Strata Hasse invariants, Hecke algebras and Galois representations
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Abstract
We construct grouptheoretical generalizations of the Hasse invariant on strata closures of the stacks \(\mathop {G\hbox {}{} \mathtt{Zip}}\nolimits ^{\mu }\). Restricting to zip data of Hodge type, we obtain a grouptheoretical Hasse invariant on every Ekedahl–Oort stratum closure of a general Hodgetype Shimura variety. A key tool is the construction of a stack of zip flags \(\mathop {G\hbox {}{} \mathtt{ZipFlag}}\nolimits ^\mu \), fibered in flag varieties over \(\mathop {G\hbox {}{} \mathtt{Zip}}\nolimits ^{\mu }\). It provides a simultaneous generalization of the “classical case” homogeneous complex manifolds studied by Griffiths–Schmid and the “flag space” for Siegel varieties studied by Ekedahl–van der Geer. Four applications are obtained: (1) Pseudorepresentations are attached to the coherent cohomology of Hodgetype Shimura varieties modulo a prime power. (2) Galois representations are associated to many automorphic representations with nondegenerate limit of discrete series Archimedean component. (3) It is shown that all Ekedahl–Oort strata in the minimal compactification of a Hodgetype Shimura variety are affine, thereby proving a conjecture of Oort. (4) Part of Serre’s letter to Tate on mod p modular forms is generalized to general Hodgetype Shimura varieties.
1 Introduction
 (A)
Automorphic Algebraicity: The inherent algebrogeometric properties of automorphic representations, particularly those conjectured by the Langlands correspondence.
 (B)
GZip geometricity: The geometry engendered by the theory of GZips, including the Ekedahl–Oort (EO) stratification of Shimura varieties, their flag spaces and Hasse invariants.
 (C)
Griffiths–Schmid Algebraicity: Is there an algebrogeometric framework which applies to Griffiths–Schmid manifolds?
The common pursuit of (A), (B) and (C) rests on two themes developed by Deligne, Serre and their collaborators. These themes are (i) geometrybygroups and (ii) characteristicshifting—back and forth between characteristic 0 and p.
Regarding (i), geometrybygroups manifests itself in two stages. The first is that geometric objects at the heart of (A), (B), (C)—Shimura varieties, stacks of GZips and Griffiths–Schmid manifolds—are all constructed from the same grouptheoretic template: A pair \((G, \mu )\) consisting of a reductive group G and a cocharacter \(\mu \). The second stage is guided by the more general hypothesis that all objects constructed from reductive groups should admit a closeknit relationship with algebraic geometry. In our setting, two fundamental testcases are Automorphic Algebraicity and Griffiths–Schmid Algebraicity.
As for (ii), it is wellknown that the method of characteristicshifting applies throughout algebraic geometry; e.g., the characteristic p approach of Deligne–Illusie to Kodaira vanishing and the degeneration of the Hodge–de Rham spectral sequence. Inspired by Deligne–Serre, this paper applies characteristicshifting in tandem with geometrybygroups: The most basic application is to GZip Geometricity. Joined to the latter, the duo is next applied to Automorphic Algebraicity.
Further elaboration of our program is given in our papers [36, 37] and joint work in progress with Brunebarbe et al. [13]. Building on the current work, all of these papers compound evidence that GZip Geometricity is a “mod p Hodge theory” which interacts with classical Hodge theory via characteristicshifting. Specifically, our results suggest that stacks of GZips are mod p analogues of period domains (and more generally Mumford–Tate domains, of which Griffiths–Schmid manifolds are quotients [41]). An idea along these lines was first put forth for GL(n)Zips by Moonen–Wedhorn in the introduction of [82]. The work of Griffiths–Schmid on the manifolds which bear their name [44] motivates the flag spaces which play a key role in this work, see Remark .
1.1 Grouptheoretical Hasse invariants
The main technical result of this paper is the construction of grouptheoretical Hasse invariants on strata closures in the stacks \(\mathop {G\hbox {}{} \mathtt{Zip}}\nolimits ^{\mu }\) (Theorem I.1.1). As recalled below, the theory of the EO stratification has evolved in three stages, progressively shifting from a combinatorial viewpoint to a grouptheoretic one. Our work develops a fourth stage in this progression.
1.1.1 The Ekedahl–Oort stratification
Initially, Oort defined a stratification of \({\mathcal A}_g \otimes {{\mathbf {F}}}_p\), the moduli space of principally polarized abelian varieties in characteristic \(p>0\), by isomorphism classes of the ptorsion [85]. He parametrized strata combinatorially by “elementary sequences”. Later on, Moonen [81] used the canonical filtration of a \(BT_1\) with PELstructure to give a grouptheoretical classification of these. He described isomorphism classes of \(BT_1\)’s over an algebraically closed field k of characteristic p as a certain subset \({}^I W\) of the Weyl group W of the reductive group attached to the PELstructure.
In the third stage, in a series of papers, Moonen, Wedhorn, Pink and Ziegler defined the algebraic stack \(\mathop {G\hbox {}{} \mathtt{Zip}}\nolimits ^\mu \), [82, 87, 88, 106], whose kpoints parametrize isomorphism classes of \(BT_1\)’s with Gstructure over k. Viehmann and Wedhorn showed that the special fiber \(S_{{\mathcal K}}\) of any PELtype Shimura variety admits a universal Gzip of type \(\mu \), which gives rise to a faithfully flat morphism of stacks \(\zeta :S_{{\mathcal K}}\rightarrow \mathop {G\hbox {}{} \mathtt{Zip}}\nolimits ^\mu \), [104]. By definition, the fibers of \(\zeta \) are the EO strata of \(S_{{\mathcal K}}\); this definition agrees with those of Ekedahl–Oort and Moonen in the Siegel and PEL cases respectively. In his thesis, Zhang constructed a universal Gzip over the special fiber of a general Hodgetype Shimura variety. He proved that the induced map \(\zeta \) is smooth [112] (see also [109]).
1.1.2 Hasse invariants
The theory of Hasse invariants in its modern form goes back to the algebrogeometric pursuit of modular forms modulo p, pioneered by Deligne, Katz, Serre and others in the late 1960s and early 1970s. Since then, many people have studied generalizations and applications of the classical Hasse invariant of an abelian scheme cf. [29, 40, 52].
The work [39] by the first author and Nicole was the first to produce a Hasse invariant on a general class of Shimura varieties—those of PELtype A—whose classical ordinary locus is often empty. The second author and Wedhorn extended this result to all Shimura varieties of Hodgetype, [62]. Their work was the first to construct a grouptheoretical generalization of the Hasse invariant on the stack of GZips. A partial grouptheoretical result concerning smaller strata was also obtained by the second author [61]. It is used as a starting point for the construction of Hasse invariants in this paper.
One advantage of the GZip approach to Hasse invariants is that sections obtained by pullback from this stack to a Shimura variety are automatically Heckeequivariant. Another is that geometric properties of such sections can be read off from a root datum of G.
1.1.3 New results on Hasse invariants
Let p be a prime (\(p=2\) allowed). Let G be a connected, reductive \({{\mathbf {F}}}_p\)group and \(\mu \in X_*(G)\) a cocharacter. The works of Moonen–Wedhorn [82] and Pink–Wedhorn–Ziegler [87, 88] define the stack \(\mathop {G\hbox {}{} \mathtt{Zip}}\nolimits ^\mu \) of Gzips of type \(\mu \) (Definition ) and attach to the pair \((G,\mu )\) the zip groupE (Sect. ). This group acts naturally on G and one has \(\mathop {G\hbox {}{} \mathtt{Zip}}\nolimits ^{\mu } \cong [E\backslash G]\). The Eorbits in G are locally closed subsets \(G_w\), parametrized by elements \(w\in {}^I W\) (Sect. ). Let \(L=\mathrm{Cent}(\mu )\) be the centralizer of \(\mu \) in G. Every character \(\chi \in X^*(L)\) gives rise to a line bundle \({\mathscr {V}}(\chi )\) on \(\mathop {G\hbox {}{} \mathtt{Zip}}\nolimits ^\mu \) (Sect. N.4.1). It satisfies \({\mathscr {V}}(N\chi )={\mathscr {V}}(\chi )^N\) for all \(N\ge 1\).
Theorem I.1.1
(Grouptheoretical Hasse invariants, Theorem ) Let \(G_w\subset G\) be an Eorbit; \(\overline{G}_w\) its Zariski closure.^{1} Assume \(\chi \in X^*(L)\) is (p, L)admissible (Definition N.5.3). Then there exists \(N_w\ge 1\) and a section \(h_w\in H^0([E\backslash \overline{G}_w],{\mathscr {V}}(\chi )^{N_w})\) whose nonvanishing locus is exactly the substack \([E\backslash G_w]\).
We call the sections \(h_w\) afforded by Theorem I.1.1grouptheoretical Hasse invariants. The space \(H^0([E\backslash G_w],{\mathscr {V}}(\chi ))\) has dimension \(\le 1\) for any \(\chi \in X^*(L)\), hence the sections \(h_w\) of Theorem I.1.1 are unique up to scalar.
Let \(f:(G_1,\mu _1)\rightarrow (G_2,\mu _2)\) be a finite morphism of cocharacter data and \(\widetilde{f}:\mathop {G_1\hbox {}{} \mathtt{Zip}}\nolimits ^{\mu _1}\rightarrow \mathop {G_2\hbox {}{} \mathtt{Zip}}\nolimits ^{\mu _2}\) the induced map of stacks (Sect. ). Put \(L_i=\mathrm{Cent}(\mu _i)\).
Corollary I.1.2
(Discrete fibers, Theorem ) Assume there exists a \((p,L_2)\)admissible \(\chi \in X^*(L_2)\), whose restriction to \(L_1\) is orbitally pclose (Definition N.5.3). Then \(\widetilde{f}\) has discrete fibers on the underlying topological spaces.
Remark I.1.3
The corollary generalizes [61, Corollary 3] about the \(\mu \)ordinary locus. In [36, Theorem 2], we generalized Corollary I.1.2 above as follows: We proved that if \(f:(G_1,\mu _1)\rightarrow (G_2,\mu _2)\) is a morphism with central schemetheoretic kernel, then \(\widetilde{f}\) has discrete fibers. The extra assumption of Corollary I.1.2 is thus unnecessary.
1.2 The EO stratification of Hodgetype Shimura varieties
For the rest of the introduction, assume \(p>2\). Suppose \(({\mathbf {G}}, {\mathbf {X}})\) is a Shimura datum of Hodge type. Assume \(\mathbf {G}\) is unramified at some prime p and \({\mathcal K}={\mathcal K}_p{\mathcal K}^p\) is an open compact subgroup of \({\mathbf {G}}({{\mathbf {A}}}_f)\) with \({\mathcal K}_p \subset {\mathbf {G}}(\mathbf {Q}_p)\) hyperspecial and \({\mathcal K}^p \subset {\mathbf {G}}({{\mathbf {A}}}_f^p)\). Let \({\mathscr {S}}_{{\mathcal K}}\) be the Kisin–Vasiu integral model at p, of level \({\mathcal K}\), of the associated Shimura variety \({{\,\mathrm{Sh}\,}}({\mathbf {G}}, {\mathbf {X}})\) (Sect. ). Write \(S_{{\mathcal K}}\) for the special fiber of \({\mathscr {S}}_{{\mathcal K}}\) at a a prime \({\mathfrak p}\) of the reflex field dividing p.
We first describe the key applications of Theorem I.1.1 to the EO stratification of \(S_{{\mathcal K}}\) and then discuss extensions to compactifications. In particular, note that the following three corollaries are completely independent of the theory of compactifications.
Let \(\omega \) be the Hodge line bundle on \(S_{{\mathcal K}}\) associated to a symplectic embedding \(\varphi :({\mathbf {G}}, {\mathbf {X}})\rightarrow (GSp(2g), {\mathbf {X}}_{g})\). Let \(\eta _{\omega }\) be the character of L satisfying \(\omega ={\mathscr {V}}(\eta _{\omega })\) (Sect. ). We have shown [38, Theorem 1.4.4] that \(\eta _{\omega }\) is quasiconstant [Definition N.5.3(a)]. Moreover, \(\eta _{\omega }\) is easily seen to be Lample (Definition N.5.1). A quasiconstant character is orbitally pclose for all p (Remark N.5.4). Since (p, L)admissible is defined as Lample and orbitally pclose (Definition N.5.3), \(\eta _{\omega }\) is (p, L)admissible for all p. Hence:
Corollary I.2.1
(Corollary ) Suppose \((G, \mu )\) arises by reduction modulo p from a Hodgetype Shimura datum and \(\chi =\eta _{\omega }\). Then the conclusion of Theorem I.1.1 holds for all primes p.
Remark I.2.2
There is another proof of Corollary I.2.1 that does not use the notion of “quasiconstant characters” and the result of [38, Theorem 1.4.4]. By [36, Corollary 1], Corollary holds for a more general class of pairs \((G,\mu )\), namely the “maximal” ones (loc. cit., Sect. ). The proof is based on the discrete fiber theorem (loc. cit., Theorem 2).
Pulling back the sections of Corollary I.2.1 along \(\zeta \), we obtain Hasse invariants for all EO strata of \(S_{{\mathcal K}}\).
Corollary I.2.3
(EO Hasse invariants, Corollary ) For every EO stratum \(S_w \subset S_{{\mathcal K}}\), there exists \(N_w\ge 1\) and a section \(h_w\in H^0(\overline{S}_w,\omega ^{N_w})\) whose nonvanishing locus is precisely \(S_w\).
