Q, {\mathcal {H}}}^{{\text {ord}}, +}\); and the subset of \(\mathbf {K}_{Q, {\mathcal {H}}}^{{\text {ord}}, +}\) consisting of \(\kappa = (\widehat{{\mathcal {H}}}, \widehat{\varSigma }^{\text {ord}})\) with \(\widehat{{\mathcal {H}}}_\mathrm {G}= {\mathcal {H}}\) and \(\widehat{{\mathcal {H}}} = {\mathcal {H}}\ltimes \widehat{{\mathcal {H}}}_{\widehat{\mathrm {U}}}\) is denoted \(\mathbf {K}_{Q, {\mathcal {H}}}^{\text {ord}}\) (see definitions 7.1.1.7 and 7.1.1.11 of [44]). For \(? = \emptyset \), \(+\) or \(++\), any pair \(\kappa = (\widehat{{\mathcal {H}}}, \widehat{\varSigma })\) in \(\mathbf {K}_{Q, {\mathcal {H}}}^?\) introduced earlier in Sect. B.7 such that \(\widehat{{\mathcal {H}}}\) is of the form allowed here induces a pair \((\widehat{{\mathcal {H}}}, \widehat{\varSigma }^{\text {ord}})\) in \(\mathbf {K}_{Q, {\mathcal {H}}}^{{\text {ord}}, ?}\), and conversely any pair in \(\mathbf {K}_{Q, {\mathcal {H}}}^{{\text {ord}}, ?}\) extends to a pair in \(\mathbf {K}_{Q, {\mathcal {H}}}^?\) (see proposition 7.1.1.21 of [44]). For any compatible collection \(\varSigma ^{\text {ord}}\) of admissible projective smooth rational polyhedral cone decompositions for \(\vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\), and for \(?=\emptyset \), \(+\), \(++\), the subset of \(\mathbf {K}_{Q, {\mathcal {H}}}^{{\text {ord}}, ?}\) consisting of \(\kappa = (\widehat{{\mathcal {H}}}, \widehat{\varSigma }^{\text {ord}})\) such that \(\widehat{\varSigma }^{\text {ord}}\) is compatible with \(\varSigma ^{\text {ord}}\) in the sense that each \(\widehat{\rho } \in \widehat{\varSigma }_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}\) is mapped into some \(\sigma \in \varSigma _{\varPhi _{\mathcal {H}}}\) (see condition 7.1.1.17 and definition 7.1.1.19 of [44]) is denoted \(\mathbf {K}_{Q, {\mathcal {H}}, \varSigma ^{\text {ord}}}^{{\text {ord}}, ?}\). This notion of compatibility agrees with the one in this paper (see Sect. 5.2).
Each such \(\kappa = (\widehat{{\mathcal {H}}}, \widehat{\varSigma }^{\text {ord}})\) in \(\mathbf {K}_{Q, {\mathcal {H}}}^{{\text {ord}}, ++}\) induces a pair
in \({\mathcal {J}}_n^{(m),{{\text {tor}}},{{\text {ord}}}}\) (with \(U^p(N_1,N_2)=\widehat{{\mathcal {H}}})\) in Sect. 5.2 of this paper, because, in order to define \((U^p(N_1),\varSigma )\) as in Sect. 5.2, it suffices to define the admissible cone decomposition \(\varSigma (g)_0\) for \(X_*({\mathcal {S}}_{n,(i),(gU^pg^{-1}\cap P_{n,(i)}^{(m),+}({\mathbb {A}}^{\infty ,p}))(N_1)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\succ 0}\), for each \(g\in G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times } = \widehat{\mathrm {G}}({\mathbb {A}^{\infty , p}}) \times \widehat{\mathrm {P}}^{\text {ord}}_{\texttt {D}}(\mathbb {Z}_p)\), which can be taken to be the pull-back of the subcollection \(\{ \widehat{\varSigma }_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}} \}_{[(\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}})]}\) of \(\widehat{\varSigma }^{\text {ord}}\) indexed by the ordinary cusp labels \([(\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}})]\) with underlying \(\breve{{\texttt {Z}}}_{\widehat{{\mathcal {H}}}}\) equal to \({\breve{{\texttt {Z}}}}^{(i, g)}_{\widehat{{\mathcal {H}}}}\). In fact, \({\mathcal {J}}_n^{(m),{{\text {tor}}},{{\text {ord}}}}\) is exactly the set of such induced pairs (as \(U^p(N_1,N_2)=\widehat{{\mathcal {H}}}\) varies). (As before, it is hard to explicitly describe the set \({\mathcal {J}}_n^{(m),{{\text {tor}}},{{\text {ord}}}}\) when \(m>0\), because they are induced by auxiliary choices of compatible collections \(\widetilde{\varSigma }^{\text {ord}}\) for \(\vec {\mathsf {M}}^{\text {ord}}_{\widetilde{{\mathcal {H}}}}\). Nevertheless, this is unnecessary for our purpose.)
Each \(\widehat{\varSigma }_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}\) defines an affine toroidal embedding
over \(\vec {\widehat{C}}^{\text {ord}}_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}}}\). The formal completion of \(\overline{\vec {\widehat{\varXi }}}_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}}, \widehat{\varSigma }_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}}^{\text {ord}}\) along the union of the \(\widehat{\rho }\)-strata \(\vec {\widehat{\varXi }}^{\text {ord}}_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}}, \widehat{\rho }}\) for all \(\widehat{\rho } \in \widehat{\varSigma }_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}\) such that \(\widehat{\rho } \cap \widehat{\mathbf {P}}_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}^+ \ne \emptyset \) is denoted
(see (7.1.2.12), (7.1.2.13), (7.1.2.14) and (7.1.2.16) of [44]). The schemes
of this paper (see Sect. 4.4) are identified with
Moreover, \({\mathcal {T}}^{(m),{{\text {ord}}},+,\wedge }_{n,(i),U_g^{p,\prime }(N_1,N_2),\varSigma (g)_0}\) is identified with
and \({\mathcal {T}}^{(m),{{\text {ord}}},\natural ,\wedge }_{n,(i),U_g^{p,\prime }(N_1),\varSigma (g)_0}\) is identified with
where the second disjoint union is over cusp labels with underlying \(\breve{{\texttt {Z}}}_{\widehat{{\mathcal {H}}}}\) equal to \({\breve{{\texttt {Z}}}}^{(i, \widehat{g})}_{\widehat{{\mathcal {H}}}}\) for a fixed \(\widehat{g}\) in \(\widehat{\mathrm {G}}({\mathbb {A}^{\infty , p}})\times \widehat{\mathrm {P}}^{\text {ord}}_{\texttt {D}}(\mathbb {Z}_p)\) corresponding to \(g \in G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\). (Again \(\widehat{{\mathcal {H}}}=U^p(N_1,N_2)\).)
For each \(\kappa = (\widehat{{\mathcal {H}}}, \widehat{\varSigma }^{\text {ord}}) \in \mathbf {K}_{Q, {\mathcal {H}}}^{{\text {ord}}, ++}\), we have a smooth quasi-projective scheme
containing \(\vec {\mathsf {N}}^{\text {ord}}\) (of Sect. B.14, which we will henceforth write as \(\vec {\mathsf {N}}^{\text {ord}}_\kappa \) to emphasize the dependence on \(\widehat{{\mathcal {H}}}\)) as an open fibrewise dense subscheme. The scheme \({\vec {\mathsf {N}}}^{{\text {ord}}, \text {tor}}_\kappa \) is called a partial toroidal compactification of \(\vec {\mathsf {N}}^{\text {ord}}_\kappa \) (see theorem 7.1.4.1 of [44]). Its characteristic zero fibre \({\vec {\mathsf {N}}}^{{\text {ord}}, \text {tor}}_\kappa \otimes _{\mathbb {Z}} {\mathbb {Q}}\) is an open subscheme of \({\vec {\mathsf {N}}}^{\text {tor}}_{\kappa '}\) for any \({\kappa '}= (\widehat{{\mathcal {H}}}, \widehat{\varSigma }) \in {\mathbf K}_{Q, {\mathcal {H}}}^{++}\) extending \(\kappa = (\widehat{{\mathcal {H}}}, \widehat{\varSigma }^{\text {ord}}) \in {\mathbf K}_{Q, {\mathcal {H}}}^{{\text {ord}}, ++}\), which is identified with the union of the strata \(\widehat{\mathsf {Z}}_{[(\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}}, \widehat{\rho })]}\) indexed by equivalence classes \([(\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}}, \widehat{\rho })]\) whose underlying cusp labels \([(\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}})]\) are ordinary (see theorem 7.1.4.1(6) of [44]). When \(U^p(N_1,N_2)=\widehat{{\mathcal {H}}}\), and when \((U^p(N_1,N_2),\varSigma ) \in {\mathcal {J}}_n^{(m),{{\text {tor}}}}\) is induced by \({\kappa '}= (\widehat{{\mathcal {H}}}, \widehat{\varSigma })\) as in Sect. B.7, the partial toroidal compactification
(by gluing along their common open subscheme described above) is the
in this paper (see Sect. 5.3). In this case, the \((U^p(N_1),\varSigma ^{{\text {ord}}}) \in {\mathcal {J}}_n^{(m),{{\text {tor}}},{{\text {ord}}}}\) induced by \((U^p(N_1,N_2),\varSigma )\) is induced by the \(\kappa = (\widehat{{\mathcal {H}}}, \widehat{\varSigma }^{\text {ord}})\) induced by \({\kappa '}= (\widehat{{\mathcal {H}}}, \widehat{\varSigma })\). The partial toroidal compactifications \(\vec {\mathsf {N}}^{\text {ord}}_\kappa \hookrightarrow {\vec {\mathsf {N}}}^{{\text {ord}}, \text {tor}}_\kappa \cup {\mathsf {N}}^{\text {tor}}_{\kappa '}\) are compatible with the actions of \(\widehat{\mathrm {G}}({\mathbb {A}^{\infty , p}}) \times \widehat{\mathrm {P}}^{\text {ord}}_{\texttt {D}}(\mathbb {Z}_p)\) and of the element of \(\widehat{\mathrm {P}}^{\text {ord}}_{\texttt {D}}(\mathbb {Q}_p)\) corresponding to \(\varsigma _p\) (see theorem 7.1.4.1, (4) and (6), of [44]), and they are compatible with the actions of \(G_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\) on the partial toroidal compactifications \({\mathcal {A}}_{n,U^p(N_1,N_2)}^{(m),{{\text {ord}}}} \hookrightarrow {\mathcal {A}}_{n,U^p(N_1,N_2),\varSigma }^{(m),{{\text {ord}}}}\).