The sections \(h_w\) are also \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant, see Corollary . Since the nonvanishing locus of a section of an ample line bundle on a proper scheme is affine, we deduce:
Corollary I.2.4
(Affineness, compact case; Corollary ) Assume \(({\mathbf {G}}, {\mathbf {X}})\) is a Shimura datum of compact type. Then all EO strata in \(S_{{\mathcal K}}\) are affine.
The following results concern extending the EO stratification and its Hasse invariants to compactifications. Let \(S_{{\mathcal K}}^{\Sigma } \) be one of the proper toroidal compactifications of \(S_{{\mathcal K}}\) constructed by MadapusiPera (Sect. ). In the next theorem, we do not assume \(S_{{\mathcal K}}^{\Sigma } \) is smooth, but we do require a technical logintegrality assumption related to \(\Sigma \), see Sects. and .
Theorem I.2.5
 (a)
The map \(\zeta \) admits an extension \(\zeta ^{\Sigma }_{{\mathcal K}}: S_{{\mathcal K}}^{\Sigma } \longrightarrow \mathop {G\hbox {}{} \mathtt{Zip}}\nolimits ^{\mu }\).
 (b)
Let \(G_w \subset G\) be an Eorbit; put \(S_w^{\Sigma }=(\zeta ^{\Sigma }_{{\mathcal K}})^{1}([E\backslash G_w])\) and \(S_w^{\Sigma ,*}=(\zeta ^{\Sigma }_{{\mathcal K}})^{1}([E\backslash \overline{G}_w])\). Then the Hasse invariant \( h_w\in H^0(\overline{S}_w, \omega ^{N_w}))\) of Corollary I.2.3 extends to \(h_w^{\Sigma } \in H^0(S_w^{\Sigma ,*}, \omega ^{N_w})\) with nonvanishing locus \(S_w^{\Sigma }\).
Furthermore, the extension \(\zeta ^{\Sigma }_{{\mathcal K}}\) is \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant (6.2.2). Let \(S_{{\mathcal K}}^{\min }\) be the minimal compactification of \(S_{{\mathcal K}}\) (Sect. ). Define \(S_w^{\min }\) as the image of \(S_w^{\Sigma }\) by the natural map \(S_{{\mathcal K}}^{\Sigma } \rightarrow S_{{\mathcal K}}^{\min } \). We call \(S_w^{\min }\) the extended EO strata in \(S_{{\mathcal K}}^{\min } \). Combining Theorem I.2.5 with a Stein factorization argument, we deduce:
Corollary I.2.6
(Affineness, noncompact case; Proposition ) Suppose \(({\mathbf {G}}, {\mathbf {X}})\) is a Shimura datum of noncompact, Hodge type. Then the extended EO strata \(S_w^{\min }\) are affine for all \(w \in {}^I W\).
In the Siegel case—for \({\mathcal A}_g \otimes {{\mathbf {F}}}_p\) with a suitable level structure—Corollary I.2.6 was conjectured by Oort almost 20 years ago [85, 14.2]. As far as we know, even this special case of Corollary I.2.6 is new. The affineness of the generic EO stratum for Hodgetype Shimura varieties was proved by the second author and Wedhorn [62, Corollary 2].
In the restricted special case of Shimura varieties of PELtype A and C, the thesis of Boxer [10, 11] obtained the Corollaries I.2.3, I.2.4, I.2.6 and a variant of Theorem I.2.5 simultaneously and independently from us.
1.3 Applications to the Langlands correspondence
1.3.1 Galois representations associated to automorphic representations
Most of the work to date on (Gal) has been restricted to cases when the Archimedean component \(\pi _{\infty }\) is regular. Prior to this work, the limited number of results which considered irregular \(\pi _{\infty }\) were all in cases when \(\pi _{\infty }\) is a holomorphic limit of discrete series (LDS); this is the mildest possible type of irregularity [32, 39, 54, 100]. For a detailed classification of Archimedean components in terms of (Gal), see [34].
1.3.2 Pseudorepresentations associated to torsion
Starting with Ash [6], a number of torsion analogues of the Langlands correspondence have been proposed, where automorphic representations are replaced by systems of Hecke eigenvalues appearing in the cohomology (Betti or coherent) of a locally symmetric space with mod \(p^n\) coefficients and the Galois representations (Gal) are replaced by mod \(p^n\)valued pseudorepresentations. The interest in such “torsion Langlands correspondences” has grown considerably due to the pivotal role that they play in the Calegari–Geraghty program of pushing the Taylor–Wiles method beyond the regular case [15].
Scholze achieved a breakthrough in the Betti case, by associating pseudorepresentations to the cohomology of the locally symmetric spaces of GL(n) over a CM field, with mod \(p^n\) coefficients [93]. By contrast, much less was known prior to this work concerning the coherent cohomology of Shimura varieties mod \(p^n\). The only case where pseudorepresentations were associated to higher coherent cohomology (i.e., \(H^i\), \(i>0\)) was that of Hilbert modular varieties, due to Emerton–Reduzzi–Xiao [29].
1.3.3 New results about the Langlands correspondence
In this paper, we prove general results which associate pseudorepresentations to the coherent cohomology of Hodgetype Shimura varieties modulo a prime power. Consequently, we deduce results about the existence of \(r \circ R_{p,\iota }(\pi )\) when G is a \(\mathbf {Q}\)group admitting a Hodgetype Shimura variety and \(\pi _{\infty }\) is an arbitrary nondegenerate LDS.
Let \({\mathscr {S}}_{{\mathcal K}}^{\Sigma }\) be a toroidal compactification of a Hodge type Shimura variety \({\mathscr {S}}_{{\mathcal K}}\) as in Sect. I.2. We now add the assumption that \({\mathscr {S}}_{{\mathcal K}}^{\Sigma }\) is smooth. Write \({\mathscr {S}}_{{\mathcal K}}^{\Sigma ,n} \) for its reduction mod \(p^n\), so that \(S_{{\mathcal K}}^{\Sigma }={\mathscr {S}}_{{\mathcal K}}^{\Sigma , 1}\) is its special fiber. Let \({\mathscr {V}}^{\mathrm{sub}}(\eta )\) be the subcanonical extension to \({\mathscr {S}}_{{\mathcal K}}^{\Sigma } \) of an automorphic vector bundle \({\mathscr {V}}({\eta })\) on \({\mathscr {S}}_{{\mathcal K}}\) (Sect. ). Let \({\mathcal H}\) denote the (global, unramified, primetop) Hecke algebra of \(\mathbf {G}\) and \({\mathcal H}^{i,n}(\eta )\) its image in \(\mathrm{End}(H^i({\mathscr {S}}_{{\mathcal K}}^{\Sigma ,n} , {\mathscr {V}}^{\mathrm{sub}}(\eta )))\) (Sect. ).
The following two theorems summarize our results; see Theorems , , for the precise statements. Unconditional analogues of Theorem I.3.2 for unitary groups are given in Theorems and .
Theorem I.3.1
For Shimura varieties of PEL type A or C, Boxer independently and simultaneously obtained the weaker statement that \({\mathcal H}\twoheadrightarrow {\mathcal H}^{i,n}(\eta )\) factors through \({\mathcal H}\twoheadrightarrow {\mathcal H}^{0,n}(\eta +a\eta _{\omega })\) for infinitely many a, [10, 11]. In contrast with Theorem I.3.1, the weights \(\eta +a\eta _{\omega }\) may all be singular; see also Remark .
Theorem I.3.2
 (a)For every triple \((i, n, \eta )\), there exists a continuous Galois pseudorepresentationwhich for unramified v, \(v \ne p\), maps \(\mathrm{Frob}_v^j\) to the Hecke operators \(T_v^{(j)}\) defined in Sect. .$$\begin{aligned}R_{p,\iota }(r;i,n, \eta ): \mathrm{Gal}(\overline{{{\mathbf {Q}}}}/ \mathbf {Q})\longrightarrow {\mathcal H}^{i,n}(\eta ),\end{aligned}$$
 (b)
Assume \(\pi \) is a cuspidal automorphic representation of \(\mathbf {G}\) with \(\pi _{\infty }\) a nondegenerate, Calgebraic LDS and \(\pi _p\) unramified. Then \(r \circ R_{p, \iota }(\pi )\) exists too.
The case of cohomological degree \(i>0\) reveals genuinely new obstacles (for both Theorems I.3.1 and I.3.2). In terms of automorphic representations, \(i>0\) corresponds to the condition that the nondegenerate LDS \(\pi _{\infty }\) is nonholomorphic. When \(i>0\), it is apparent that the original Deligne–Serre method of multiplication by a single mod p automorphic form is insufficient.
Our new idea to overcome this problem is to use not just one mod p automorphic form, but rather a whole family of such viz. the strata Hasse invariants \(h_w\) and their toroidal extensions \(h_w^{\Sigma }\). These are ‘generalized mod p automorphic forms’, in that they aren’t defined on the whole special fiber of the Shimura variety, but only on EO strata closures. As pullbacks of sections on strata of \(\mathop {G\hbox {}{} \mathtt{Zip}}\nolimits ^\mu \), the \(h_w, h_w^{\Sigma }\) are \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant (Sect. , Theorem ). The reduction to the case \(i=0\) is accomplished by studying the long exact sequences in coherent cohomology associated to technical modifications of the \(h_w,h_w^{\Sigma }\).
Results similar to Theorems I.3.1 and I.3.2 were obtained simultaneously and independently by Pilloni–Stroh [86]. Their method is completely different: It is based on Scholze’s theory of perfectoid Shimura varieties. The analogue of Theorem I.3.2(a) in loc. cit. concerns the coherent cohomology of Scholze’s integral models of \({{\,\mathrm{Sh}\,}}({\mathbf {G}}, {\mathbf {X}})\) rather than the modular Kisin–Vasiu models studied here. Boxer also announced applications of his thesis to the construction of Galois representations, but as far as we know, no preprint containing such results has appeared.
1.4 Serre’s letter to Tate on mod p modular forms
Let N be prime to p and let X(N) (resp. \((X(N)^{\mathrm{ss}}\)) be the modular curve of full level N (resp. its supersingular locus) in characteristic p. In his letter to Tate [94], Serre first showed that the systems of Hecke eigenvalues which appear in \( \bigoplus _{k \in \mathbf {N}} H^0(X(N), \omega ^k)\) are the same as those which appear in \(\bigoplus _{k \in \mathbf {N}} H^0(X(N)^{\mathrm{ss}}, \omega ^{k})\). He went on to show that these systems of Hecke eigenvalues are also precisely those that appear in Gross’ algebraic modular forms for the quaternion algebra \(B_{p,\infty }\) ramified at \(\{p,\infty \}\).
As a further application of grouptheoretical Hasse invariants, we generalize Serre’s first result to arbitrary Shimura varieties of Hodge type. Let D be the boundary divisor of \({\mathscr {S}}_{{\mathcal K}}\) in \({\mathscr {S}}_{{\mathcal K}}^{\Sigma }\). Let \(S_e\) be the (unique) zerodimensional EO stratum of \(S_{{\mathcal K}}\).
Theorem I.4.1
(see Theorem ) If \(({\mathbf {G}}, {\mathbf {X}})\) is neither of compacttype, nor of PELtype, then assume that there exists a a \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant Cartier divisor \(D'\) such that \(D'_{\mathrm{red}}=D\) and \(\omega ^k(D')\) is ample on \({\mathscr {S}}_{{\mathcal K}}^{\Sigma }\) for all \(k \gg 0\). As \(\eta \) ranges over all weights, the systems of Hecke eigenvalues appearing in each of \(\bigoplus _{\eta }H^0(S_{{\mathcal K}}, {\mathscr {V}}({\eta }))\), \(\bigoplus _{\eta }H^0(S_{{\mathcal K}}^{\Sigma }, {\mathscr {V}}^{\mathrm{sub}}(\eta ))\) and \( \bigoplus _{\eta }H^0(S_e, {\mathscr {V}}({\eta }))\) are the same. In particular the number of such systems is finite.
This application lies on the border between (A) Automorphic Algebraicity and (B) GZip Geometricity. The existence of such \(D'\) is known in the PEL case by Lan [66, Theorem 7.3.3.4] (see also Remark below) and it should follow similarly in the general Hodge case from the work of MadapusiPera [77]. In future work, we hope to return to Serre’s second result in this level of generality. It is likely that it follows from the recent preprint [110], but we have not checked this in detail.
No cases of Theorem I.4.1 were previously known where the classical superspecial locus of \(S_{{\mathcal K}}\) is empty.^{2} Serre’s two results had previously been generalized by Ghitza to Siegel modular varieties [30] and by Reduzzi [90] to a (rather restricted) class of PELtype Shimura varieties. Unfortunately, these works seem to contain a nontrivial error, see Remark . Both were limited to PEL cases where the classical superspecial locus of \(S_{{\mathcal K}}\) is nonempty. Analogous to the distinction between the classical ordinary locus and the \(\mu \)ordinary locus (=unique open EO stratum), examples of both PEL and Hodgetype abound where the classical superspecial locus is empty. For example, if \(({\mathbf {G}}, {\mathbf {X}})\) is of unitary type with reflex field \(E \ne \mathbf {Q}\) and p splits completely in E, then the classical superspecial locus of \(S_{{\mathcal K}}\) is empty.