The partial toroidal compactification \({\vec {\mathsf {N}}}^{{\text {ord}}, \text {tor}}_\kappa \) admits a stratification by locally closed subschemes
(see theorem 7.1.4.1(1) of [44]) indexed by equivalence classes \([(\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}}, \widehat{\rho })]\) as in lemma 1.2.4.42 of [44] whose underlying cusp labels \([(\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}})]\) are ordinary. Each stratum \(\vec {\widehat{\mathsf {Z}}}^{\text {ord}}_{[(\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}}, \widehat{\rho })]}\) is canonically isomorphic to \(\vec {\widehat{\varXi }}^{\text {ord}}_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}}, \widehat{\rho }}\), and the formal completion of \({\vec {\mathsf {N}}}^{{\text {ord}}, \text {tor}}_\kappa \) along the union of the strata \(\vec {\widehat{\mathsf {Z}}}^{\text {ord}}_{[(\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}}, \widehat{\rho })]}\) labelled by equivalence classes \([(\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}}, \widehat{\rho })]\) with the same underlying ordinary cusp label \([(\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}})]\) is canonically isomorphic to \(\vec {\widehat{\mathfrak {X}}}^{\text {ord}}_{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}}} / \varGamma _{\breve{\varPhi }_{\widehat{{\mathcal {H}}}}}\) (see theorem 7.1.4.1(1) of [44]).
When \(U^p(N_1,N_2)=\widehat{{\mathcal {H}}}\), and when \((U^p(N_1),\varSigma ) \in {\mathcal {J}}_n^{(m),{{\text {tor}}},{{\text {ord}}}}\) is induced by \(\kappa = (\widehat{{\mathcal {H}}}, \widehat{\varSigma }^{\text {ord}})\) as above, the formal completion of \({\vec {\mathsf {N}}}^{{\text {ord}}, \text {tor}}_\kappa \) along the union of all strata \(\vec {\widehat{\mathsf {Z}}}^{\text {ord}}_{[(\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}}, \widehat{\rho })]}\) labelled by equivalence classes \([(\breve{\varPhi }_{\widehat{{\mathcal {H}}}}, \breve{\delta }_{\widehat{{\mathcal {H}}}}, \widehat{\rho })]\) with underlying \(\breve{{\texttt {Z}}}_{\widehat{{\mathcal {H}}}}\) equal to \({\breve{{\texttt {Z}}}}^{(i, \widehat{g})}_{\widehat{{\mathcal {H}}}}\) for some \(\widehat{g} \in \widehat{\mathrm {G}}({\mathbb {A}^{\infty , p}})\times \widehat{\mathrm {P}}^{\text {ord}}_{\texttt {D}}(\mathbb {Z}_p)\) is
where the indices g and h run over \(P_{n,(i)}^{(m),+}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times } \backslash G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times } / U^p(N_1,N_2)\) and \(L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {Z}}_{(p)}) \backslash L^{(m)}_{n,(i),{{\text {lin}}}}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }/U_g^{p,\prime }(N_1)\), respectively, in this paper (see Sect. 5.3).
If \(\kappa \in \mathbf {K}_{Q, {\mathcal {H}}, \varSigma ^{\text {ord}}}^{{\text {ord}}, ++}\), then the canonical morphism
extends to a canonical log smooth morphism
(see theorem 7.1.4.1(2) of [44]). When \(U^p(N_1,N_2)=\widehat{{\mathcal {H}}}\) and \((U')^p(N_1',N_2')={\mathcal {H}}\), and when \((U^p(N_1,N_2),\varSigma ) \in {\mathcal {J}}_n^{(m),{{\text {tor}}}}\) and \(((U')^p(N_1',N_2'),\varDelta ) \in {\mathcal {J}}_n^{{\text {tor}}}\) are induced by some \({\kappa '}= (\widehat{{\mathcal {H}}}, \widehat{\varSigma })\) and \(({\mathcal {H}}, \varSigma )\) extending \(\kappa = (\widehat{{\mathcal {H}}}, \widehat{\varSigma }^{\text {ord}})\) and \(({\mathcal {H}}, \varSigma ^{\text {ord}})\), respectively, we have
(see Sect. 5.2 in this paper) and the union of the above morphism with
(see Sect. B.7) is the log smooth morphism
in this paper (see Sect. 5.3).
These identifications are all Hecke equivariant (see theorem 7.1.4.1(4) of [44]).
The formal completion of \({\vec {\mathsf {N}}}^{{\text {ord}}, \text {tor}}_\kappa \) along
is denoted
in [44] (see definition 3.4.4.2). When \(U^p(N_1,N_2)=\widehat{{\mathcal {H}}}\), and when \((U^p(N_1),\varSigma ) \in {\mathcal {J}}_n^{(m),{{\text {tor}}},{{\text {ord}}}}\) is induced by \(\kappa = (\widehat{{\mathcal {H}}}, \widehat{\varSigma }^{\text {ord}})\), these are the
and
respectively, in this paper (see Sect. 5.3). (Their independence of \(N_2={r_0}\) is explained in theorem 7.1.4.1(4h) of [44].) For the Hecke actions on these formal schemes, see theorem 7.1.4.1, (4g)–(4j), of [44].
1.16 Automorphic bundles in mixed characteristics
We first recall how some of the notation of [44] specializes in our case. In the beginning of section 8.1.1 of [44], we can take:
-
(1)
\({R_0}= {\mathbb {Z}_{(p)}}\) and \({\tilde{R}_0}= \mathbb {Z}_p\).
-
(2)
\({\text {Gr}}_{{\texttt {D}}, 0}^{-1}= {\text {Gr}}_{{\texttt {D}}^\#, 0}^{-1}= \mathcal {O}_{F, (p)}^n\) with its canonical \(\mathcal {O}\otimes _{\mathbb {Z}} {{\mathbb {Z}_{(p)}}}\)-module structure, with \(\phi _{{\texttt {D}}, 0}^{-1}: {\text {Gr}}_{{\texttt {D}}, 0}^{-1}\mathop {\rightarrow }\limits ^{\sim }{\text {Gr}}_{{\texttt {D}}^\#, 0}^{-1}\) given by the identity morphism.
Then we have
and, for each \({\mathbb {Z}_{(p)}}\)-algebra R,
which is canonically isomorphic to the one in definition 8.1.1.1 of [44] because of the simpler setting here. (We will not need the other groups \(\mathrm {G}^{\text {ord}}_{{\texttt {D}}, 0}(R)\) and \(\mathrm {P}^{\text {ord}}_{{\texttt {D}}, 0}(R)\) in this paper.) This can be canonically identified with the group \(L_{n,(n)}(R)\) in this paper (see Sect. 1.2). If R is a \(\mathbb {Q}\)-algebra, then \(\mathrm {M}^{\text {ord}}_{{\texttt {D}}, 0}(R) \cong \mathrm {M}_0(R)\) (see Sect. B.8).
Let \({\mathcal {H}}\) be any open compact subgroup of \(\mathrm {G}({\hat{\mathbb {Z}}})\) of the form \({\mathcal {H}}= {\mathcal {H}}^p{\mathcal {H}}_p\), where \({\mathcal {H}}^p\) is a neat open compact subgroup of \(\mathrm {G}({\hat{\mathbb {Z}}^{p}})\), and where \({\mathcal {H}}_p= \mathcal {U}_{p, 1, 0}(p^{r_1}, p^{r_0})\) for some integers \(0 \le {r_1}\le {r_0}\), so that \(\vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\) and \({\vec {\mathsf {M}}}^{{\text {ord}}, \text {min}}_{\mathcal {H}}\) are defined over \({{\text {Spec}}\,}{\mathbb {Z}_{(p)}}\) as in Sects. B.10 and B.12. Then the tautological abelian scheme \(A\) over \(\vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\) defines a locally free sheaf
(where \(e_A\) denotes the identity section), which is the
in this paper (see Sect. 3.4.3), with \(U^p(N_1,N_2)={\mathcal {H}}\). We can similarly define \({\underline{{\text {Lie}}}}^{\vee }_{{A}^{\vee }/\vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}}\). The action of \(\mathrm {G}({\mathbb {A}^{\infty , p}}) \times \mathrm {P}^{\text {ord}}_{\texttt {D}}(\mathbb {Z}_p)\) (resp. of the element of \(\mathrm {P}^{\text {ord}}_{\texttt {D}}(\mathbb {Q}_p)\) corresponding to \(\varsigma _p\)) on \(\{ \vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\}_{\mathcal {H}}\) is defined by respecting their tautological abelian schemes up to canonical \({{\mathbb {Z}}^\times _{(p)}}\)-isogenies (resp. \({\mathbb {Q}}^\times \)-isogenies). Therefore, such an action induces actions on \(\{ {\underline{{\text {Lie}}}}^{\vee }_{A/\vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}} \}_{\mathcal {H}}\) and \(\{ {\underline{{\text {Lie}}}}^{\vee }_{{A}^{\vee }/\mathsf {M}_{\mathcal {H}}} \}_{\mathcal {H}}\) covering the one on \(\{ \vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\}_{\mathcal {H}}\), which are compatible with the isomorphisms
induced by the tautological polarizations \(\lambda : A\rightarrow {A}^{\vee }\). Here the formal Tate twist requires (as before) the Hecke action to be twisted by the similitude character \(\nu \), which corresponds to the tensor product with
in this paper (see Sect. 3.4.3).