Using Theorems I.3.1 and I.4.1, we deduce:
Corollary I.4.2
1.5 Outline
Following a preliminary Sect. N to fix notation, this paper is naturally divided into three parts: The primary goal of Part 1 (Sects. 1–3) is the construction of grouptheoretical Hasse invariants (Theorem I.1.1). To this end, we introduce the stack of zip flags (Sect. 2). Parts 2 and 3 are concerned with applications. Part 2 contains those applications which are directly concerned with the geometry of the EO stratification of Shimura varieties, but which avoid Hecke operators. By contrast, Part 3 regards three applications involving the Hecke algebra: (i) Factorization of the Hecke algebra action on the coherent cohomology of automorphic vector bundles (Sects. 8 and 9), (ii) Association of Galois pseudorepresentations to coherent cohomology modulo \(p^n\) and Galois representations to automorphic representations of nondegenerate LDStype (Sect. 10), (iii) Generalization of Serre’s Letter to Tate (Sect. 11). We refer to the beginning of each section for a more detailed description of its contents.
2 Notation
2.1 Ring theory
N.1.1. Throughout, k denotes an algebraically closed field.
2.1.1 Adeles
The adele ring of \(\mathbf {Q}\) is written \(\mathbf {A}\), the finite adeles \(\mathbf {A}_f\). Given a prime p, \(\mathbf {A}_f^p\) denotes the finite adeles with trivial padic component.
2.1.2 CM fields
A number field is a finite extension of \(\mathbf {Q}\). In this paper, ‘CM field’ means a number field which is either totally real or a totally imaginary quadratic extension of a totally real field.
2.2 Scheme theory
2.2.1 Base change
If A is a commutative ring with 1, M is an Amodule and B an Aalgebra, we write \(M_B\) for the Bmodule \(M \otimes _A B\).
If S is a scheme, \(f:X \rightarrow S\) is an Sscheme and \(g:T \rightarrow S\) is a morphism of schemes, write \(X_T\) for the base change \(X \times _S T\). If \(S={{\,\mathrm{Spec}\,}}(A)\) and \(T={{\,\mathrm{Spec}\,}}(B)\), then we also write \(X_B\) in place of \(X_T\).
2.2.2 Frobenius twist
Let k be a field of characteristic p, and \(\sigma :k\rightarrow k\) the map \(x\mapsto x^p\). For a kscheme X, we denote by \(X^{(p)}\) the fiber product \(X^{(p)}:=X\otimes _{k,\sigma } k\).
2.2.3 Reduction modulo \(p^n\)
Let p be a prime, \(K/\mathbf {Q}_p\) a finite extension with ring of integers \({\mathcal O}_K\) and maximal ideal \({\mathfrak p}\). If \({\mathcal X}\) is an \({\mathcal O}_K\)scheme, write \({\mathcal X}^n\) for the base change of \({\mathcal X}\) along \({{\,\mathrm{Spec}\,}}{\mathcal O}_K/{\mathfrak p}^n \rightarrow {{\,\mathrm{Spec}\,}}{\mathcal O}_K\). In particular, \({\mathcal X}^1\) is the special fiber of \({\mathcal X}\).
Occasionally it will be useful to consider \({\mathcal X}^n\) for all \(n \in \mathbf {Z}_{\ge 1}\), as well \({\mathcal X}\) and its generic fiber \({\mathcal X}\otimes _{{\mathcal O}_K}K\). For this purpose we adopt the convention of writing \({\mathcal X}^+:={\mathcal X}\) and \({\mathcal X}^0:={\mathcal X}\otimes _{{\mathcal O}_K}K\).
2.2.4 Sections of line bundles
If X is a scheme, \({\mathscr {L}}\) is a line bundle on X and \(s \in H^0(X, {\mathscr {L}})\), write Z(s) for the scheme of zeroes of s in X (cf. [51, App. A, C6]) and \({{\,\mathrm{nonvanish}\,}}(s)\) for the open subset \(X {\setminus } Z(s)\).
We say \(s \in H^0(X, \mathcal L)\) is injective if the map of sheaves \(\mathcal O_X\rightarrow \mathcal L\) given by multiplication by s is injective.^{3} Equivalently s is injective if and only if Z(s) is an effective Cartier divisor in X.
2.2.5 Associated reduced scheme
If X is any scheme, \(X_{\mathrm{red}}\) will denote the associated reduced subscheme.
2.3 Structure theory and root data
2.3.1 Root data
Let G be an algebraic kgroup. Write \(X^*(G)\) (resp. \(X_*(G)\)) for the group of characters (resp. cocharacters) of G. If T is a ktorus, write \(\langle , \rangle \) for the perfect pairing \(X^*(T)\times X_*(T)\rightarrow \mathbf {Z}\).
For a connected, reductive kgroup G and a maximal torus \(T\subset G\), write \(\Phi :=\Phi (G,T)\) for the Troots in G, \(\Phi ^{\vee }:=\Phi ^{\vee }(G,T)\) for the Tcoroots in G and \({\mathcal R}{\mathcal D}(G,T)=(X^*(T), \Phi , X_*(T), \Phi ^{\vee })\) for the root datum of T in G. If L is a Levi subgroup of G containing T, set \(\Phi _L:=\Phi (L,T)\) (resp. \(\Phi _L^{\vee }:=\Phi ^{\vee }(L,T)\)).
2.3.2 Based root data
2.3.3 Weyl group
Let \(W:=W(G,T)\) be the Weyl group of T in G. For \(\alpha \in \Phi \), let \(s_{\alpha }\) denote the root reflection about \(\alpha ^{\vee }\). Write \(\ell :W \rightarrow \mathbf {Z}_{\ge 0}\) for the length function of the Coxeter group \((W, \{s_{\alpha }\}_{\alpha \in \Delta })\). The longest element of W is denoted by \(w_0\) and the identity element by e.
If \(L\subset G\) is a Levi subgroup containing T, denote by \(W_L\subset W\) the Weyl group of L. Write \(w_{0,L}\) for the longest element in \(W_L\). If \(P\subset G\) is a parabolic subgroup, denote its type by \({{\,\mathrm{type}\,}}(P)\subset \Delta \). It is normalized as follows: If P contains B with Levi subgroup L containing T, then \({{\,\mathrm{type}\,}}(P):=\Delta _L\).
2.3.4 Weyl chambers
Write \(\alpha ^{\perp }\) for the hyperplane in \(X^*(T)_{\mathbf {R}}\) orthogonal to \(\alpha ^{\vee }\). A Weyl chamber is a connected component C of \(X^*(T)_{\mathbf {R}}\backslash \bigcup _{\alpha \in \Phi }\alpha ^{\perp }\); thus C is an open cone for the Archimedean topology. Write \(\overline{C}\) for the closure of C and \(\partial C:=\overline{C}\backslash C\) for its boundary. One says that \(\alpha \in \Phi \) (resp. \(\alpha ^{\vee } \in \Phi ^{\vee }\)) is Cpositive if \(\langle \chi , \alpha ^{\vee } \rangle >0\) for any (equivalently every) \(\chi \in C\). The Cpositive roots (resp. coroots) form a system of positive roots (resp. coroots).
2.3.5 Galois action
If B is a Borel subgroup of \(G_{ \kappa '}\) for some finite extension \(\kappa '/\kappa \), the based root datum of (G, B, T) is that of the triple \((G_k, B_k, T_k)\).
2.3.6 Ramification
In the notation of Sect. N.3.5, suppose \(\kappa \) is a number field. The set of finite places v of \(\kappa \) where G ramifies, together with all the infinite places is denoted \({{\,\mathrm{Ram}\,}}(G)\). Similarly, if \(\pi \) is an admissible representation of \(G(\mathbf {A}_{\kappa })\) given by a restricted tensor product of components \(\pi _v\), then \({{\,\mathrm{Ram}\,}}(\pi )\) denotes the set of places v where \(\pi _v\) is ramified, together with all the Archimedean places.
2.4 Quotients
If an algebraic kgroup G acts on a kscheme X, we denote by \([G\backslash X]\) the associated quotient stack. Denote by \(\pi :X\rightarrow [G\backslash X]\) the natural projection. If \(\cdot :X\times G \rightarrow X\) is a right action, we define [X / G] as \([G\backslash X]\) for the action \(g\star x := x\cdot g^{1}\).
2.4.1 Vector bundles on quotient stacks
2.5 Conditions on characters
Let (G, B, T) as in Sect. N.3.5 and \(L\subset G\) a Levi subgroup containing T. Let \(I\subset \Delta \) be the type of the parabolic \(P=LB\).
Definition N.5.1
We say that \(\chi \in X^*(L)\) is Lample if it satisfies \(\langle \chi , \alpha ^{\vee } \rangle <0\) for all \(\alpha \in \Delta {\setminus } I\).
Remark N.5.2
 (a)
\(\chi \) is Lample.
 (b)
The associated line bundle \({\mathscr {L}}(\chi )\) is antiample on G / P.
Definition N.5.3
 (a)
Orbitally pclose if \(\max {{\,\mathrm{Orb}\,}}(\chi , \alpha ^{\vee })\le p1\) for all \(\alpha \in \Phi \) with \(\langle \chi , \alpha ^{\vee } \rangle \ne 0\).
 (b)
Quasiconstant if \({{\,\mathrm{Orb}\,}}(\chi , \alpha ^{\vee }) \subset \{0,1\}\) for all \(\alpha \in \Phi \) with \(\langle \chi , \alpha ^{\vee } \rangle \ne 0\).
 (c)
psmall if \(\langle \chi , \alpha ^{\vee } \rangle  \le p1\) for all \(\alpha \in \Phi \).
 (d)
(p, L)admissible if \(\chi \) is orbitally pclose and Lample.
Remark N.5.4
For more on quasiconstant (co)characters, including a general classification and applications, see [38].
Definition N.5.5
Let \(\delta \in \mathbf {R}_{\ge 0}\). We say that \(\chi \in X^*(T)\) is \(\delta \)regular if \(\langle \chi , \alpha ^{\vee } \rangle  >\delta \) for all \(\alpha \in \Phi \).
The case \(\delta =0\) gives the usual notion of regularity.
2.6 Hodge structures
Let \(\mathbf {S}:={{\,\mathrm{Res}\,}}_{{{\mathbf {C}}}/{\mathbf {R}}}{\mathbf {G}}_{m, {\mathbf {C}}}\) be the Deligne torus. An \(\mathbf {R}\)Hodge structure is a morphism of \(\mathbf {R}\)algebraic groups \(\mathbf {S}\rightarrow GL(V_{\mathbf {R}})\), where \(V_{\mathbf {R}}\) is an \(\mathbf {R}\)vector space.
2.6.1 Hodge bigrading convention
An \(\mathbf {R}\)Hodge structure \(h: \mathbf {S}\rightarrow GL(V_{\mathbf {R}})\) induces a bigrading \(V_{\mathbf {C}} = \oplus _{a,b \in \mathbf {Z}}V^{a,b}\). Our convention is that of Deligne [24, 1.1.6]: \(z \in \mathbf {S}(\mathbf {R})\) acts on \(V^{a,b}\) via \(z^{a}{\bar{z}}^{b}\). In particular, a complex structure on \(V_{\mathbf {R}}\) is an \(\mathbf {R}\)Hodge structure of type \(\{(0,1),(1,0)\}\).
2.6.2 Associated cocharacter
There is a natural isomorphism \( \mathbf {S}_\mathbf {C}\simeq \prod _{{{\,\mathrm{Gal}\,}}(\mathbf {C}/\mathbf {R})} {\mathbf {G}}_{m, {\mathbf {C}}}. \) Let \(\mu _0 : {\mathbf {G}}_{m, {\mathbf {C}}}\rightarrow \mathbf {S}_\mathbf {C}\) be the natural injection onto the factor corresponding to \(\mathrm{Id}\in {{\,\mathrm{Gal}\,}}(\mathbf {C}/\mathbf {R})\). If G is a connected, reductive \(\mathbf {R}\)group and \(h: \mathbf {S}\rightarrow G\) is a morphism of \(\mathbf {R}\)groups, the cocharacter \(\mu \in X_*(G)\) associated to h is \(\mu :=h_{\mathbf {C}} \circ \mu _0\).
3 Part 1. Grouptheoretical Hasse invariants
Part 1 is devoted to grouptheoretical objects. Section recalls basic facts about morphisms of quotient stacks, used throughout this part. In Sect. , we recall the definition of the stack of GZips and introduce the notion of cocharacter data. Those of Hodgetype are singled out in Sect. . In Sect. , we recall the parametrization and properties of zip strata.
The topic of Sect. 2 is the stack of zip flags \(\mathop {G\hbox {}{} \mathtt{ZipFlag}}\nolimits ^\mu \), it is key for constructing Hasse invariants. The definition is given in Sect. . We recall the definition and the properties of the ‘Schubert stack’ in Sect. . It is used to define a stratification of \(\mathop {G\hbox {}{} \mathtt{ZipFlag}}\nolimits ^\mu \) in Sect. . Special strata termed ‘minimal’ and ‘cominimal’ are studied in Sect. .
We apply these tools in Sect. 3 to construct grouptheoretical Hasse invariants. First, Sect. studies line bundles on these various stacks. The main result of Part 1 is Theorem , proved in Sect. . We discuss functoriality and deduce Theorem in Sect. . Finally, we introduce the cone of global sections in Sect. ; it will be used later in Sect. .
4 The stack of Gzips
4.1 Morphisms of quotient stacks
4.2 The stack of Gzips
4.2.1 Cocharacter datum
We denote by k an algebraic closure of \({{\mathbf {F}}}_p\). A cocharacter datum is a pair \((G,\mu )\) consisting of a connected reductive \({{\mathbf {F}}}_p\)group G and a cocharacter \(\mu :\mathbf {G}_{m,k}\rightarrow G_k\). We will denote by \(\varphi :G\rightarrow G\) the Frobenius homomorphism.