Let \(\varSigma ^{\text {ord}}\) be a compatible collection of admissible projective smooth rational polyhedral cone decompositions for \(\vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\), so that \({\vec {\mathsf {M}}}^{{\text {ord}}, \text {tor}}_{{\mathcal {H}}, \varSigma ^{\text {ord}}}\) is defined over \({{\text {Spec}}\,}{\mathbb {Z}_{(p)}}\) as in Sect. B.13. Then the tautological semi-abelian scheme \(G\) over \({\vec {\mathsf {M}}}^{{\text {ord}}, \text {tor}}_{{\mathcal {H}}, \varSigma ^{\text {ord}}}\) defines a locally free sheaf
(where \(e_G\) denotes the identity section), which is
in this paper (see Sect. 5.4), with \(U^p(N_1,N_2)={\mathcal {H}}\), and with \(\varDelta \) induced by any extension \(\varSigma \) of \(\varSigma ^{\text {ord}}\) as in Sect. B.13. We can similarly define \({\underline{{\text {Lie}}}}^{\vee }_{{G}^{\vee }/{\vec {\mathsf {M}}}^{{\text {ord}}, \text {tor}}_{{\mathcal {H}}, \varSigma ^{\text {ord}}}}\), where \({G}^{\vee }\) denotes the tautological dual semi-abelian scheme over \({\vec {\mathsf {M}}}^{{\text {ord}}, \text {tor}}_{{\mathcal {H}}, \varSigma ^{\text {ord}}}\) extending \({A}^{\vee }\). The action of \(\mathrm {G}({\mathbb {A}^{\infty , p}}) \times \mathrm {P}^{\text {ord}}_{\texttt {D}}(\mathbb {Z}_p)\) (resp. of the element of \(\mathrm {P}^{\text {ord}}_{\texttt {D}}(\mathbb {Q}_p)\) corresponding to \(\varsigma _p\)) on \(\{ {\vec {\mathsf {M}}}^{{\text {ord}}, \text {tor}}_{{\mathcal {H}}, \varSigma ^{\text {ord}}} \}_{({\mathcal {H}}, \varSigma ^{\text {ord}})}\) is defined by respecting their tautological semi-abelian schemes up to canonical \({{\mathbb {Z}}^\times _{(p)}}\)-isogenies (resp. \({\mathbb {Q}}^\times \)-isogenies). Therefore, such an action induces actions on \(\{ {\underline{{\text {Lie}}}}^{\vee }_{G/{\vec {\mathsf {M}}}^{{\text {ord}}, \text {tor}}_{{\mathcal {H}}, \varSigma ^{\text {ord}}}} \}_{({\mathcal {H}}, \varSigma ^{\text {ord}})}\) and \(\{{\underline{{\text {Lie}}}}^{\vee }_{{G}^{\vee }/{\vec {\mathsf {M}}}^{{\text {ord}}, \text {tor}}_{{\mathcal {H}}, \varSigma ^{\text {ord}}}} \}_{({\mathcal {H}}, \varSigma ^{\text {ord}})}\) covering the one on \(\{ {\vec {\mathsf {M}}}^{{\text {ord}}, \text {tor}}_{{\mathcal {H}}, \varSigma ^{\text {ord}}} \}_{({\mathcal {H}}, \varSigma ^{\text {ord}})}\), which are compatible with the isomorphisms
induced by the tautological polarizations \(\lambda : G\rightarrow {G}^{\vee }\). Here the formal Tate twist requires (as before) the Hecke action to be twisted by the similitude character \(\nu \), which corresponds to the tensor product with
in this paper (see Sect. 5.4).
Then we have the principal \(\mathrm {M}^{\text {ord}}_{{\texttt {D}}, 0}\)-bundle
which is an \(\mathrm {M}^{\text {ord}}_{{\texttt {D}}, 0}\)-torsor over \(\vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\) (see definition 8.1.2.4 and lemma 8.1.2.6 of [44]), which canonically extends (as an \(\mathrm {M}^{\text {ord}}_{{\texttt {D}}, 0}\)-torsor) to a principal \(\mathrm {M}^{\text {ord}}_{{\texttt {D}}, 0}\)-bundle
over \({\vec {\mathsf {M}}}^{{\text {ord}}, \text {tor}}_{{\mathcal {H}}, \varSigma ^{\text {ord}}}\) (see (8.1.3.11) and lemma 8.1.3.12 of [44]). These are the restrictions (to \(\vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\) and \({\vec {\mathsf {M}}}^{{\text {ord}}, \text {tor}}_{{\mathcal {H}}, \varSigma ^{\text {ord}}}\), respectively) of the
in this paper (see Sects. 3.4.3, 5.4), with \(U^p(N_1,N_2)={\mathcal {H}}\), and with \(\varDelta \) induced by some extension \(\varSigma \) of \(\varSigma ^{\text {ord}}\) as in Sect. B.13.
For each \({\mathbb {Z}_{(p)}}\)-algebra \(R\), we denote by \(\mathrm {Rep}_R(\mathrm {M}^{\text {ord}}_{{\texttt {D}}, 0})\) the category of \(R\)-modules with algebraic actions of \(\mathrm {M}^{\text {ord}}_{{\texttt {D}}, 0} \otimes _{{\mathbb {Z}_{(p)}}} {R}\) (see definition 8.1.2.7 of [44]). Then we also define, for each \(W\in \mathrm {Rep}_R(\mathrm {M}^{\text {ord}}_{{\texttt {D}}, 0})\) that is locally free of finite rank as an \(R\)-module, the automorphic bundle
over \(\vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\otimes _{{\mathbb {Z}_{(p)}}} {R}\) (see definition 8.1.2.8 of [44]), which extends to the canonical extension
and the subcanonical extension
over \({\vec {\mathsf {M}}}^{{\text {ord}}, \text {tor}}_{{\mathcal {H}}, \varSigma ^{\text {ord}}} \otimes _{{\mathbb {Z}_{(p)}}} {R}\) (see definition 8.1.3.13 of [44]), where \(\mathscr {I}_{\vec {\mathsf {D}}^{\text {ord}}_\infty }\) is the \(\mathscr {O}_{{\vec {\mathsf {M}}}^{{\text {ord}}, \text {tor}}_{{\mathcal {H}}, \varSigma ^{\text {ord}}}}\)-ideal defining the boundary divisor \(\vec {\mathsf {D}}^{\text {ord}}_\infty := {\vec {\mathsf {M}}}^{{\text {ord}}, \text {tor}}_{{\mathcal {H}}, \varSigma ^{\text {ord}}} - \vec {\mathsf {M}}^{\text {ord}}_{\mathcal {H}}\) (with its canonical reduced subscheme structure). These are restrictions of the vector bundles
in this paper (see Sects. 3.4.3, 5.4), with \(U^p(N_1,N_2)={\mathcal {H}}\), and with \(\varDelta \) induced by some extension \(\varSigma \) of \(\varSigma ^{\text {ord}}\) as in Sect. B.13. The bundles \(\vec {\mathcal {E}}^{\text {ord}}_{\mathrm {M}^{\text {ord}}_{{\texttt {D}}, 0}, R}(W)\) and \(\vec {\mathcal {E}}_{\mathrm {M}^{\text {ord}}_{{\texttt {D}}, 0}, R}^{{\text {ord}}, {\mathrm {can}}}(W)\) admit compatible actions of \(\mathrm {G}({\mathbb {A}^{\infty , p}}) \times \mathrm {P}^{\text {ord}}_{\texttt {D}}(\mathbb {Z}_p)\) (see proposition 8.1.4.1 of [44]), which are compatible with the compatible actions of \(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\) on \({\mathcal {E}}_{U^p(N_1,N_2),\rho }^{{\text {ord}}}\) and \({\mathcal {E}}_{U^p(N_1,N_2),\varDelta ,\rho }^{{{\text {ord}}},{{\text {can}}}}\), covering the ones on their respective base schemes. The base extensions of these bundles from \({\mathbb {Z}_{(p)}}\) to \(\mathbb {Q}\) are canonically isomorphic to restrictions of the corresponding bundles introduced in Sect. B.8.
Beyond the ordinary loci, we still have the tautological abelian scheme \(\vec {A}\) and the principal polarization \(\vec {\lambda }: \vec {A} \mathop {\rightarrow }\limits ^{\sim }{\vec {A}}^{\vee }\) over \(\vec {\mathsf {M}}_{\mathcal {H}}\) (see proposition 2.2.1.1 of [44]). Hence, we can still define the principal bundle
(see (8.3.1.2) and lemma 8.3.1.4 of [44]), for any \({\mathcal {H}}\), and accordingly the automorphic bundle
over \(\vec {\mathsf {M}}_{\mathcal {H}}\) (see definition 8.3.2.1 of [44]), for any \(W\in \mathrm {Rep}_R(\mathrm {M}^{\text {ord}}_{{\texttt {D}}, 0})\) that is locally free of finite rank as an \(R\)-module.
For simplicity, assume that \(R\) is just \({\mathbb {Z}_{(p)}}\). Then we can still define some canonical and subcanonical extensions
over \({\vec {\mathsf {M}}}^{\text {min}}_{\mathcal {H}}\), which are \(\mathscr {O}_{{\vec {\mathsf {M}}}^{\text {min}}_{\mathcal {H}}}\)-torsion free coherent sheaves extending \(\vec {\mathcal {E}}_{\mathrm {M}^{\text {ord}}_{{\texttt {D}}, 0}, R}(W)\), whose pull-backs to \({\mathsf {M}}^{\text {min}}_{\mathcal {H}}\) (resp. \({\vec {\mathsf {M}}}^{{\text {ord}}, \text {min}}_{\mathcal {H}}\)) are canonically isomorphic to the push-forwards from \({\mathsf {M}}^{\text {tor}}_{{\mathcal {H}}, \varSigma }\) (resp. \({\vec {\mathsf {M}}}^{{\text {ord}}, \text {tor}}_{{\mathcal {H}}, \varSigma ^{\text {ord}}}\)) (for any \(\varSigma \) inducing \(\varSigma ^{\text {ord}}\)) of the corresponding canonical and subcanonical extensions for the automorphic bundle associated with \(W\otimes _{\mathbb {Z}} {\mathbb {Q}}\) (resp. \(W\)). (See definition 8.3.5.1, lemma 8.3.5.2, corollary 8.3.5.4, lemma 8.3.5.7, and corollary 8.3.5.8 of [44].) The above \(\vec {\mathcal {E}}_{\mathrm {M}^{\text {ord}}_{{\texttt {D}}, 0}, R}^{{\mathrm {sub}}, \text {min}}(W)\) is the
in lemma 5.5 of this paper, with \(U^p(N_1,N_2)={\mathcal {H}}\), with \(R={\mathbb {Z}_{(p)}}\), and with \(W_\rho =W\). Such coherent sheaves admit compatible actions of \(\mathrm {G}({\mathbb {A}^{\infty , p}}) \times \mathrm {G}(\mathbb {Z}_p)\) (see proposition 8.3.6.5 of [44]), and the identifications in this paragraph are all Hecke equivariant.
Index of Notations
-
\({\mathcal {A}}^{{\text {univ}}}_\varDelta \), 174
-
\(({\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\Sigma })^0\), 175
-
\({\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N_1,N_2),\Sigma }\), 204
-
\({\mathcal {A}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\), 129
-
\(\{{\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2)} \}\), 100
-
\([({\mathcal {A}}^{{\text {univ}}},i^{{\text {univ}}},\lambda ^{{\text {univ}}},[\eta ^{{\text {univ}}}])]\), 90
-
\({\mathcal {A}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+}\), 129
-
\({\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\Sigma }\), 173
-
\({\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N_1),\Sigma ^{{\text {ord}}}}\), 205
-
\({\mathfrak {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1),\Sigma ^{{\text {ord}}}}\), 176
-
\({\mathfrak {A}}^{(m),{{\text {ord}}}}_{U^p(N_1)}\), 105
-
\(\alpha _*T\), 46
-
\(A_n\), 16
-
\(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1),\Sigma ^{{\text {ord}}}}\), 176
-
(A, i), 84
-
\(A_{n,(i)}\), 25
-
\(A_{n,(i),{{\text {herm}}}}\), 25
-
\((A,i,\lambda )\), 84
-
\((A,i,\lambda ,[(\eta _0,\eta _1)])g\), 89
-
\((A,i,\lambda ,[(\eta _0^p,\eta _1^p,C,\eta _p)])g\), 89
-
\(A_{n,(i),{{\text {lin}}}}\), 25
-
\(A^\perp \), 53
-
\(A^+_{n,(i),U}\), 126
-
\(A^{(m)}_{n,U,\Sigma }\), 171
-
\(\overline{{A}}^{(m),{{\text {ord}}}}_{n,U^p(N)}\), 105
-
\((A^\vee ,i^\vee )\), 84
-
\(A^{{\text {univ}}}\), 172
-
\(A_{n,(i),U}^{(m),+}\), 126
-
\(A_{n,U,\Sigma ,i}^{(m),\wedge }\), 172
-
\(A_{n,U}^{(m)}\), 94
-
\([(A^{{\text {univ}}},i^{{\text {univ}}},\lambda ^{{\text {univ}}},[\eta ^{{\text {univ}}}])]\), 90
-
\(A^\vee \), 53
-
\(A^{\vee ,0}\), 53
-
\(\Vert \,\,\, \Vert _F\), 10
-
\(\vert \,\,\, \vert _v\) , 10
-
\(B_n\), 16
-
\(B_n^1\), 29
-
\({{\text {BC}}}(\Pi )_w\), 29, 30
-
\(B_{GL_m}\), 28
-
\({\mathfrak {C}}\), 34
-
\({\mathfrak {C}}_{=i}\), 34
-
\({\mathfrak {C}}^{(m)}\), 34
-
\({\mathfrak {C}}^{(m)}_{=i}\), 34
-
\({\mathfrak {C}}^{(m)}(W)\), 32
-
\({\mathfrak {C}}(W)\), 32
-
\({\mathfrak {C}}^{>0}(W)\), 33
-
\({\mathfrak {C}}^{\ge 0}(W)\), 33
-
\({\mathfrak {C}}^{(m),>0}(W)\), 33
-
\({\mathfrak {C}}^{(m),\ge 0}(W)\), 33
-
\({\mathfrak {C}}^{(m),\succ 0}(W)\), 33
-
\({\mathfrak {C}}^{\succ 0}(W)\), 33
-
\(C_n\), 12
-
\({\mathcal {D}}_F^{-1}\), 10
-
\(\varDelta _n\), 16
-
\(\varDelta _{(n)}\), 25
-
\(\delta _P\), 11
-
\(D^{(s)}\), 51
-
\(D_S\), 51
-
\(d_v^{(i)}\), 195
-
\(\partial A^{(m)}_{n,U,\Sigma }\), 171
-
\(\partial \overline{{X}}_{n,U^p(N_1)}^{{{\text {ord}}},{{\text {min}}}}\), 162
-
\(\partial {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N_1),\Sigma ^{{\text {ord}}}}\), 205
-
\(\partial {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N_1,N_2),\Sigma }\), 204
-
\(\partial {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\Sigma }\), 173
-
\(\partial {\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}\), 160
-
\(\partial _iX_{n,U}^{{\text {min}}}\), 160
-
\(\partial ^{(s)} {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N_1,N_2),\Sigma }\), 204
-
\(\partial ^{(s)} {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N_1),\Sigma ^{{\text {ord}}}}\), 205
-
\(\partial T_\sigma \), 54
-
\(\partial T_\Sigma \), 57, 80
-
\(\partial _{\{0\}}^0T_\Sigma \), 58
-
\(\partial _{[\sigma ]} A_{n,U,\Sigma }^{(m)}\), 173
-
\(\partial _{[\sigma ]} \overline{{A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1),\Sigma ^{{\text {ord}}}}\), 177
-
\(\partial _{[\sigma ]} {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N_1),\Sigma ^{{\text {ord}}}}\), 205
-
\(\partial _{[\sigma ]} {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N_1,N_2),\Sigma }\), 204
-
\(\partial _{[\sigma ]} {\mathfrak {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1),\Sigma ^{{\text {ord}}}}\), 177
-
\(\partial _{[\sigma ]} {\mathcal {A}}^{(m),{{\text {ord}}}}_{n,U^p(N_1,N_2),\Sigma }\), 175
-
\(\partial _i A_{n,U,\Sigma }^{(m)}\), 172
-
\(\partial _i \overline{{A}}_{n,U^p(N),\Sigma ^{{\text {ord}}}}^{(m),{{\text {ord}}}}\), 177
-
\(\partial _i\overline{{X}}_{n,U^p(N)}^{{{\text {ord}}},{{\text {min}}}}\), 162
-
\(\partial _i {\mathcal {A}}_{n,U^p(N_1,N_2),\Sigma }^{(m),{{\text {ord}}}}\), 174
-
\(\partial _i{\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}\), 161
-