Definition 1.2.1
Let \((G_1,\mu _1)\) and \((G_2,\mu _2)\) be two cocharacter data. A morphism \(f:(G_1,\mu _1)\rightarrow (G_2,\mu _2)\) is a morphism of groups \(f:G_1\rightarrow G_2\) defined over \({{\mathbf {F}}}_p\), such that \(\mu _2=f\circ \mu _1\).
We say that f is an embedding (resp. finite) if the underlying map \(f:G_1\rightarrow G_2\) is an embedding (resp. finite). Cocharacter data form a category.
Definition 1.2.2
[88, Definition 1.4] A Gzip of type \(\mu \) over S is a tuple \(\underline{I}=(I,I_P,I_Q,\iota )\) where I is a Gtorsor over S, \(I_P\subset I\) is a Ptorsor, \(I_Q\subset I\) is a Qtorsor, and \(\iota : (I_P)^{(p)}/U^{(p)} \rightarrow I_Q/V\) an isomorphism of Mtorsors.
The category of Gzips over S is denoted by \(\mathop {G\hbox {}{} \mathtt{Zip}}\nolimits ^{\mu }(S)\). This gives rise to a fibered category \(\mathop {G\hbox {}{} \mathtt{Zip}}\nolimits ^{\mu }\) over the category of kschemes, which is a smooth algebraic stack of dimension 0 [88, Theorem 1.5].
4.2.2 Representation as a quotient stack
4.2.3 Conjugation
Assumption 1.2.3
There exists a Borel pair (B, T) in G defined over \({{\mathbf {F}}}_p\) such that \(B\subset P\).
If \((G,\mu )\) is an arbitrary cocharacter datum, we can find \(g\in G(k)\) such that \((G,{}^g\mu )\) satisfies Assumption . By (1.2.4), it is harmless to assume that it is satisfied.
4.3 Cocharacter data of Hodge type
Definition 1.3.1
 (a)
We say that \((G,\mu )\) is of Hodgetype if there exists a Siegeltype cocharacter datum \((GSp(W,\psi ),\mu _{\mathcal D})\) and an embedding \(\iota : (G,\mu )\rightarrow (GSp(W,\psi ),\mu _{\mathcal D})\).
 (b)
Given an embedding \(\iota \) as in (a), we denote again by \(\eta _\omega \) the restriction of \(\eta _\omega \) to L, and call it the Hodge character of \((G, \mu )\) relative to \(\iota \).
A posteriori, at least when \(\iota \) arises from an embedding of Shimura data, the dependence of \(\eta _{\omega }\) on \(\iota \) is very minimal, see Remark .
4.4 Stratification
Let \((G,\mu )\) be a cocharacter datum, and let P, Q, L, M, E be the attached groups, as defined in Sect. . We assume that there exists a Borel pair (B, T) satisfying Assumption . Denote by \(I,J\subset \Delta \) the type of P, Q respectively (as defined in Sect. N.3.3).
5 The stack of zip flags
5.1 Definition
Fix a cocharacter datum \((G,\mu )\) and a Borel pair (B, T) satisfying Assumption .
Definition 2.1.1
A Gzip flag of type \(\mu \) over a kscheme S is a pair \(\hat{I}=(\underline{I},J)\) where \(\underline{I}=(I,I_P,I_Q,\iota )\) is a Gzip of type \(\mu \) over S, and \(J\subset I_P\) is a Btorsor.
Theorem 2.1.2
Proof
The proof is similar to that of [88, Proposition 3.11]. Let S be a scheme over k. To \((g,r)\in (G\times P)(S)\), we attach a “standard” Gzip flag \(\hat{I}_{(g,r)}=(\underline{I}_{g}, J_{(g,r)})\) over S. Let \(\underline{I}_g\) be the Gzip over S attached to g by Construction 3.4 in loc. cit. Define \(J_{(g,r)}\) as the image of \(B\times S \subset P\times S\) under left multiplication by r.
We claim that \((g,r)\mapsto \hat{I}_{(g,r)}\) is then a fully faithful functor \({\mathcal Y}(S) \rightarrow \mathop {G\hbox {}{} \mathtt{ZipFlag}}\nolimits ^\mu (S)\), i.e. that there is a natural bijection between the set of morphisms \(\phi :\hat{I}_{(g,r)}\rightarrow \hat{I}_{(g',r')}\) and \({{\,\mathrm{Transp}\,}}((g,r),(g'r'))\). A morphism \(\phi :\hat{I}_{(g,r)}\rightarrow \hat{I}_{(g',r')}\) consists in particular of a morphism \(\varphi :\underline{I}_g\rightarrow \underline{I}_{g'}\). By [88, Lemma 3.10], one can attach to \(\varphi \) a pair \(\epsilon =(p_+,p_)\in E(S)\), (using the notation of loc. cit.). By compatibility, the map \(J_{(g,r)}\rightarrow J_{(g',r')}\) must be induced by left multiplication by \(p_+\). Since \(J_{(g,r)}=r(B\times S)\) and \(J_{(g',r')}=r'(B\times S)\), we have \(b:=r'^{1}p_+r \in B(S)\). We obtain a map \(\phi \mapsto (\epsilon , b)\), and it is easy to check that it is a bijection.
To show that we obtain an isomorphism \(\mathop {G\hbox {}{} \mathtt{ZipFlag}}\nolimits ^\mu \simeq [(E\times B) \backslash (G\times P)]\), it remains to prove that any Gzip flag is étale locally of the form \(\hat{I}_{(g,r)}\). Let \(\hat{I}=(\underline{I},J)\) be a Gzip flag over a kscheme S, where \(\underline{I}=(I,I_P,I_Q,\iota )\) is a Gzip. By [88, Lemma 3.5], the Gzip \(\underline{I}\) is étale locally of the form \(\underline{I}_g\). Furthermore, we may choose an étale extension which trivializes the Btorsor J. Since \(J\subset I_P\) by definition, there exists an element \(r\in P(S)\) such that J is the image of \(B\times S\) under left multiplication by r. Thus \(\underline{I}=\hat{I}_{(g,r)}\). This completes the proof of the isomorphism. From our construction, it is clear that the diagram commutes. \(\square \)
5.2 The Schubert stack
Theorem 2.2.1
 (a)
\(H^0\left( {{\,\mathrm{Sbt}\,}}_w,{\mathscr {L}}_{{{\,\mathrm{Sbt}\,}}}(\lambda ,\mu )\right) \ne 0\Longleftrightarrow \mu = w^{1} \lambda \).
 (b)
\(\dim _k H^0\left( {{\,\mathrm{Sbt}\,}}_w,{\mathscr {L}}_{{{\,\mathrm{Sbt}\,}}}(\lambda ,w^{1} \lambda ) \right) =1\).
 (c)For any nonzero \(f\in H^0\left( {{\,\mathrm{Sbt}\,}}_w,{\mathscr {L}}_{{{\,\mathrm{Sbt}\,}}}(\lambda ,w^{1} \lambda ) \right) \) viewed as a rational function on \(\overline{C}_w\), one has$$\begin{aligned} \mathrm{div}(f)=\sum _{\alpha \in E_w} \langle \lambda , w\alpha ^\vee \rangle \overline{C}_{w s_\alpha }. \end{aligned}$$(2.2.3)
Proof
5.3 Schubert stratification of \(\mathop {G\hbox {}{} \mathtt{ZipFlag}}\nolimits ^\mu \)
Lemma 2.3.1
 (a)
The closed flag strata are normal and irreducible.
 (b)
The closed flag strata coincide with the closures of the flag strata.
 (c)
For all \(w\in W\), one has \(\dim (H_w)=\ell (w)+\dim (P)\).
Proof
Since \(\overline{C}_w\) is normal and \(\psi \) is smooth, \(\overline{{\mathcal X}}_w\) is normal. The smooth morphism \(\widetilde{\psi } : G\times P\rightarrow G\) has all fibers isomorphic to P. Since \(C_w\) is irreducible and \(\widetilde{\psi }\) is open with irreducible fibers, \(\widetilde{\psi }^{1}(C_w)\subset G\times P\) is irreducible. Hence \(H_w\) is irreducible. Finally, the last two assertions follow from the smoothness of \(\psi \). \(\square \)
5.4 Minimal, cominimal strata
Lemma 2.4.1
For all \(w\in W\), \(\tilde{\pi }(H_w)\) is the union of Eorbits intersecting \(B\dot{w}\dot{z}^{1}\).
Proof
It is clear that \(\tilde{\pi }(H_w)\) is the union of Eorbits intersecting \(C_w\dot{z}^{1}\). One has \(C_w\dot{z}^{1}=B( \dot{w}\dot{z}^{1}) {}^z\!B\), so any such Eorbit intersects \(B\dot{w}\dot{z}^{1}\) and conversely. \(\square \)
Definition 2.4.2
We call \({}^I W\) (resp. \(W^J\)) the set of minimal (resp. cominimal) elements of W (Sect. N.3.3).
Similarly, the flag stratum \({\mathcal X}_w\) is called minimal (resp. cominimal) if w is minimal (resp. cominimal). Note that the longest minimal (resp. cominimal) element is \(w_{0,I}w_0\) (resp. \(w_0w_{0,J}\)). The identity element e is both the shortest minimal and cominimal element.
Proposition 2.4.3
 (a)
One has \(\tilde{\pi }(H_w)=G_w\) and \(\tilde{\pi }(\overline{H}_w)=\overline{G}_w\).
 (b)
The Eaction on \(H_w\) is transitive.
 (c)
The preimage of \(G_w\) by the morphism \(\tilde{\pi }:\overline{H}_w\rightarrow \overline{G}_w\) is exactly \(H_w\).
 (d)
The map \(\tilde{\pi }:H_w\rightarrow G_w\) is finite.
Proof
The first part of (a) follows from [87, Theorem 5.14] for w minimal and from [87, Theorem 11.3] for w cominimal [with appropriate modifications due to our choice of frame \(({}^zB,T,\dot{z}^{1})\)]. The second part follows from the properness of \(\tilde{\pi }\). To show (b), note that \(\dim (H_w)=\dim (G_w)\) (Lemma (c) and (1.4.4)). Since the map \(\tilde{\pi }:H_w\rightarrow G_w\) is Eequivariant and \(G_w\) is an Eorbit, all its fibers are isomorphic. In particular, it is quasifinite. Hence \(H_w\) contains finitely many Eorbits. But if \(Z\subset H_w\) is an Eorbit, it must map surjectively onto \(G_w\) (as \(G_w\) is an Eorbit). We deduce \(\dim (Z)\ge \dim (G_w)=\dim (H_w)\). Hence any Eorbit of \(H_w\) has dimension \(\dim (H_w)\), which clearly shows that \(H_w\) is an Eorbit. To prove (c), assume there exists \(y\in \overline{H}_w{\setminus } H_w\) such that \(\tilde{\pi }(y)\in G^w\). Let \(w'\in W\) such that \(y\in H_{w'}\). Then \(H_{w'}\subset \overline{H}_w\) and \(\pi (\overline{H}_{w'})=\overline{G}_w\). But this is impossible since \(\dim H_{w'} < \dim (H_w)=\dim (G_w)\). Finally, (c) shows that \(\pi :H_w\rightarrow G_w\) is proper and quasifinite, so it is finite, which proves (d). \(\square \)
Remark 2.4.4
It is proved in [60, Proposition 2.2.1(ii)] that the map \(\tilde{\pi }:H_w\rightarrow G_w\) is also étale.
Corollary 2.4.5
For all Eorbits \(S\subset G\), there is a unique minimal stratum H and a unique cominimal stratum \(H'\) such that \(\pi (H)=\pi (H')=S\).
6 Hasse invariants
6.1 Line bundles
 (1)
Identify \(X^*(E)=X^*(P)=X^*(L)\) via the first projection \(E\rightarrow P\) and the inclusion \(L\subset P\). By Sect. N.4.1, a character \(\lambda \in X^*(L)\) gives rise to a line bundle \({\mathscr {V}}(\lambda )\) on the quotient stack \(\mathop {G\hbox {}{} \mathtt{Zip}}\nolimits ^\mu \simeq \left[ E \backslash G\right] \).
 (2)
Similarly, identify \(X^*(E')=X^*(B)=X^*(T)\) using the first projection \(E'\rightarrow B\) and the inclusion \(T\subset B\). For a character \(\lambda \in X^*(T)\), we obtain by Sect. N.4.1 a line bundle \({\mathscr {L}}(\lambda )\) on \(\mathop {G\hbox {}{} \mathtt{ZipFlag}}\nolimits ^\mu \simeq [E'\backslash G]\).
Lemma 3.1.1
 (a)
For all \(\lambda \in X^*(L)\), one has \(\pi ^*{\mathscr {V}}(\lambda )={\mathscr {L}}(\lambda )\).
 (b)For \(\lambda ,\nu \in X^*(T)\), one haswhere \(\sigma :k\rightarrow k\) denotes the inverse of the map \(x\mapsto x^p\).$$\begin{aligned} \psi ^*{\mathscr {L}}_{{{\,\mathrm{Sbt}\,}}}(\lambda ,\nu )={\mathscr {L}}(\lambda +p \ {}^\sigma (z\nu )), \end{aligned}$$(3.1.1)
Proof
In view of Theorem (a), we now restrict ourselves to line bundles of the form \({\mathscr {L}}_{{{\,\mathrm{Sbt}\,}}}(\lambda , \nu )\) where \(\nu = w^{1}\lambda \) for \(w\in W\). For \(w\in W\) and \(r\ge 1\), define \(w^{(0)}=e\) and by induction \(w^{(r)}:={}^\sigma (w^{(r1)}w)\) for all \(r\ge 1\).
Lemma 3.1.2
 (a)
For all \(r,s\ge 1\) and \(w\in W\), one has \({}^{\sigma ^s}(w^{(r)})w^{(s)}=w^{(r+s)}\).