\(\partial _i^0 A_{n,U,\Sigma }^{(m)}\), 172
-
\(\partial _i^0 \overline{{A}}_{n,U^p(N),\Sigma ^{{\text {ord}}}}^{(m),{{\text {ord}}}}\), 177
-
\(\partial _i^0\overline{{X}}_{n,U^p(N)}^{{{\text {ord}}},{{\text {min}}}}\), 162
-
\(\partial _i^0X_{n,U}^{{\text {min}}}\), 160
-
\(\partial _i^0 {\mathcal {A}}_{n,U^p(N_1,N_2),\Sigma }^{(m),{{\text {ord}}}}\), 175
-
\(\partial _i^0{\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}\), 161
-
\(\partial _i T_\Sigma \), 59, 80
-
\(\partial _{\Sigma _0}T_\Sigma \), 56
-
\(\partial _{\Sigma _0} T^{(m),\natural }_{n,(i),U}\), 146
-
\(\partial _{\Sigma _0} \overline{{T}}^{(m),{{\text {ord}}},\natural }_{n,(i),U^p(N)}\), 148
-
\(\partial _{\Sigma _0} {\mathcal {T}}^{(m),{{\text {ord}}},\natural }_{n,(i),U^p(N_1,N_2)}\), 146
-
\(\partial _{\Sigma _0} T_\sigma \), 54
-
\(\partial _{\Sigma _0} T^{(m),+}_{n,(i),U}\), 144
-
\(\partial _\sigma T_\Sigma \), 58, 80
-
\(\partial _{\Sigma _0} T\), 81
-
\(\partial _{\Sigma _0}T\), 65
-
\(\partial _{\Sigma _0} T_\Sigma \), 79
-
\(\partial _{\Sigma _0} {\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\), 144
-
\(\partial ^0_\sigma T_\Sigma \), 58
-
\(\partial X_{n,U}^{{\text {min}}}\), 159
-
\({\mathcal {E}}_{(i),U^p(N_1,N_2)}^{{{\text {ord}}},+}\), 125
-
\({\mathcal {E}}^{+}_{(i),U,\rho }(W)\), 123
-
\({\mathcal {E}}^{{\text {can}}}_{U,\varDelta ,\rho }\), 182
-
\({\mathcal {E}}^{{\text {can}}}_{U,\varDelta }\), 182
-
\({\mathcal {E}}^{{{\text {can}}},\dag }_{U^p(N),\varDelta ,\rho }\), 206
-
\({\mathcal {E}}^{{{\text {ord}}},+}_{(i),U^p(N_1,N_2),\rho }\), 125, 130
-
\({\mathcal {E}}^{{{\text {ord}}},{{\text {sub}}}}_{U^p(N),\varDelta ^{{\text {ord}}},\rho }\), 186
-
\({\mathcal {E}}^{{{\text {ord}}},{{\text {sub}}}}_{U^p(N),\rho }\), 186
-
\({\mathcal {E}}^{{{\text {ord}}},{{\text {can}}}}_{U^p(N),\varDelta ^{{\text {ord}}},\rho }\), 186
-
\({\mathcal {E}}^{(m),{{\text {ord}}}}_{U^p(N_1,N_2),\rho }\), 116
-
\({\mathcal {E}}^{(m)}_{U,\rho }\), 112
-
\({\mathcal {E}}^{{{\text {sub}}},\dag }_{U^p(N),\varDelta ,\rho }\), 206
-
\({\widetilde{{\mathcal {E}}}}_{(i),U}^{+}\), 128
-
\({\mathcal {E}}_{(i),U}^{+}\), 123
-
\({\mathcal {E}}_{(i),U,\varDelta _0,\rho }^\natural \), 151
-
\({\mathcal {E}}_{(i),U^p(N_1,N_2),\varDelta _0,\rho }^{{{\text {ord}}},\natural }\), 151
-
\({\mathcal {E}}_{U^p(N_1,N_2)}^{{{\text {ord}}}}\), 114
-
\({\mathcal {E}}_{U}\), 111
-
\({\mathcal {E}}_{U,\varDelta ,\rho }^{{\text {sub}}}\), 183
-
\({\mathcal {E}}_{U,\rho }\), 111
-
\({\mathcal {E}}_{U,\rho }^{{\text {sub}}}\), 185
-
\({\mathcal {E}}_{U^p(N_1,N_2),\varDelta ,\rho }^{{{\text {ord}}},{{\text {can}}}}\), 182
-
\({\mathcal {E}}_{U^p(N_1,N_2),\varDelta }^{{{\text {ord}}},{{\text {can}}}}\), 182
-
\({\mathcal {E}}_{U^p(N_1,N_2),\rho }^{{{\text {ord}}},{{\text {sub}}}}\), 185
-
\({\mathcal {E}}_{U^p(N_1,N_2),\varDelta ,\rho }^{{{\text {ord}}},{{\text {sub}}}}\), 183
-
\({\mathcal {E}}_{U}^{(m)}\), 112
-
\({\mathcal {E}}_{U^p(N_1,N_2),\rho }^{{{\text {ord}}}}\), 115
-
\({\mathcal {E}}_{U^p(N_1,N_2)}^{(m),{{\text {ord}}}}\), 116
-
\({\widetilde{{\mathcal {E}}}}_{(i),U^p(N_1,N_2)}^{{{\text {ord}}},+}\), 130
-
\({\mathfrak {E}}_\rho \), 18
-
\({\mathfrak {E}}_{U,\rho }\), 111
-
\({\mathfrak {E}}_\rho \), 25
-
\([\eta ]\), 85
-
\([\eta ]\), 85
-
\((\eta _0^p,\eta _1^p,C,\eta _p)\), 85
-
\(\Vert \eta _0^p\Vert \), 86
-
\(\Vert \eta _0\Vert \), 85
-
e, 15
-
\({\mathcal {F}}^\dag \), 202
-
\(f^\dag \), 203
-
\({\widetilde{{\mathcal {G}}}}^{{\text {univ}}}\), 130
-
\(({\mathcal {G}}^{{\text {univ}}}, i^{{\text {univ}}}, j^{{\text {univ}}}, \lambda ^{{\text {univ}}}, [\eta ^{{\text {univ}}}])\), 101
-
\(\gamma (G,i,j,\lambda ,[(\eta _0^p,\eta _1^p,C,D,\eta _p)])\), 100
-
\(\gamma (G,i,j,\lambda ,[(\eta _0,\eta _1)])\), 94
-
\(G_n\), 11
-
\(G_n^1\), 28
-
\(G_F\), 10
-
\(G_F^S\), 10
-
(G, i, j), 92
-
(G, i, j), 97
-
\((G,i,j,\lambda ,[(\eta ^p_0,\eta ^p_1,C,D,\eta _p)])g\), 99
-
\((G,i,j,\lambda ,[(\eta _0^p,\eta _1^p,C,D,\eta _p)])\varsigma _p\), 100
-
\((G,i,j,\lambda ,[(\eta _0,\eta _1)])g\), 94
-
\(G_n^{(m)}\), 13
-
\((G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)}_{=i})^{{\text {ord}}}\), 35
-
\((G_n^{(m)}({\mathbb {A}}^\infty ) \times \pi _0(G_n({\mathbb {R}})) \times {\mathfrak {C}}^{(m)})^{{\text {ord}}}\), 35
-
\(G_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\), 28
-
\(\widetilde{{G}}_n^{(m)}({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\), 28
-
\(\widetilde{{G}}_n^{(m)}({\mathbb {A}}^\infty )^{{\text {ord}}}\), 28
-
\(\widetilde{{G}}^{(m)}_n\), 16
-
\(G_n({\mathbb {A}}^\infty )^{{{\text {ord}}},\times }\), 28
-
\([(G^{{\text {univ}}},i^{{\text {univ}}},j^{{\text {univ}}},\lambda ^{{\text {univ}}},[\eta ^{{\text {univ}}}])]\), 95
-
\(\widetilde{{G}}^{{\text {univ}}}/A^+_{n,(i),U}\), 128
-
\(\ge \), 166
-
\(\ge \), 167
-
\(\ge \), 167
-
\({\mathcal {H}}{{o}}{{m}}\), 9
-
\({\mathfrak {H}}^\pm _n\), 18
-
\(H(R)^+\), 17
-
\({{\text {Hasse}}}_{M,U}\), 193
-
\({{\text {Hasse}}}_{M,U^p(N)}\), 193
-
\({{\text {Hasse}}}_{U}\), 163
-
\({{\text {Herm}}}_X^{>0}\), 31
-
\({{\text {Herm}}}_X^{\ge 0}\), 31
-
\({{\text {Herm}}}^{(m)}\), 14
-
\({{\text {Herm}}}_X\), 13
-
\(H^i_{{\text {Int}}}\), 36
-
\(H^i_{{\text {Int}}}({\mathfrak {T}}_{(n)}^{(m)},\overline{{{\mathbb {Q}}}}_p)\), 36
-
\( H^i_{{\text {Int}}}({\mathfrak {T}}_{=n}^{(m),{{\text {ord}}}},\overline{{{\mathbb {Q}}}}_p)\), 36
-
\(H^i_{{\text {Int}}}({\mathfrak {T}}_{(n)},{\mathcal {L}}_\rho )\), 41
-
\({{\text {Hom}}}^{(m)}_n\), 12
-
\(H^0({\mathcal {X}}^{{{\text {ord}}},{{\text {min}}}}_n, {\mathcal {E}}_\rho ^{{{\text {ord}}},{{\text {sub}}}} \otimes {\mathbb {Z}}/p^r{\mathbb {Z}})\), 186
-
\(H^0({\mathcal {X}}_{U^p(N)}^{{{\text {ord}}},{{\text {min}}},\dag }, {\mathcal {E}}_{U^p(N),\rho }^{{{\text {sub}}},\dag })_{\overline{{{\mathbb {Q}}}}_p,\le a}\), 216
-
\(H^0({\mathcal {X}}^{{{\text {ord}}},\dag }, {\mathcal {E}}^{{{\text {sub}}}}_\rho )\), 206