 (b)
The set \(R:=\{r\ge 0 \  \ w^{(r)}=e\}\) is a nontrivial submonoid of \(\mathbf {Z}_{\ge 0}\).
 (c)
If \(w^{(r)}=e\) for \(r\ge 1\), then \(w^{(r1)}=w^{1}\).
Proof
The first part follows from an easy induction. Hence R is stable under addition. Since W is finite, there exists \(r>s\ge 0\) such that \(w^{(r)}=w^{(s)}\). By (a), we have \(w^{(rs)}=e\). Finally, (c) is clear from the definition. \(\square \)
Lemma 3.1.3
 (a)
The map \(D_w\) induces a \(\mathbf {Q}\)linear automorphism of \(X^*(T)_\mathbf {Q}\).
 (b)The inverse of \(D_w\) is given as follows: Let \(\chi \in X^*(T)\). Fix \(r\ge 1\) such that \((zw^{1})^{(r)}=e\) and let \(m\ge 1\) such that \(\chi \) is defined over \(\mathbf {F}_{p^m}\). Then one has \(D_w(\lambda )=\chi \) for the quasicharacterFurthermore, the summand corresponding to \(i=rm1\) is \(wz^{1}({}^{\sigma ^{1}}\chi )\).$$\begin{aligned} \lambda =\frac{1}{p^{rm}1} \sum _{i=0}^{rm1} p^i (zw^{1})^{(i)} ({}^{\sigma ^i}\chi ) \end{aligned}$$(3.1.3)
Proof
6.2 Grouptheoretical Hasse invariants

\(r\ge 1\) such that \(w^{(r)}=e\) for all \(w\in W\).

\(m\ge 1\) such that T splits over \(\mathbf {F}_{p^m}\).

\(N\ge 1\) such that \(\dim _k H^0(\left[ E\backslash G_w\right] ,{\mathscr {V}}(N\chi ))= 1\) for all \(w\in {}^IW\) and \(\chi \in X^*(L)\).

For \(w\in {}^I W\) and \(\chi \in X^*(L)\), let \(h_{w,\chi }\) be a nonzero element of the line \(H^0(\left[ E\backslash G_w\right] ,{\mathscr {V}}(N\chi ))\).

Similarly, for all \(w\in W\) and \(\lambda \in X^*(T)\), let \(f_{w,\lambda }\) be a nonzero element of the line \(H^0({{\,\mathrm{Sbt}\,}}_w,{\mathscr {L}}_{{{\,\mathrm{Sbt}\,}}}(\lambda ,w^{1}\lambda ))\) (see Theorem ).

For \(w\in W\) and \(\lambda \in X^*(T)\), set \(f'_{w,\lambda }:=\psi ^*(f_{w,\lambda })\in H^0({\mathcal X}_w,\psi ^*{\mathscr {L}}_{{{\,\mathrm{Sbt}\,}}})\).
Proposition 3.2.1
 (a)
There exists \(d\ge 1\) such that \(h_{w,\chi }^d\) extends to \(\left[ E\backslash \overline{G}_w\right] \) with nonvanishing locus \(\left[ E\backslash G_w\right] \).
 (b)For all \(\alpha \in E_w\), one has:$$\begin{aligned} \sum _{i=0}^{rm1} \langle (zw^{1})^{(i)}({}^{\sigma ^i}\chi ),w\alpha ^\vee \rangle p^i >0. \end{aligned}$$(3.2.1)
Proof
Let \(\lambda \in X^*(T)_\mathbf {Q}\) given by (3.1.3) of Lemma . Put \(C:=p^{rm}1\), so that \(C\lambda \in X^*(T)\). It satisfies \(\psi ^*({\mathscr {L}}_{{{\,\mathrm{Sbt}\,}}}(C\lambda ,w^{1}C\lambda ))={\mathscr {L}}(C\chi )\), so we have \(f'_{w,C\lambda }\in H^0({\mathcal X}_w,{\mathscr {L}}(C\chi ))\). Equation (3.2.1) is equivalent to \(\langle \lambda , w\alpha ^\vee \rangle >0\). Hence by Theorem (c), Property (b) is satisfied if and only if \(f_{w,C\lambda }\) extends to \(\overline{{{\,\mathrm{Sbt}\,}}}_w\) with nonvanishing locus \({{\,\mathrm{Sbt}\,}}_w\). Equivalently, it holds if and only if \(f'_{w,C\lambda }\) extends to \(\overline{{\mathcal X}}_w\) with nonvanishing locus \({\mathcal X}_w\) (by smoothness of \(\psi \)). By Proposition (a) combined with [62, Proposition 1.18], we have \(\dim _k H^0({\mathcal X}_w,{\mathscr {L}}(dC\chi )) \le 1\), so \(\pi ^*(h_{w,\chi })^{dC}=(f'_{w,C\lambda })^{d}\) up to a nonzero scalar. Since \(\overline{{\mathcal X}}_w\) is normal, \(f'_{w,C\lambda }\) extends to \(\overline{{\mathcal X}}_w\) with nonvanishing locus \({\mathcal X}_w\) if and only if some power of it does, so we deduce that (b) is equivalent to the fact that \(\pi ^*(h_{w,\chi })\) extends to \(\overline{{\mathcal X}}_w\) with nonvanishing locus \({\mathcal X}_w\). In particular, (a) implies (b). The converse follows from the next lemma. \(\square \)
Lemma 3.2.2
Let \(f:X\rightarrow Y\) a proper surjective morphism of integral schemes of finitetype over k. Let \({\mathscr {L}}\) be a line bundle on Y. Let \(U\subset Y\) be a normal open subset and \(h\in {\mathscr {L}}(U)\) a nonvanishing section over U. Assume that the section \(f^*(h)\in H^0(f^{1}(U),f^*{\mathscr {L}})\) extends to X with nonvanishing locus \(f^{1}(U)\). Then there exists \(d \ge 1\) such that \(h^d\) extends to Y, with nonvanishing locus U.
Proof
We will reduce to the case when X, Y are affine, \({\mathscr {L}}={\mathcal O}_Y\) and f is the normalization of Y.
First, after replacing X by its normalization \(\widetilde{X}\rightarrow X\), we may assume that X is normal. In this case, the map f factors through the normalization \(\pi :\widetilde{Y}\rightarrow Y\). Hence there exists \(f':X\rightarrow \widetilde{Y}\) such that \(f=\pi \circ f'\). The map \(f'\) is again proper, so it is surjective. Write \(\mathrm{div}(\pi ^*(h))=\sum _{i=1}^r n_iZ_i\) where \(n_i\in \mathbf {Z}\) and \(Z_i\subset \widetilde{Y}{\setminus }\pi ^{1}(U)\) are codimension one irreducible subvarieties. If \(n_i<0\) for some i, then \(f^*(h)\) would have a pole (because \(f'\) is surjective), hence \(\pi ^*(h)\) extends to \(\widetilde{Y}\) with nonvanishing locus \(\pi ^{1}(U)\). Thus we may assume \(X=\widetilde{Y}\). Also, we may reduce to the case \(Y={{\,\mathrm{Spec}\,}}(A)\), \(X={{\,\mathrm{Spec}\,}}(B)\) for an integral domain A and B its integral closure, and \({\mathscr {L}}={\mathcal O}_Y\). Write \(I\subset A\) for the ideal sheaf of \(Z:= X{\setminus } U\), endowed with the reduced structure. Hence \(U=D(I)=\{{\mathfrak p}\in {{\,\mathrm{Spec}\,}}A, I\nsubseteq {\mathfrak p}\}\). Replacing h by a power, we may further assume that \(h\in IB\).
We claim that for any \(s\in IB\), there exists \(n\ge 1\) such that \(s^{p^n}\) lies in A. Since k has characteristic p, we may assume that \(s=gx\) for some \(g\in I\) and \(x\in B\), because the \(p^n\) power map is additive. Since U is normal, \(f^{1}(U)\rightarrow U\) is an isomorphism, so the map \(A_g\rightarrow B_g\) is an isomorphism (since \(D(g)\subset U\)). Hence we can find \(m\ge 1\) such that \(g^m x \in A\). Since A[x] is generated as an Amodule by \(1,x,\ldots ,x^r\) for some \(r\ge 1\), it follows that we can find m such that \(g^m x^d\in A\) for all \(d\ge 0\). Increasing m, we may assume that it is a power of p, say \(m=p^n\). Taking \(d=m\) gives \(g^m x^m \in A\), which proves the claim. We have showed that there exists \(d\ge 1\) such that \(h^d\in A\). If \(V\subset Y\) is the nonvanishing locus of \(h^d\), then \(f^{1}(V)=f^{1}(U)\), hence \(U=V\). \(\square \)
Now we come to the main theorem of this part, the existence of grouptheoretical Hasse invariants (Theorem I.1.1):
Theorem 3.2.3
If \(\chi \in X^*(L)\) is (p, L)admissible [Definition N.5.3(d)], then there exists \(d\ge 1\) such that for all \(w\in W^J\), the section \(h_{w,\chi }^d\) extends to \(\left[ E\backslash \overline{G}_w\right] \) with nonvanishing locus \(\left[ E\backslash G_w\right] \).
Proof
Remark 3.2.4
When \(G_w\) is the open stratum of G, Theorem was proved in [62] for \(\chi \) any Lample character. The assumption that \(\chi \) is orbitally pclose is superfluous in this case.
Lemma 3.2.5
Let \((G,\mu )=(GSp(W,\psi ),\mu _{\mathcal D})\) be a cocharacter datum of Siegeltype (Sect. ). The Hodge character \(\eta _\omega \) is Lample and quasiconstant. In particular, \(\eta _\omega \) is (p, L)admissible for all p (Definition N.5.3).
Proof
We prove first that \(\eta _\omega \) is Lample. Choose a Borel subgroup \(B\subset P_{\mathcal D}\) and a maximal torus \(T\subset B\). Since \(P_{\mathcal D}\) is a maximal parabolic subgroup, the set \(\Delta {\setminus } I\) consists of a single simple root \(\alpha \). It is then easy to see from the definition of \(\eta _\omega \) that \(\langle \eta _\omega , \alpha ^\vee \rangle =1\), which shows that \(\eta _\omega \) is Lample.
We now show that \(\eta _\omega \) is quasiconstant. For any positive root \(\beta \) not in \(\Phi (L_{\mathcal D},T)\), one has \(\langle \eta _\omega , \beta ^\vee \rangle =1\) if \(\beta \) is a long root, and \(\langle \eta _\omega , \beta ^\vee \rangle =2\) if \(\beta \) is a short root. Since \(W\rtimes {{\,\mathrm{Gal}\,}}(k/\mathbf {F}_p)\) preserves the length of roots, we deduce that \({{\,\mathrm{Orb}\,}}(\eta _\omega ,\beta ^\vee )=\{0,1\}\) in all cases, which proves the result. \(\square \)
Corollary 3.2.6
Let \((G,\mu )=(GSp(W,\psi ),\mu _{\mathcal D})\) be a cocharacter datum of Siegeltype. There exists \(d\ge 1\) such that for all \(w\in W^J\), there exists a section \(h_w \in H^0(\left[ E\backslash \overline{G}_w\right] , \omega ^d)\) with nonvanishing locus \(\left[ E\backslash G_w\right] \).
We will explain later in Sect. that \(\eta _\omega \) is (p, L)admissible for all character data \((G,\mu )\) of Hodgetype coming from a Shimura datum of Hodgetype. In [36], we have shown the analogue of Corollary for any \((G,\mu )\) of Hodgetype (not necessarily attached to a Shimura datum), and also for more general \((G,\mu )\) (those of maximal type).
6.3 Functoriality of zip strata
Let \(f:(G_1,\mu _1)\rightarrow (G_2,\mu _2)\) be a finite morphism of cocharacter data and \(\widetilde{f}:\mathop {G_1\hbox {}{} \mathtt{Zip}}\nolimits ^{\mu _1}\rightarrow \mathop {G_2\hbox {}{} \mathtt{Zip}}\nolimits ^{\mu _2}\) the induced map of stacks. The underlying topological spaces of \(\mathop {G_1\hbox {}{} \mathtt{Zip}}\nolimits ^{\mu _1}\) and \(\mathop {G_2\hbox {}{} \mathtt{Zip}}\nolimits ^{\mu _2}\) are finite, and the map \(\widetilde{f}\) is continuous.
We denote by \(P_1,L_1,E_1\) (resp. \(P_2,L_2,E_2\)) the subgroups attached to \(\mu _1\) (resp. \(\mu _2\)), as in Sect. . Let \(f^*:X^*(L_2)\rightarrow X^*(L_1)\) denote the induced morphism.
Theorem 3.3.1
(Discrete Fibers) Assume there exists a \((p,L_2)\)admissible character \(\chi \in X^*(L_2)\), such that \(f^*\chi \) is orbitally pclose. Then the map \(\widetilde{f}\) has discrete fibers on the underlying topological spaces.
Proof
We claim first that \(f:G_1\rightarrow G_2\) induces a finite morphism \(G_1/P_1\rightarrow G_2/P_2\). It suffices to show that this morphism is quasifinite. By definition, one has \(P_1\subset f^{1}(P_2)\), so it follows that \(H:=f^{1}(P_2)_{\mathrm{red}}\) is a parabolic subgroup of \(G_1\) containing \(P_1\). Using (1.2.1), we find also \(R_u(P_1)\subset f^{1}(R_u(P_2))\) hence \(R_u(P_1)\subset f^{1}(R_u(P_2))_\mathrm{red}=R_u(H)\). It follows that \(H=P_1\), so the map \(G_1/P_1\rightarrow G_2/P_2\) is injective on kpoints.
In particular, Remark N.5.2 shows that the \(L_2\)ample character \(\chi \) restricts to an \(L_1\)ample character of \(X^*(L_1)\) which we denote again by \(\chi \). Hence we may apply Theorem to \(\chi \) on both stacks \(\mathop {G_1\hbox {}{} \mathtt{Zip}}\nolimits ^{\mu _1}\) and \(\mathop {G_2\hbox {}{} \mathtt{Zip}}\nolimits ^{\mu _2}\). If the statement were false, there would exist two \(E_1\)orbits \(C_1\) and \(C'_1\) in \(G_1\) such that \(C'_1\subset \overline{C}_1\) and \(f(C_1)\subset C_2\), \(f(C'_1)\subset C_2\) for some \(E_2\)orbit \(C_2\subset G_2\). There exists an integer \(N\ge 1\) and a section \(h_i\in H^0(\overline{C}_i, {\mathscr {V}}(\chi )^{ N})\) (for \(i=1,2\)), whose nonvanishing locus is \(C_i\). Since \(\dim _k H^0(C_1, {\mathscr {V}}(\chi )^{ N})=1\), one has \(f^*(h_2)=h_1\) (up to nonzero scalar). This contradicts the fact that the nonvanishing locus of \(h_1\) is \(C_1\). \(\square \)
We will see that Theorem applies in the context of embeddings of Shimura data of Hodgetype (Corollary ). As explained in Remark I.1.3 of the introduction, the statement remains true without the cumbersome assumption that there exists a \((p,L_2)\)admissible character \(\chi \in X^*(L_2)\), such that \(f^*\chi \) is orbitally pclose. This is proved in [36, Theorem 2]. The trick is to replace the Frobenius \(\varphi \) by higher powers of \(\varphi \).
6.4 The cone of global sections
We will construct nonzero global sections for certain line bundles \({\mathscr {L}}(\lambda )\). Actually, not all line bundles \({\mathscr {L}}(\lambda )\) admit nonzero global sections. In general, it is hard to characterize the set of \(\lambda \in X^*(T)\) which do.
Lemma 3.4.1
Proof
By Theorem (c), the divisor of any nonzero element \(f\in H^0({{\,\mathrm{Sbt}\,}}_{w_0}, {\mathscr {L}}_{{{\,\mathrm{Sbt}\,}}}(\lambda ,w_0\lambda ))\) (viewed as a rational function on G) is a sum of \(B\times B\)orbits in G with multiplicities \(\langle \lambda , w_0 \alpha ^\vee \rangle \) with \(\alpha \in E_{w_0}\). One has \(E_{w_0}=\Delta \), hence f extends to G if and only if \(\langle \lambda , w_0 \alpha ^\vee \rangle \ge 0\) for all positive roots \(\alpha \). Since \(w_0\) maps bijectively the positive roots to the negative ones, this is equivalent to \(\lambda \in X^*_+(T)\). \(\square \)
Lemma 3.4.2
The subset \({\mathcal C}_{w_0}\subset X^*(T)\) is a cone of maximal rank (i.e \({{\,\mathrm{Span}\,}}_\mathbf {Q}({\mathcal C}_{w_0})=X^*(T)_\mathbf {Q}\)).
Proof
Since \(D_{w_0}\) is linear and \(X^*_+(T)\) is a cone, it is clear that \({\mathcal C}_{w_0}\) is a cone. It is of maximal rank because \(X_{+}^*(T)\) is of maximal rank and \(D_{w_0}\) induces an automorphism of \(X^*(T)_\mathbf {Q}\) [Lemma (a)]. \(\square \)
Proposition 3.4.3
For any \(\chi \in {\mathcal C}\), there exists a nonzero global section \(h_\chi \in H^0(\mathop {G\hbox {}{} \mathtt{ZipFlag}}\nolimits ^{\mu },{\mathscr {L}}(\chi ))\).
Proof
We already showed this for \(\chi \in {\mathcal C}_{w_0}\). Hence it suffices to show it for \(N\eta _\omega \), i.e that \(\omega ^N\) admits a nonzero global section. The embedding \(\iota \) induces a map of stacks \(\mathop {G\hbox {}{} \mathtt{Zip}}\nolimits ^\mu \rightarrow \mathop {GSp(2g)\hbox {}{} \mathtt{Zip}}\nolimits ^{\mu _{\mathcal D}}\). The image of the unique open stratum of \(\mathop {G\hbox {}{} \mathtt{Zip}}\nolimits ^\mu \) maps to some stratum \(S\subset \mathop {GSp(2g)\hbox {}{} \mathtt{Zip}}\nolimits ^{\mu _{\mathcal D}}\) (not necessarily open). In particular, the image of \(\mathop {G\hbox {}{} \mathtt{Zip}}\nolimits ^\mu \) is contained in \(\overline{S}\), by continuity.
By Corollary , there exists a section \(h_S\in H^0(\overline{S}, \omega ^d)\) (some \(d\ge 1\)) with nonvanishing locus S. The pullback of \(h_S\) to \(\mathop {G\hbox {}{} \mathtt{Zip}}\nolimits ^\mu \) is then nonzero. This gives a nonzero global section h of \(\omega ^d\) over \(\mathop {G\hbox {}{} \mathtt{Zip}}\nolimits ^\mu \). Now we claim that \(\omega ^N\) also admits a section. For this, let \(f\in H^0([E \backslash G_{w_0}],\omega ^N)\ne 0\) be nonzero. Then \(f^d\) and \(h^N\) are both sections of \(\omega ^{Nd}\) over the open stratum \([E \backslash G_{w_0}]\), hence coincide up to a nonzero scalar, as \(\dim _k H^0([E \backslash G_{w_0}], \omega ^{Nd})=1\). We deduce that \(f^d\) (viewed as a rational function on G) extends to G. Since G is normal, f extends to G, which shows that \(\omega ^N\) also has a nonzero global section over \(\mathop {G\hbox {}{} \mathtt{Zip}}\nolimits ^\mu \), as claimed. This terminates the proof. \(\square \)
If G is split over \(\mathbf {F}_p\) then one has \(N\eta _\omega \in {\mathcal C}_{w_0}\), hence \({\mathcal C}={\mathcal C}_{w_0}\), but we don’t know if this remains true in general. To summarize, the above results provide nonzero global sections for \({\mathscr {L}}(\chi )\) for \(\chi \) in a cone \({\mathcal C}\subset X^*(T)\) of maximal rank, which we tried to make as large as possible. This will be used later in Theorem (b).
7 Part 2. Strata Hasse invariants of Shimura varieties
In Part 2, we apply the general results of Part 1 on grouptheoretical Hasse invariants to Hodgetype Shimura varieties and subschemes related to their EO stratification. Section 4 concerns Hodgetype Shimura varieties. Section introduces notation for integral models of Hodgetype Shimura varieties, to be used throughout Parts 2 and 3. Section deduces the corollaries of Sect. I.1 about Hasse invariants for the EO stratification.
In Sect. 6, we study the extension of the EO stratification to compactifications. In Sect. , we show that the universal GZip over \(S_{{\mathcal K}}\) admits an extension to a GZip over a toroidal compactification \(S_{{\mathcal K}}^{\Sigma } \). This is applied to the minimal compactification in Sect. where we prove the affineness statement of Corollary I.2.6. The notion of length stratification is introduced in Sect. for a general scheme \(X \rightarrow \mathop {G\hbox {}{} \mathtt{Zip}}\nolimits ^{\mu }\). In Sect. , the general considerations of Sect. are applied to \(S_{{\mathcal K}}^{\Sigma } \). The length stratification will play a key role in Sect. 8, in the proof of the factorization theorem for Hecke algebras (Theorems I.3.1, ).
The last two sections of Part 2 record auxiliary results that will be used in Part 3. In Sect. 7, we note how vanishing theorems for \({\mathscr {S}}_{{\mathcal K}}^{\Sigma } \) generalize to strata. Section 5 describes results about lifting and gluing powers of EO Hasse invariants. A key technical point in our arguments is working with the Cohen–Macaulay property to avoid embedded components.
8 Shimura varieties of Hodge type
8.1 Background
8.1.1 Rational theory
Let \(({\mathbf {G}}, {\mathbf {X}})\) be a Shimura datum [24, 2.1.1]. Write \(E=E({\mathbf {G}}, {\mathbf {X}})\) for the reflex field of \(({\mathbf {G}}, {\mathbf {X}})\) and \({\mathcal O}_E\) for its ring of integers. Given an open compact subgroup \({\mathcal K}\subset {\mathbf {G}}({{\mathbf {A}}}_f)\), write \({{\,\mathrm{Sh}\,}}({\mathbf {G}}, {\mathbf {X}})_{{\mathcal K}}\) for Deligne’s canonical model at level \({\mathcal K}\) over E (see loc. cit.). Every inclusion \({\mathcal K}' \subset {\mathcal K}\) induces a finite étale projection \(\pi _{{\mathcal K}'/{\mathcal K}}:{{\,\mathrm{Sh}\,}}({\mathbf {G}}, {\mathbf {X}})_{{\mathcal K}'} \rightarrow {{\,\mathrm{Sh}\,}}({\mathbf {G}}, {\mathbf {X}})_{{\mathcal K}}\). Let \({{\,\mathrm{Sh}\,}}({\mathbf {G}}, {\mathbf {X}})\) be the resulting tower of Eschemes.^{5} It admits a right \({\mathbf {G}}({{\mathbf {A}}}_f)\)action given by a compatible system of isomorphisms \(g: {{\,\mathrm{Sh}\,}}({\mathbf {G}}, {\mathbf {X}})_{{\mathcal K}} {\mathop {\rightarrow }\limits ^{\sim }} {{\,\mathrm{Sh}\,}}({\mathbf {G}}, {\mathbf {X}})_{g^{1}{\mathcal K}g}\) for \(g \in {\mathbf {G}}({{\mathbf {A}}}_f)\). Define d to be the common dimension of all the \({{\,\mathrm{Sh}\,}}({\mathbf {G}}, {\mathbf {X}})_{{\mathcal K}}\).
8.1.2 Symplectic embedding
Let \(g\ge 1\) and let \((V,\psi )\) be a 2gdimensional, nondegenerate symplectic space over \(\mathbf {Q}\). Write \(GSp(2g)=GSp(V, \psi )\) for the group of symplectic similitudes of \((V,\psi )\). Write \(\mathbf {X}_g\) for the double Siegel halfspace [24, 1.3.1]. The pair \((GSp(2g), {\mathbf {X}}_{g})\) is the Siegel Shimura datum; it has reflex field \(\mathbf {Q}\). Recall that \(({\mathbf {G}}, {\mathbf {X}})\) is of Hodge type if there exists an embedding of Shimura data \(\varphi :({\mathbf {G}}, {\mathbf {X}})\hookrightarrow (GSp(2g), {\mathbf {X}}_{g})\) for some \(g \ge 1\). Henceforth, assume \(({\mathbf {G}}, {\mathbf {X}})\) is of Hodgetype.
8.1.3 Integral model
For the rest of this paper, fix a prime \(p \not \in {{\,\mathrm{Ram}\,}}(\mathbf {G}) \cup \{2\}\) (Sect. N.3.6). Let \({\mathcal G}\) be a reductive, \(\mathbf {Z}_{(p)}\)model of \(\mathbf {G}\) and \({\mathcal K}_p:={\mathcal G}(\mathbf {Z}_p) \subset {\mathbf {G}}(\mathbf {Q}_p)\) the associated hyperspecial subgroup.
Let \({\mathfrak p}\) be a prime of E above p and let \({\mathcal O}_{E,{\mathfrak p}}\) be the localization of \({\mathcal O}_E\) at \({\mathfrak p}\). Write \(E_{\mathfrak p}\) for the completion of E at \({\mathfrak p}\) and \({\mathcal O}_{{\mathfrak p}}\) for its ring of integers. By Vasiu [102, Theorem 0] and Kisin [58, Theorem 1], as \({\mathcal K}^p\) ranges over sufficiently small open compact subgroups of \(G(\mathbf {A}_f^p)\), the subtower of Eschemes \(({{\,\mathrm{Sh}\,}}({\mathbf {G}}, {\mathbf {X}})_{{\mathcal K}^p{\mathcal K}_p})_{{\mathcal K}^p}\) admits an integral canonical model \(({\mathscr {S}}_{{\mathcal K}^p{\mathcal K}_p})_{{\mathcal K}^p}\) with \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)action over \({\mathcal O}_{E,{\mathfrak p}}\) in the sense of Milne [80].
For short, say that \({\mathcal K}\subset {\mathbf {G}}({{\mathbf {A}}}_f)\) is a phyperspecial level when \({\mathcal K}\) is an open, compact subgroup of \({\mathbf {G}}({{\mathbf {A}}}_f^p)\) of the form \({\mathcal K}={\mathcal K}^p{\mathcal K}_p\) with \({\mathcal K}_p \subset {\mathbf {G}}(\mathbf {Q}_p)\) hyperspecial and \({\mathcal K}^p \subset {\mathbf {G}}({{\mathbf {A}}}_f^p)\). The projections between levels \(\pi _{{\mathcal K}'/{\mathcal K}}:{\mathscr {S}}_{{\mathcal K}'} \rightarrow {\mathscr {S}}_{{\mathcal K}}\) and the right \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)action \(g:{\mathscr {S}}_{{\mathcal K}} \rightarrow {\mathscr {S}}_{g^{1}{\mathcal K}g}\) are denoted the same way as for the canonical model (Sect. ).