-
\(H^0({\mathcal {X}}^{{{\text {ord}}},{{\text {min}}},\dag },{\mathcal {E}}_\rho ^{{\text {sub}}})_{\overline{{{\mathbb {Q}}}}_p,\le a}\), 216
-
\(H^0({\mathfrak {X}}^{{{\text {ord}}},{{\text {min}}}}, {\mathcal {E}}_\rho ^{{{\text {ord}}},{{\text {sub}}}})_{\overline{{{\mathbb {Q}}}}_p}\), 186
-
\(H^0({\mathfrak {X}}^{{{\text {ord}}},{{\text {min}}}}_n, {\mathcal {E}}_\rho ^{{{\text {ord}}},{{\text {sub}}}})\), 186
-
\(H^i({\mathcal {A}}^{(m),{{\text {ord}}},\dag } , \Omega ^j(\log \infty )\otimes {\mathcal {I}})\), 206
-
\(H^i(\partial ^{(s)}{\mathcal {A}}^{(m),{{\text {ord}}},\dag } , \Omega ^j)\), 206
-
\(H^i_{c-\partial }(\overline{{A}}^{(m),{{\text {ord}}}})\), 220
-
\(H^i_{c-\partial }(\overline{{A}}^{(m),{{\text {ord}}}})_{\le a}\), 220
-
\(H^i_{c-\partial }(\overline{{A}}^{(m),{{\text {ord}}}}_{U^p(N),\Sigma })\), 220
-
\(H^i_{{\text {rig}}}(\overline{{Y}})\), 202
-
\(H^i(X_n^{{\text {min}}}, {\mathcal {E}}_\rho ^{{\text {sub}}})\), 186
-
\(H^0({\mathcal {X}}^{{{\text {ord}}},{{\text {min}}},\dag },{\mathcal {E}}_\rho ^{{\text {sub}}})_{\overline{{{\mathbb {Q}}}}_p}\), 212
-
\(H^i({\mathcal {A}}^{(m),{{\text {ord}}},\dag } , \Omega ^s(\log \infty )\otimes {\mathcal {I}}_{\partial {\mathcal {A}}^{(m),{{\text {ord}}},\dag }})_{\le a}\), 219
-
\({\mathcal {I}}_{\partial _{\Sigma _0} T_\Sigma }\), 56
-
\({\mathcal {I}}_{\partial T_\Sigma }\), 57
-
\({\mathcal {I}}^{(m),\natural +,\wedge }_{\partial , n,(i),U,\Sigma _0}\), 149
-
\({\mathcal {I}}^{(m),\natural +,\wedge }_{\partial , n,(i),U,\Sigma _0}\), 149
-
\({\mathcal {I}}^{(m),{{\text {ord}}},\natural +,\wedge }_{\partial , n,(i),U^p(N_1,N_2),\Sigma _0}\), 149
-
\({\mathcal {I}}^{(m),{{\text {ord}}},+,\wedge }_{\partial , n,(i),U^p(N_1,N_2),\Sigma _0}\), 149
-
\({\mathcal {I}}_{\partial A_{U,\Sigma }^{(m)} \times {{\text {Spec}}\,}R_0}\), 179
-
\({\mathcal {I}}_{\partial , \Sigma _0}^\wedge \), 66, 81
-
\({\mathcal {I}}_{\partial T_\Sigma }\), 80
-
\({\mathcal {I}}_{\partial X_{n,U,\varDelta }}\), 179
-
\({\mathcal {I}}_{\partial A_{U,\Sigma }^{(m)}}\), 179
-
\({\mathcal {I}}_{\partial X_{n,U}^{{\text {min}}}\times {{\text {Spec}}\,}R_0}\), 179
-
\({\mathcal {I}}_{\partial X_{n,U,\varDelta } \times {{\text {Spec}}\,}R_0}\), 179
-
\({\mathcal {I}}_{T_{\Sigma _0}^\wedge }\), 66, 81
-
\({\mathcal {I}}_{\partial X_{n,U}^{{\text {min}}}}\), 179
-
\({{\text {Ind}}}\), 10
-
\(i^{(s)}\), 204
-
\(i^{(m)}_{A^{{\text {univ}}}}\), 96
-
\(i^{(m)}_{{\mathcal {A}}^{{\text {univ}}}}\), 102
-
\(i_\lambda ^{(m)}\), 97, 104
-
\([\infty ]\), 9
-
\([\infty ^p]\), 9
-
\({\mathcal {J}}^{(m),{{\text {tor}}},{{\text {ord}}}}_n\), 169
-
\({\mathcal {J}}^{(m),{{\text {tor}}}}_n\), 169
-
\(J_n\), 11
-
\(j_U^{{\text {min}}}\), 159
-
\(j_{U,\Sigma }^{(m)}\), 171
-
\(j_{U^p(N_1,N_2),\Sigma }^{(m),{{\text {ord}}}}\), 173
-
\(j_{U^p(N_1,N_2)}^{{\text {min}}}\), 160
-
\({{\text {KS}}}\), 26
-
k(v), 10
-
\({\mathcal {L}}_{\rho ,U}\), 41
-
\({\mathcal {L}}^{(m)}_{n,\widetilde{{U}}}(\chi )\), 106
-
\({\mathcal {L}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}^p(N_1,N_2)}(\chi )\), 107
-
\({\mathcal {L}}_T(\chi )\), 45
-
\({\mathcal {I}}_{\partial {\mathcal {X}}^{{{\text {ord}}},\dag }_{U^p(N),\varDelta }}\), 206
-
\({\mathcal {L}}_\psi \), 59
-
\({\mathcal {L}}_\psi ^\wedge \), 66
-
\({\mathcal {L}}_{U}^+(a)\), 138
-
\({\mathcal {L}}_{U^p(N_1,N_2)}^+(a)\), 139
-
\(\Lambda _n\), 11
-
\(\Lambda _{n,(i)}\), 12
-
\(\Lambda _{n,(i)}'\), 20
-
\(\Lambda _n^{(m)}\), 12
-
\(L_{n,(i),{{\text {herm}}}}\), 23
-
\(L_{n,(i),{{\text {herm}}}}^{-}\), 24
-
\(L_{n,(i)}\), 24
-
\(L_{n,(i)}^{(m)}\), 24
-
\(\widetilde{{L}}^{(m)}_{n,(i)}\), 24
-
\(L_{n,(i),{{\text {lin}}}}\), 22
-
\(L^{(m)}_{n,(i),{{\text {lin}}}}\), 22
-
\(\widetilde{{L}}^{(m)}_{n,(i),{{\text {lin}}}}\), 22
-
\({\mathcal {M}}(D)\), 47
-
\({\mathcal {M}}_{\Sigma ,i}^\wedge \), 175
-
\({\mathcal {M}}^\wedge _{\Sigma _0}\), 65
-
\({\mathcal {M}}_\Sigma \), 57, 80
-
\({\mathcal {M}}_{\Sigma ,i}^\wedge \), 172
-
\(({\mathcal {M}}^\wedge ,\alpha ^\wedge )\), 48
-
\({\mathcal {M}}^\wedge _{\Sigma _0}\), 81
-
\({\text{ N }}\), 30
-
\(\nu \), 12
-
\(N_n\), 16
-
\(N_{n,(i)}\), 20
-
\(N_{n,(i)}^+\), 19
-
\(N_{n,(i)}^{(m)}\), 20
-
\(N_{n,(i)}^{(m),+}\), 19
-
\(\widetilde{{N}}_{n,(i)}^{(m)}\), 20
-
\(\widetilde{{N}}_{n,(i)}^{(m),+}\), 19
-
\({{\text {n-Ind}}}\), 11
-
\(N(L^{(m)}_{n,(i),{{\text {lin}}}})\), 22
-
\(N_n^{(m)}\), 15
-
\(N(R_{n,(n),(i)})\), 24
-
\({\mathcal {O}}_{T_{\Sigma _0}^\wedge }\), 66
-
\(\Omega ^j_{ \partial ^{(s)} {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\Sigma } }\), 206
-
\(\Omega ^j_{{\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\Sigma }}(\log \infty ) \otimes {\mathcal {I}}_{\partial {\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\Sigma }}\), 205
-
\(\Omega ^j_{{\mathcal {A}}^{(m),{{\text {ord}}},\dag }_{U^p(N),\Sigma }}(\log \infty )\), 205
-
\(\Omega ^{{\text {ord}}}_{n,U^p(N_1,N_2),\varDelta }\), 180
-
\(\Omega ^{{{\text {ord}}},\natural }_{n,(i),U^p(N_1,N_2)}\), 126
-
\(\Omega ^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\), 125
-
\(\Omega ^1_{A^{{\text {univ}}}/X_{n,U}}\), 109
-
\(\Omega ^1_{{\mathcal {A}}^{{\text {univ}}}/{\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}}\), 113
-
\(\Omega ^1_{{\mathcal {G}}^{{\text {univ}}}/{\mathcal {A}}_{n,U^p(N_1,N_2)}^{(m),{{\text {ord}}}}}\), 115
-
\(\Omega ^1_{T_{\Sigma _0}^\wedge /{{\text {Spf}}\,}R_0}(\log \infty )\), 81
-
\(\Omega ^1_{T_\Sigma /{{\text {Spec}}\,}R_0}(\log \infty )\), 80
-
\(\Omega ^1_{T_{\Sigma _0}^\wedge /{{\text {Spf}}\,}R_0}(\log \infty )\), 66
-
\(\Omega ^i_{{\mathfrak {X}}/{\mathfrak {Y}}}(\log {\mathcal {M}}/{\mathcal {N}})\), 50
-
\(\Omega ^i_{{\mathfrak {X}}/{{\text {Spf}}\,}R_0}\), 50
-
\(\Omega ^i_{{\mathfrak {X}}/{{\text {Spf}}\,}R_0}(\log {\mathcal {M}})\), 50
-
\(\Omega ^i_X(\log {\mathcal {M}})\), 48
-
\(\Omega ^i_{X/Y}(\log {\mathcal {M}}/{\mathcal {N}})\), 48
-
\(\omega _{n,U^p(N_1,N_2)}\), 113
-
\(\widetilde{\varOmega }^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\), 130
-
\(\widetilde{\varOmega }^{+}_{n,(i),U}\), 128
-
\(\Omega _{n,U,\varDelta }\), 180
-
\(\Omega _{n,U}\), 109
-
\(\omega _{n,U}\), 109
-
\(\Omega _{n,U}^{(m)}\), 112
-
\(\Omega _{n,U^p(N_1,N_2)}^{(m),{{\text {ord}}}}\), 115
-
\(\Omega _{n,U^p(N_1,N_2)}^{{\text {ord}}}\), 113
-
\(\Omega _{T_\Sigma /(\alpha _* T)_{\Sigma '}}^1(\log \infty )\), 82
-
\(\Omega _{T_\Sigma /(\alpha _* T)_{\Sigma '}}^1(\log \infty )\), 60
-
\(\Omega _{T_{\Sigma _0}^\wedge /(\alpha _* T)_{\Sigma _0'}^\wedge }^1(\log \infty )\), 82
-
\(\omega _{U}\), 163
-
\(\omega _U\), 109
-
\(\omega _{U,\varDelta }\), 181
-
\(\omega _{U^p(N)}\), 193
-
\(\omega _{U^p(N_1,N_2),\varDelta }^{{\text {ord}}}\), 181
-
\(\omega _{U^p(N_1,N_2)}\), 113
-
\(\Omega ^{\natural }_{n,(i),U}\), 124
-
\(\Omega ^{+}_{n,(i),U}\), 123
-
\(\widetilde{\varOmega }_{n,(i),U,\varDelta _0}^\natural \), 151
-
\(\widetilde{\varOmega }_{n,(i),U^p(N_1,N_2),\varDelta _0}^{{{\text {ord}}},\natural }\), 151
-
\({{\text {ord}}}\), 28
-
\({{\text {ord}}},\times \), 28
-
\({\mathfrak {p}}_n\), 18
-
\({\mathfrak {p}}_n^{\pm }\), 18
-
\(\Phi _n\), 16
-
\(\Phi _n^+\), 16
-
\(\Phi _{n,(n)}\), 25
-
\(\Phi _{n,(n)}^+\), 25
-
\(\varpi _v\), 10
-
\(\pi _{m,m'}\), 94
-
\((\pi _{A^+/X^\natural ,*} {\mathcal {L}}\otimes {\mathcal {F}})^+_U(a^\natural )\), 138
-
\((\pi _{A^+/X^+,*} {\mathcal {L}})^+_U(a)\), 138
-
\(\pi _{A^{(m),{{\text {tor}}}}/X^{{\text {tor}}}}\), 171
-
\(\pi _{A^{(m),{{\text {tor}}}}/X^{{\text {min}}}}\), 171
-
\(\pi _{A^{(m)}_n/A^{(m')}_n}\), 95
-
\(\pi _{A^{(m)}_n/X_n}\), 95
-
\(\pi _{{\mathcal {A}}^{(m),{{\text {ord}}}}_n/{\mathcal {A}}^{(m'),{{\text {ord}}}}_n}\), 101
-
\(\pi _{{\mathcal {A}}^{(m),{{\text {ord}}},{{\text {tor}}}}/{\mathcal {X}}^{{{\text {ord}}},{{\text {min}}}}}\), 174
-
\(\pi _{{\mathcal {A}}^{(m),{{\text {ord}}},{{\text {tor}}}}/{\mathcal {X}}^{{{\text {ord}}},{{\text {tor}}}}}\), 174
-
\((\pi _{{\mathcal {A}}^{{{\text {ord}}},+}/{\mathcal {X}}^{{{\text {ord}}},\natural },*}{\mathcal {L}}\otimes {\mathcal {F}})_{U^p(N_1,N_2)}^+(a^\natural )\), 139
-
\(\pi _{m,m'} (G,i,j,\lambda ,[(\eta ^p_0,\eta ^p_1,C,D,\eta _p)])\), 100
-
\(\pi _{(U,\Sigma ), (U',\Sigma ')}\), 172
-
\(\Psi _n\), 11
-
\(\psi _\Sigma \), 60
-
\(P_{n,(i)}\), 19
-
\(P_{n,(i)}^+\), 19
-
\(P_{n,(i)}^{(m)}\), 19
-
\(P_{n,(i)}^{(m),+}\), 19
-
\(P_{n,(i)}^{(m),-}\), 24
-
\(\widetilde{{P}}_{n,(i)}^{(m)}\), 19
-
\(\widetilde{{P}}_{n,(i)}^{(m),+}\), 19
-
\({\mathfrak {Q}}_n\), 18
-
\({\mathfrak {q}}_n\), 18
-
\(Q_{m,(m')}\), 24
-
\(Q_{m,(m')}'\), 24
-
\(\varrho _n\), 16
-
\(\rho _\lambda \), 16
-
\(\varrho _{n,(n)}\), 25
-
\(\rho _{(n),\lambda }\), 25
-
\(\rho ^{i,j}_{m,s}\), 217
-
\({\text {rec}}\), 10
-
\(R_{n,(n),(i)}\), 24
-
\(R_{n,(n)}^{(m)}\), 24
-
\(r_{p,\imath }(\Pi )\), 31
-
\({{\text {RS}}}\), 9
-
\(R_{p,\imath }(\pi ,N)\), 223
-
\(R_p({\Pi })\), 217, 219
-
\(R_{p}({\pi })\), 223
-
\(R_p({\Pi })\), 198, 222
-
\(\vert {\mathcal {S}}\vert \), 10
-
\(\vert {\mathcal {S}}(D)\vert \), 51
-
\(\vert {\mathcal {S}}(D,\ge )\vert \), 51
-
\(\vert {\mathcal {S}}(\partial A_{n,U,\Sigma }^{(m)})\vert _{=i}\), 173
-
\(\vert {\mathcal {S}}(U,\Sigma )\vert _{=i}\), 170
-
\(\vert {\mathcal {S}}(U^p(N),\Sigma )^{{\text {ord}}}\vert _{=i}\), 170
-
\(\widehat{{{\mathcal {S}}}}_{n,(i),\widetilde{{U}}^p(N)}^{(m),{{\text {ord}}},+}\), 134
-
\({\mathcal {S}}(D)\), 51
-
\({\mathcal {S}}(D,\ge )\), 51
-
\({\mathcal {S}}_{n,\widetilde{{U}}^p}^{(m),{{\text {ord}}}}\), 105
-
\({\mathcal {S}}^+_{n,(i),U^p(N)}\), 134
-
\({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+}\), 134
-
\({\mathcal {S}}(U,\Sigma )\), 170
-
\({\mathcal {S}}(U,\Sigma )_{\le i}\), 170
-
\({\mathcal {S}}(U^p(N),\Sigma )^{{\text {ord}}}_{\le i}\), 170
-
\(\widetilde{{{\mathcal {S}}}}^{(m),{{\text {ord}}},+}_{n,(i),\widetilde{{U}}^p(N)}\), 132
-
\(\vert \Sigma \vert \), 55, 79
-
\(\vert \Sigma \vert ^*\), 79
-
\(\vert \Sigma \vert ^{\vee }\), 79
-
\(\vert \Sigma \vert ^{\vee ,0}\), 79
-
\(\vert \Sigma _0\vert \), 57, 65, 80
-
\(\vert \Sigma _0\vert ^0\), 54
-
\(\vert \Sigma _0\vert ^0\), 65, 80
-
\(\vert \Sigma _0\vert ^0\), 57
-
\(\vert \Sigma _0\vert ^*\), 80
-
\(\vert \Sigma _0\vert ^\vee \), 80
-
\(\vert \Sigma _0\vert ^{\vee ,0}\), 80
-
\(\overline{\varSigma }_0(\sigma )\), 59
-
\(\overline{\varSigma }(\sigma )\), 59
-
\(\Sigma (\sigma )\), 58
-
\(\Sigma ^{{\text {ord}}}\), 167
-
\(\widetilde{\varSigma }_0\), 65, 80
-
\(\varsigma _p\), 27
-
\(\widehat{{S}}_{n,(i),\widetilde{{U}}}^{(m),+}\), 133
-
\(S(F^m)^{>0}\), 31
-
\(S_n\), 9
-
\(S_{n,\widetilde{{U}}}^{(m)}\), 105
-
\(S^+_{n,(i),U}\), 133
-
\(S(\sigma )\), 58
-
\(S_{n,(i),U}^{(m),+}\), 133
-
\({{\text {Std}}}\), 26
-
\({{\text {Std}}}_\tau \), 26
-
\({{\text {sw}}}\), 14
-
S(X), 14
-
\(\widetilde{{S}}^{(m),+}_{n,(i),\widetilde{{U}}}\), 131
-
\({\mathbb {T}}_{U^p(N),\rho }^{{{\text {ord}}},S}(W)\), 197
-
\({\mathbb {T}}_{U^p(N_1,N_2),\rho }^{S}\), 196
-
\({\mathbb {T}}_{U^p(N_1,N_2),\rho }^{{{\text {ord}}},S}(W)\), 197
-
\({\widetilde{{\mathcal {T}}}}_{n,(i),\widetilde{{U}}^p(N_1,N_2)}^{(m),{{\text {ord}}},+}\), 137
-
\(({\mathcal {T}}^{(m),{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2),\widetilde{\varSigma }_0},{\mathcal {M}}_{\widetilde{\varSigma }_0})\), 142
-
\(({\mathcal {T}}^{(m),{{\text {ord}}},\natural +,\wedge }_{n,(i),U^p(N_1,N_2),\Sigma _0},{\mathcal {M}}_{\Sigma _0}^\wedge )\), 146
-
\(({\mathcal {T}}^{(m),{{\text {ord}}},\natural ,\wedge }_{n,(i),U^p(N_1,N_2),\Sigma _0},{\mathcal {M}}_{\Sigma _0}^\wedge )\), 145
-
\(({\mathcal {T}}^{(m),{{\text {ord}}},+,\wedge }_{n,(i),U^p(N_1,N_2),\Sigma _0},{\mathcal {M}}_{\Sigma _0}^\wedge )\), 144
-
\({\mathcal {T}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+}\), 137
-
\({\mathcal {T}}^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\), 137
-