8.1.4 Integral symplectic embedding
8.1.5 The cocharacter \(\mu \)
Given \(h \in \mathbf {X}\), let \(\mu \in X_*(\mathbf {G})\) be the associated minuscule cocharacter, given by \(\mu (z)=h_\mathbf {C}\circ \mu _0\) (Sect. N.6.2). The reflex field E of \(({\mathbf {G}}, {\mathbf {X}})\) is the field of definition of the \(\mathbf {G}(\mathbf {C})\)conjugacy class \([\mu ]\) of \(\mu \).
For any algebraically closed extension K / E, the conjugacy class \([\mu ]\) defines a unique conjugacy class \([\mu ]_K\) of cocharacters of \(\mathbf {G}_K\) defined over E. In particular, we obtain a conjugacy class \([\mu ]_{\overline{E}_{\mathfrak p}}\) for a choice of an algebraic closure \(\overline{E}_{\mathfrak p}\) of \(E_{\mathfrak p}\). Since \(\mathbf {G}_{\mathbf {Q}_p}\) is unramified, it is in particular quasisplit. Thus, there exists a representative \(\mu \in X_*(\mathbf {G}_{E_{{\mathfrak p}}})\) of \([\mu ]_{\overline{E}_{\mathfrak p}}\) defined over \(E_{\mathfrak p}\).
 (1)
A pair of opposite parabolic subgroups \((\mathbf {P}^,\mathbf {P}^+)\) in \(\mathbf {G}_{E_{\mathfrak p}}\) attached to \(\mu _{E_{\mathfrak p}}\), and a common Levi subgroup \(\mathbf {L}:=\mathbf {P}^\cap \mathbf {P}^+=\mathrm{Cent}(\mu _{E_{\mathfrak p}})\). We set \(\mathbf {P}:=\mathbf {P}^\).
 (2)
A pair of opposite parabolic subgroups \((P_g^, P_g^+)\) in \(GSp(2g)_{E_{\mathfrak p}}\) attached to \(\mu _{g, E_{\mathfrak p}}\), and a common Levi subgroup \(L_g:=P_g^\cap P_g^+=\mathrm{Cent}(\mu _{g,E_{\mathfrak p}})\). We set \(P_g:=P_g^\). One has \(\mathbf {P}^\pm =P^\pm _g\cap \mathbf {G}\) and \(L_g \cap \mathbf {G}=\mathbf {L}\).
8.1.6 Compatibility with the complex theory
In Sect. 10, we will fix an isomorphism \(\iota : \overline{\mathbf {Q}}_{p}{\mathop {\rightarrow }\limits ^{\sim }} \mathbf {C}\). We choose \(\iota \) compatibly with \(\mathbf {P}\) so that the \(\mathbf {C}\)parabolic \(\iota \mathbf {P}_{\overline{\mathbf {Q}}_{p}}\) deduced via \(\iota \) is the stabilizer of the Hodge filtration associated to the \(\mathbf {R}\)Hodge structure \({{\,\mathrm{Ad}\,}}\circ h\). Consequently, given \(\alpha \in \Phi {\setminus } \Phi _{\mathbf {L}}\), one has \(\alpha \in \Phi ^+\) if and only if the image of \(\alpha \)root space of \({{\,\mathrm{Lie}\,}}(\mathbf {G})_{\mathbf {C}}\) is nonzero in \({{\,\mathrm{Lie}\,}}(\mathbf {G})_{\mathbf {C}}/{{\,\mathrm{Lie}\,}}(\iota \mathbf {P}_{\overline{\mathbf {Q}}_{p}})\), which is also the \((1,1)\)part of the Hodge structure \({{\,\mathrm{Ad}\,}}\circ h\) (conventions N.3.2, N.6.1).
8.1.7 Integral structure theory
Since \({\mathcal G}_{\mathbf {Z}_p}\) is also quasisplit, we may assume that the representative \(\mu \) of \([\mu ]_{\overline{E}_{\mathfrak p}}\) extends to a cocharacter \(\mu : \mathbf {G}_{m,{\mathcal O}_{{\mathfrak p}}}\rightarrow {\mathcal G}_{{\mathcal O}_{{\mathfrak p}}}\). The centralizer of \(\mu \) is a Levi subgroup \({\mathcal L}\subset {\mathcal G}_{{\mathcal O}_{\mathfrak p}}\). Define \(\mu _g : \mathbf {G}_{m,{\mathcal O}_{{\mathfrak p}}} \rightarrow GSp(\Lambda ,\psi )_{{\mathcal O}_{\mathfrak p}}\) by \(\mu _g:=\varphi \circ \mu \). Since \(\mathbf {S}{\mathop {\rightarrow }\limits ^{h_g}} GSp(V_{\mathbf {R}}, \psi ) \rightarrow GL(V_{\mathbf {R}})\) is an \(\mathbf {R}\)Hodge structure of type \(\{(0,1),(1,0)\}\) (Sect. N.6.1), the cocharacter \(\mu _g\) defines a \(\mathbf {Z}\)grading \(\Lambda \otimes {\mathcal O}_{\mathfrak p}= \Lambda _0 \oplus \Lambda _{1}\) (where \(\mathbf {G}_m\) acts through \(\mu _g\) by \(z\mapsto z^{i}\) on \(\Lambda _i\) for \(i=0,1\)).
Since \({\mathbf {Z}}_p\) and \({\mathcal O}_{{\mathfrak p}}\) are discrete valuation rings, [5, ChapterXXVI, 3.5] and the valuative criterion of properness applied to the schemes of parabolic subgroups of \({\mathcal G}_{{\mathcal O}_{\mathfrak p}}\) imply that \(\mathbf {P}\) extends uniquely to a parabolic subgroup \({\mathcal P}\subset {\mathcal G}_{{\mathcal O}_{\mathfrak p}}\). Similarly, there is a unique extension of \(\mathbf {B}\) to a Borel subgroup \({\mathcal B}\subset {\mathcal G}_{\mathbf {Z}_p}\). Set \({\mathcal B}_{\mathcal L}:={\mathcal L}\cap {\mathcal B}\), it is the unique extension of the Borel subgroup \(\mathbf {B}_\mathbf {L}\) of \(\mathbf {L}\).
8.1.8 Hodge bundles
In general put \(H^1_{\mathrm{dR}}({\mathcal A}/{\mathscr {S}}_{{\mathcal K}}):=\varphi ^*H^1_{\mathrm{dR}}({\mathcal A}_g/{\mathscr {S}}_{g,{\tilde{{\mathcal K}}}})\), \(\Omega =\varphi ^*\Omega _g\) and \(\omega =\varphi ^*\omega _g\) (since the embedding \(\varphi \) was fixed in Sect. , we omit \(\varphi \) from the notation).
Remark 4.1.1
(Dependence on \(\varphi \)) A priori both \(\Omega \) and its determinant \(\omega \) depend on the embedding \(\varphi \). A posteriori it follows from Theorem below that when \(\mathbf {G}^{{{\,\mathrm{ad}\,}}}\) is \(\mathbf {Q}\)simple, the ray generated by \(\omega \) in \({{\,\mathrm{Pic}\,}}({\mathscr {S}}_{{\mathcal K}})\) is independent of the embedding \(\varphi \), see [38, Corollary 1.4.5]. However, this independence of \(\varphi \) is not used in this paper. Embeddings of products show that the above invariance is best possible. It remains unclear to us to what extent the Hodge vector bundle \(\Omega \) depends on \(\varphi \).
8.1.9 Torsors and automorphic bundles
Let N be a finite extension of \(E_{{\mathfrak p}}\) with ring of integers \(\mathcal O_N\) and prime \(\wp \) lying over \({\mathfrak p}\). Since \(I_{{\mathcal P}}/{\mathcal P}\cong {\mathscr {S}}_{{\mathcal K}}\), every algebraic representation of \({\mathcal P}\) on a finite free \({\mathcal O}_{N}\)module W gives rise to a vector bundle on \({\mathscr {S}}_{{\mathcal K}}\) as in Sect. N.4.1. By setting \(R_u({\mathcal P})\) to act trivially, any representation of \({\mathcal L}\) on W gives rise to one of \({\mathcal P}\).
Let \(\eta \in X^*_{+,\mathbf {L}}(\mathbf {T})\). Let \({\mathscr {L}}(\eta )\) be the associated \(\mathbf {L}_{\overline{\mathbf {Q}}_{p}}\)equivariant (or \(\mathbf {L}_{\overline{\mathbf {Q}}_{p}}\)linearized) line bundle on the flag variety \((\mathbf {L}/\mathbf {B}_\mathbf {L})_{\overline{\mathbf {Q}}_{p}}\). Then there exists an extension N as above such that \({\mathscr {L}}(\eta )\) descends to an \({\mathcal L}\)equivariant line bundle on \({\mathcal L}/{\mathcal B}_{\mathcal L}\times _{{\mathcal O}_{{\mathfrak p}}} {\mathcal O}_N\) over \({\mathcal O}_N\). Continue to call the descended line bundle \({\mathscr {L}}(\eta )\). Let \(V_{\eta }\) be the representation of \({\mathcal L}\) on \(H^0({\mathcal L}/{\mathcal B}_{{\mathcal L}} \times _{{\mathcal O}_{{\mathfrak p}}} {\mathcal O}_N, {\mathscr {L}}(\eta ))\). By the Borel–Weil Theorem, \(V_{\eta } \otimes \overline{\mathbf {Q}}_{p}\) is irreducible of highest weight \(\eta \). In general \(V_{\eta }\) is the (possibly reducible) highest weight, induced module denoted \(H^0(\eta )\) in [53].
The automorphic vector bundle of weight \(\eta \) is the vector bundle \({\mathscr {V}}({\eta })\) on \({\mathscr {S}}_{{\mathcal K}}\) afforded by the torsor \(I_{{\mathcal L}}\) and the representation \(V_{\eta }\). If different levels are in play, write \({\mathscr {V}}_{{\mathcal K}}(\eta )\) to specify the level.
8.1.10 \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant objects
Recall our convention Sect. N.2.3: Write \({\mathscr {S}}_{{\mathcal K}}^+:={\mathscr {S}}_{{\mathcal K}}\), \({\mathscr {S}}_{{\mathcal K}}^0={\mathscr {S}}_{{\mathcal K}}\otimes _{{\mathcal O}_{E,{\mathfrak p}}}E_{{\mathfrak p}}\) for its generic fiber and \({\mathscr {S}}_{{\mathcal K}}^n:={\mathscr {S}}_{{\mathcal K}}\otimes _{{\mathcal O}_{E,{\mathfrak p}}} {\mathcal O}_{E,{\mathfrak p}}/{\mathfrak p}^n\) for \(n\ge 1\). Let \(n \in \mathbf {Z}_{\ge 0}\cup \{+\}\). A \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant sheaf on the tower \(({\mathscr {S}}^n_{{\mathcal K}})_{{\mathcal K}^p}\) is a system of sheaves \(({\mathscr {F}}_{{\mathcal K}})_{{\mathcal K}^p}\) such that \(\pi _{{\mathcal K}'/{\mathcal K}}^*{\mathscr {F}}_{{\mathcal K}}={\mathscr {F}}_{{\mathcal K}'}\) and \(g^*{\mathscr {F}}_{g^{1}{\mathcal K}g} ={\mathscr {F}}_{{\mathcal K}}\) for all \({\mathcal K}'^{p} \subset {\mathcal K}^p\) and all \(g \in {\mathbf {G}}({{\mathbf {A}}}_f^p)\), where \({\mathcal K}'={\mathcal K}_p{\mathcal K}'_p\). Similarly a morphism of \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant sheaves \(({\mathscr {F}}_{{\mathcal K}}) \rightarrow ({\mathscr {G}}_{{\mathcal K}})\) consists of a family of morphisms \({\mathscr {F}}_{{\mathcal K}} \rightarrow {\mathscr {G}}_{{\mathcal K}}\) which is compatible with \(\pi _{{\mathcal K}'/{\mathcal K}}\) and g. This applies to both étale sheaves (in particular étale torsors) and to (quasi)coherent \(({\mathcal O}_{{\mathscr {S}}_{{\mathcal K}}})_{{\mathcal K}^p}\)modules. Given a \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant sheaf \(({\mathscr {F}}_{{\mathcal K}})\), a \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant section t is a system \((t_{{\mathcal K}} \in H^0({\mathscr {S}}^n_{{\mathcal K}}, {\mathscr {F}}))\) satisfying \(\pi _{{\mathcal K}'/{\mathcal K}}^*t_{{\mathcal K}}=t_{{\mathcal K}'}\) and \(g^* t_{g^{1}{\mathcal K}g}=t_{{\mathcal K}}\).