\({\mathcal {T}}^{(m),{{\text {ord}}}}_{n,\widetilde{{U}}^p(N_1,N_2)}\), 107
-
\({\mathfrak {T}}^{(m)}_{(i),U}\), 36
-
\({\mathfrak {T}}_{U^p(N_1),=n}^{(m),{{\text {ord}}}}\), 36
-
\({\mathfrak {T}}_{(n)}\), 41
-
\({\mathfrak {t}}_v^{(i)}\), 195
-
T, 9
-
\(T_n\), 16
-
\(T^1_n\), 29
-
\(T_1 \otimes _S T_2\), 46
-
\({{\text {TF}}}\), 9
-
\(T_{GL_m}\), 28
-
\(\widetilde{{T}}_{n,(i),\widetilde{{U}}}^{(m),+}\), 136
-
\((T^{(m),\natural +,\wedge }_{n,(i),U,\Sigma _0},{\mathcal {M}}_{\Sigma _0}^\wedge )\), 146
-
\(T^p\), 9
-
\(T^pS_{\overline{{y}}}\), 45
-
\({{\text {tr}}}_F\), 208
-
\({{\text {tr}}}_F\), 215
-
\({{\text {tr}}}_F\), 220
-
\(T^+_{n,(i),U}\), 136
-
\(TS_{\overline{{y}}}\), 45
-
\((T^{(m),+}_{n,(i),U,\widetilde{\varSigma }_0},{\mathcal {M}}_{\widetilde{\varSigma }_0})\), 142
-
\((T^{(m),+,\wedge }_{n,(i),U,\Sigma _0},{\mathcal {M}}_{\Sigma _0}^\wedge )\), 144
-
\((T^{(m),\natural ,\wedge }_{n,(i),U,\Sigma _0},{\mathcal {M}}_{\Sigma _0}^\wedge )\), 145
-
\(T^{(m)}_{n,\widetilde{{U}}}\), 106
-
\(T^\wedge _{\Sigma _0}\), 81
-
\({{\text {tr}}}_{\varsigma _p}\), 207
-
\(T_{n,(i),U}^{(m),+}\), 136
-
\(T_\Sigma \), 56, 79
-
\(T_\sigma \), 54
-
\(T_{\Sigma _0}^\wedge \), 65
-
\((T_{\Sigma _0}^\wedge )_\sigma \), 66
-
\(T_v^{(i)}\), 195
-
\(U_{n,\infty }\), 17
-
U(n), 9
-
\(U^p(N)\), 27, 28
-
\(U^p(N_1,N_2)\), 19, 27, 28
-
\(U_{K,\infty }\), 30
-
\(U_p(N_1,N_2)_n\), 19
-
\(U_{p}(N_1,N_2)^{(m)}_{n,(i)}\), 27
-
\(U_{p}(N_1,N_2)^{(m),+}_{n,(i)}\), 27
-
\(\widetilde{{U}}_{p}(N_1,N_2)^{(m),+}_{n,(i)}\), 27
-
\(U_p(N_1,N_2)^{(m)}_n\), 19
-
\(\widetilde{U}_p(N_1,N_2)_n\), 19
-
\(U_{p}(N)_{n,(i)}\), 27
-
\(U_{p}(N)^{(m)}_{n,(i)}\), 27
-
\(V_n\), 11
-
V, 9
-
\(V_{n,(i)}\), 12
-
\(V_n^{(m)}\), 12
-
\(V^p\), 9
-
\(VS_{\overline{{y}}}\), 45
-
\(V_{\le a}\), 199
-
\(V_{>a}\), 199
-
\({{\text {WD}}}\), 10
-
\(W_0 H^{i}_{c-\partial }(\overline{{A}}^{(m),{{\text {ord}}}})_{\overline{{{\mathbb {Q}}}}_p}\), 222
-
\({\mathcal {X}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},\natural }\), 124
-
\({\mathcal {X}}_{n,(i),U^p(N_1,N_2)}^{{{\text {ord}}},\natural }\), 124
-
\({\mathcal {X}}_{n,(i),U^p(N_1,N_2)}^{(m),{{\text {ord}}},+}\), 124
-
\({\mathcal {X}}_{n,U^p(N_1,N_2)}^{{{\text {ord}}},{{\text {min}}}}\), 160
-
\({\mathcal {X}}^{{{\text {ord}}},{{\text {min}}},\dag }_{U^p(N_1)}\), 210
-
\({\mathcal {X}}^{{{\text {ord}}},{{\text {min}}},\dag }_{U^p(N_1,N_2)}\), 209
-
\({\mathcal {X}}_{n,U}^{{\text {min}}}\), 163
-
\({\mathcal {X}}_{n,U^p(N_1,N_2)}^{{\text {ord}}}\), 90
-
\({\mathfrak {X}}^{{{\text {ord}}}}_{n,U^p(N_1),\Sigma ^{{\text {ord}}}}\), 176
-
\({\mathfrak {X}}^{{{\text {ord}}},\dag }_{(U^p)'(N),\varDelta }\), 205
-
\({\mathfrak {X}}^{{{\text {ord}}},{{\text {min}}}}_{n,U^p(N_1)}\), 162
-
\({\mathfrak {X}}^{{\text {ord}}}_{n,U^p(N_1)}\), 92
-
\(\widetilde{\varXi }^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\), 130
-
\(\Xi ^{+}_{n,(i),U}\), 123
-
\(\widetilde{\varXi }^+_{n,(i),U}\), 128
-
\(\Xi ^{\natural }_{n,(i),U}\), 124
-
\(\Xi ^{{\text {ord}}}_{n,U^p(N_1,N_2),\varDelta }\), 181
-
\(\Xi ^{{{\text {ord}}},\natural }_{n,(i),U^p(N_1,N_2)}\), 126
-
\(\Xi ^{{{\text {ord}}},+}_{n,(i),U^p(N_1,N_2)}\), 125
-
\(\Xi _{n,U}\), 109
-
\(\Xi _{n,U^p(N_1,N_2)}\), 113
-
\(\Xi _{n,U,\varDelta }\), 181
-
\(\overline{{X}}_{n,U}^{{\text {min}}}\), 163
-
\(\overline{{X}}^{{{\text {ord}}},{{\text {min}}}}_{n,U^p(N_1)}\), 162
-
\(\overline{{x}}^{{{\text {ord}}}}_{n,U^p(N_1),\Sigma ^{{\text {ord}}}}\), 176
-
\(\overline{{X}}_{n,U}^{{{\text {min}}},{{\text {n-ord}}}}\), 163
-
\(X^*(T_n)\), 16
-
\(X^*(T_n)^+\), 16
-
\(X^*(T_n)^+_{(n)}\), 25
-
\(X^+_{n,(i),U}\), 122
-
\(X^{(m),\natural }_{n,(i),U}\), 122
-
\(X_{n,U}^{{\text {min}}}\), 159
-
\(\overline{{X}}^{{\text {ord}}}_{n,U^p(N)}\), 92
-
\(X^*(G)\), 92
-
\(X^*\), 44
-
\(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})^\natural \), 135
-
\(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})^{>0,\natural }\), 135
-
\(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})^{>0}\), 135
-
\(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})^{\ge 0,\natural }\), 135
-
\(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})^{\ge 0}\), 135
-
\(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{>0}\), 135
-
\(X^*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\ge 0}\), 135
-
\(X^*(S_{n,(i),U}^{(m),+})^\natural \), 135
-
\(X^*(S_{n,(i),U}^{(m),+})^{>0,\natural }\), 135
-
\(X^*(S_{n,(i),U}^{(m),+})^{\ge 0,\natural }\), 135
-
\(X^*(S_{n,(i),U}^{(m),+})^{\ge 0}\), 135
-
\(X^*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{>0}\), 135
-
\(X^*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\ge 0}\), 135
-
\(X_{n,(i),U}^{(m),+}\), 122
-
\(X_{n,U}\), 90
-
\(X_*\), 44
-
\(X_*(S_{n,(i),U}^{(m),+})_{{\mathbb {R}}^{\succ 0}}\), 135
-
\(X_*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{>0}\), 135
-
\(X_*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\ge 0}\), 135
-
\(X_*({\mathcal {S}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+})_{\mathbb {R}}^{\ge 0}\), 135
-
\(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{\ge 0}\), 135
-
\(X_*(S_{n,(i),U}^{(m),+})_{\mathbb {R}}^{>0}\), 135
-
\({\mathcal {Y}}^\dag \), 202
-
\({\mathcal {Y}}^\wedge _\eta \), 202
-
\({\mathcal {Y}}_{n,(i),U^p(N)}^{(m),{{\text {ord}}},+}\), 121
-
\({\mathcal {Y}}_{n,(i),U^p(N)}^{{{\text {ord}}},\natural }\), 121
-
\(]\overline{{Y}}[\), 202
-
\(Y^{{\text {an}}}\), 202
-
\(Y^\dag \), 202
-
\(Y^{(m),\natural }_{n,(i),U}\), 121
-
\(Y_{n,(i),U}^+\), 120
-
\(Y_{n,(i),U}^{(m),+}\), 120
-
\(Z(\,\,\,)\), 9
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Cite this article
Harris, M., Lan, KW., Taylor, R. et al. On the rigid cohomology of certain Shimura varieties. Res Math Sci 3, 37 (2016). https://doi.org/10.1186/s40687-016-0078-5
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DOI: https://doi.org/10.1186/s40687-016-0078-5