More generally, define a \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)system of schemes as a system of schemes \(({\mathscr {Z}}_{{\mathcal K}})_{{\mathcal K}^p}\) such that for every inclusion \({\mathcal K}'^p \subset {\mathcal K}^p\), one has a projection \({\mathscr {Z}}_{{\mathcal K}'} \rightarrow {\mathscr {Z}}_{{\mathcal K}}\) and for every \(g\in {\mathbf {G}}({{\mathbf {A}}}_f^p)\) a system of isomorphisms \(g:{\mathscr {Z}}_{{\mathcal K}} {\mathop {\rightarrow }\limits ^{\sim }} {\mathscr {Z}}_{g^{1} {\mathcal K}g}\) (subject to the usual compatibility conditions). A \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant morphism \(\alpha : ({\mathscr {Z}}_{1,{\mathcal K}}) \rightarrow ({\mathscr {Z}}_{2,{\mathcal K}})\) is a system of morphisms \(\alpha _{{\mathcal K}}:{\mathscr {Z}}_{1,{\mathcal K}} \rightarrow {\mathscr {Z}}_{2,{\mathcal K}}\) such that \({\mathscr {Z}}_{1,{\mathcal K}'} \rightarrow {\mathscr {Z}}_{1,{\mathcal K}}\) (resp. \(g:{\mathscr {Z}}_{1,{\mathcal K}} {\mathop {\rightarrow }\limits ^{\sim }} {\mathscr {Z}}_{1,g^{1} {\mathcal K}g}\)) is the pullback of \({\mathscr {Z}}_{2,{\mathcal K}'} \rightarrow {\mathscr {Z}}_{2,{\mathcal K}}\) (resp. \(g:{\mathscr {Z}}_{2,{\mathcal K}} {\mathop {\rightarrow }\limits ^{\sim }} {\mathscr {Z}}_{2,g^{1} {\mathcal K}g}\)) along \(\alpha \). In particular, a \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant subscheme of \(({\mathscr {S}}^n_{{\mathcal K}})_{{\mathcal K}^p}\) is a system of subschemes \(({\mathscr {Z}}_{{\mathcal K}})_{{\mathcal K}^p}\), where \({\mathscr {Z}}_{{\mathcal K}}\) is a subscheme of \({\mathscr {S}}^n_{{\mathcal K}}\) such that \(\pi _{{\mathcal K}'/{\mathcal K}}^{1}({\mathscr {Z}}_{{\mathcal K}})={\mathscr {Z}}_{{\mathcal K}'}\) and \(g^{1}({\mathscr {Z}}_{g^{1}{\mathcal K}g})={\mathscr {Z}}_{{\mathcal K}}\) (both schemetheoretically). It follows directly from the definitions that a system of ideal sheaves \(({\mathcal I}_{{\mathcal K}} \subset {\mathcal O}_{{\mathscr {S}}^n_{{\mathcal K}}})\) is \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant if and only if the corresponding system of zeroschemes is.
By construction the universal abelian scheme \({\mathcal A}\rightarrow {\mathscr {S}}\), the vector bundle \(H^1_{\mathrm{dR}}({\mathcal A}/{\mathscr {S}})\) and the sections \(s_{\alpha , \mathrm{dR}}\) are \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant. It follows that also the bundles \({\mathscr {V}}(\eta )\) and the torsors \(I_{{\mathcal G}},I_{{\mathcal P}},I_{{\mathcal L}}\) are all \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant.
8.1.11 Vector bundle dictionary
Let \({{\,\mathrm{Std}\,}}: GSp(V, \psi ) \hookrightarrow GL(V)\) be the forgetful representation; it is an irreducible \(GSp(V,\psi )\)module. Let \(\eta _{g, \Omega }\) be the highest weight of \({{\,\mathrm{Std}\,}}\) relative to our choice of Borel pair \((\mathbf {B}, \mathbf {T})\) [Sect. in the case \(\mathbf {G}=GSp(V, \psi )\)] and convention on positive roots (Sect. N.3.2). Using the notation of Sect. , put \(\eta _{g,\omega }:=\det V_{\eta _{g,\Omega }}^{\vee }\). Recall that \({\mathscr {V}}(\eta _{g, \Omega })^{\vee }\cong \Omega _g\) and \({\mathscr {V}}(\eta _{g, \omega }) \cong \omega _g\) cf. [32, Proof of Theorem 5.5.1] or [21, p. 257, Example (b)–(c)]. (Consider one standard set of choices/coordinates: \(V=\mathbf {Q}^{2g}\), \(\psi =\left( \begin{array}{cc} 0 &{} I_g \\ I_g &{} 0 \end{array} \right) \), \(\mathbf {T}\) is the diagonal (\(\mathbf {Q}\)split) maximal torus, identify \(X^*(\mathbf {T})\) with \(\{(a_1, \ldots a_g;c) \in \mathbf {Z}^{g+1} \  \sum a_i \equiv c \pmod 2\}\) via \(\mathrm{diag}(t_1z, \ldots ,t_gz, t_1^{1}z, \ldots ,t_g^{1}z) \mapsto t_1^{a_1} \cdots t_g^{a_g}z^c\), \(\Delta =\{e_1e_2, \ldots , e_{g1}e_g, 2e_g\}\) and \(\Delta _L=\Delta {\setminus } \{2e_g\}\). Then \(\eta _{g, \Omega }\) (resp. \(\eta _{g, \omega }\)) is identified with \(\mathrm{diag}(0,\ldots ,0,1;1)\) (resp. \(\mathrm{diag}(1, \ldots , 1; g)\)), cf. [100, p. 306].)
In general, set \(\eta _{\omega }=\varphi ^*\eta _{g, \omega }\). Then \( {\mathscr {V}}(\eta _{\omega }) \cong \omega \) as \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant line bundles on \({\mathscr {S}}_{{\mathcal K}}\). We call \(\eta _{\omega }\) the Hodge character associated to \(\varphi \). This ‘modular’ definition agrees with the grouptheoretic one given in Definition (b).
8.2 Universal Gzip over \(S_{\mathcal K}\)
Let \(S_{{\mathcal K}}:={\mathscr {S}}_{{\mathcal K}}^1\) be the special fiber of \({\mathscr {S}}_{{\mathcal K}}\). We briefly review the construction ([112, Theorem 2.4.1] and [109, §5]) of the universal Gzip \(\underline{I}=(I,I_P,I_Q,\iota )\) over \(S_{{\mathcal K}}\). Denote by \(\kappa :={\mathcal O}_E/{\mathfrak p}\) the residue field of \({\mathfrak p}\). Define \(G:={\mathcal G}\otimes \mathbf {F}_p\) and write again \(\mu :\mathbf {G}_{m,\kappa }\rightarrow G_\kappa \) and \(\mu _g=\mu \circ \varphi \) for the reduction of \(\mu ,\mu _g\). Recall (Sect. ) that we have \(\Lambda = \Lambda _{0} \oplus \Lambda _{1}\). Define parabolic subgroups P, Q of \(G_\kappa \) as the stabilizers in \(G_\kappa \) of \(\text {Fil}_P:=\Lambda _{0,\kappa }\) and \(\text {Fil}_Q:= {}^\sigma \Lambda _{1,\kappa }\), respectively.
8.3 Hasse invariants for Ekedahl–Oort strata
We explain how to apply the general group theoretic Theorem on Hasse invariants to \(S_{{\mathcal K}}\). We give two ways to deal with the (p, L)admissible hypothesis of Theorem : (i) Show that it is satisfied by the Hodge character \(\eta _{\omega }\) for all primes p, (ii) Show that for Hodgetype Shimura varieties, one can reduce to \(G=GL(n)\), for which \(\eta _{\omega }\) is minuscule. Regarding (i), one has:
Theorem 4.3.1
([38], Theorem 1.4.4) The Hodge character \(\eta _{\omega }\) of a symplectic embedding \(\varphi ':({\mathbf {G}}, {\mathbf {X}})\hookrightarrow (GSp(2g), {\mathbf {X}}_{g})\) is quasiconstant (Definition N.5.3).
Remark 4.3.2
The theorem is proved using classical Hodge theory, specifically the methods of [24]. In particular, it is valid for any symplectic embedding \(\varphi '\), not just the special integral ones considered in Sect. .
As for (ii): Using Theorem for \(G=GL(n)\) and the generalization of the discrete fibers theorem [36, Theorem 2] mentioned in Remark I.1.3, we proved:
Theorem 4.3.3
([36], Corollary 6.2.2) Let \((G,\mu )\) be a maximal cocharacter datum ([36, §2.4], we do not assume \((G, \mu )\) arises by reduction mod p from a Shimura datum). Let \(\chi \in X^*(L)\) be a maximal character (loc. cit., Definition 2.4.3). Then there exists \(N\ge 1\) such that for every \(w \in {}^I W\), there is a section \(h_w \in H^0([E\backslash \overline{G}_w], {\mathscr {V}}(N\chi ))\) with \({{\,\mathrm{nonvanish}\,}}(h_w)=[E\backslash G_w]\).
Return to the setting where \((G,\mu )\) is associated to \(({\mathbf {G}}, {\mathbf {X}})\) and \(\varphi \) as in Sect. .
Corollary 4.3.4
There exists \(N\ge 1\) such that for every \(w \in {}^I W\), there exists a section \(h_w \in H^0([E\backslash \overline{G}_w], {\mathscr {V}}(N\eta _{\omega }))\) whose nonvanishing locus is precisely \([E\backslash G_w]\).
Proof 1: Since \(\eta _{\omega }\) is easily seen to be Lample, Theorem shows that it is (p, L)admissible for all primes p (Sect. N.5). Therefore the desired sections exist by Theorem . \(\square \)
Proof 2: The forgetful representation \(GSp(V,\psi ) \rightarrow GL(V)\) exhibits \((G, \mu )\) as a maximal cocharacter datum and \(\eta _{\omega }\) as a maximal character. Thus the corollary is a special case of Theorem . \(\square \)
Corollary 4.3.5
(EO Hasse invariants) For every EO stratum \(S_w \subset S_{{\mathcal K}}\), the section \(\zeta ^*h_w\in H^0(\overline{S}_w,\omega ^{N})\) is \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant and \({{\,\mathrm{nonvanish}\,}}(\zeta ^* h_w)=S_w\).
Proof
The section \(\zeta ^*h_w\) is \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant by (4.2.6) and \({{\,\mathrm{nonvanish}\,}}(\zeta ^* h_w)=S_w\) follows from Corollary . \(\square \)
Next we deduce the affineness of strata in the compact case (Corollary I.2.4). The generalization to the noncompact case (Corollary I.2.6) is treated in Proposition .
Corollary 4.3.6
(Affineness, compact case) Suppose \(S_{{\mathcal K}}\) is proper over k. Then the strata \(S_w\) are affine for all \(w \in {}^I W\).
Proof
The Hodge line bundle \(\omega \) is ample on \(S_{{\mathcal K}}\) [77, 5.2.11(b)]. Since the nonvanishing locus of a section of an ample line bundle on a proper scheme is affine, the result follows from Corollary . \(\square \)
Combining Theorem with Theorem gives a ‘discrete fibers’ result for symplectic embeddings of Shimura varieties:
Corollary 4.3.7
(Discrete Fibers) Let \(\varphi ':({\mathbf {G}}, {\mathbf {X}})\hookrightarrow (GSp(2g), {\mathbf {X}}_{g})\) be an arbitrary symplectic embedding of Shimura data. Then the induced morphism of stacks \(\mathop {G\hbox {}{} \mathtt{Zip}}\nolimits ^{\mu } \rightarrow \mathop {GSp(2g)\hbox {}{} \mathtt{Zip}}\nolimits ^{\mu _g}\) has discrete fibers. In other words, if two EO strata map to the same one under \(\varphi '\), then there is no closure relation between them.
For example, let \(\varphi '\) be the tautological embedding of a PELtype Shimura variety in its underlying Siegel variety. The naive stratification of \(S_{{\mathcal K}}\) is defined by taking preimages of Siegel EO strata; it is induced by the isomorphism class of the underlying \(BT_1\) without its additional structure. Corollary states that a naive EO stratum is topologically a disjoint union of (true) EO strata.
9 Lifting injective sections for pnilpotent schemes and the length stratification
9.1 Gluing sections
Theorem 5.1.1
 (a)
There exists an integer \(m_0(X,{\mathcal L},{\mathcal D})\) such that for all integers \(a\ge m_0(X,{\mathcal L},{\mathcal D})\) and every rtuple of sections \(s_i \in H^0(X_i, \mathcal L)\), \(1\le i \le r\), such that \(s_i=s_j\) on \((X_i \cap X_j)_{\mathrm{red}}\), there exists a unique section \(s \in H^0(X, \mathcal L^{p^{2a}})\) which is Zariskilocally the \(p^a\)power of a section \(s'\) of \({\mathcal L}^{p^a}\) which restricts to \(s_i^{p^{2a}}\) on \(X_i\).
 (b)
If X has no embedded components, then s is injective if and only if \(s_1, \ldots , s_r\) are all injective.
Proof
First we consider (a). Since X is Noetherian, it is covered by finitely many open affine subsets \(X=U_1\cup ... \cup U_d\) such that \({\mathcal L}_{U_j}={\mathcal O}_{U_j}\). For each \(j=1,\ldots ,d\), let \({\mathcal D}_j\) be the decomposition \((X_i\cap U_{j, \mathrm red})_i\) of \(U_{j, \text {red}}\). If we prove the result for each \((U_j, {\mathcal L}_{U_j}, {\mathcal D}_j)\), then we may take \(m_0:=\sup _j \{m_0(U_j,{\mathcal L}_j, {\mathcal D}_j)\}\). Thus we may work locally and assume that \(X={{\,\mathrm{Spec}\,}}(A)\) with A Noetherian and \({\mathcal L}={\mathcal O}_X\). By induction we reduce to \(r=2\). Assertion (a) follows from Lemma below.
For (b), we may also assume \(X={{\,\mathrm{Spec}\,}}(A)\) with A Noetherian. The result then follows from the claim that s is injective if and only if \(s_{X_\mathrm{red}}\) is injective. This is an immediate consequence of the fact that the set of zerodivisors of A is the union of the associated primes of A [79, Theorem 6.1]. \(\square \)
Lemma 5.1.2
 (a)
\(x\equiv x_1^{p^{2n}} \pmod {I_1}\) and \(x\equiv x_2^{p^{2n}} \pmod {I_2}\).
 (b)
\(x=y^{p^n}\) for some \(y\in A\).
Proof
 (1)
\(p^h=0\) in A,
 (2)
\((\sqrt{I_1+I_2})^{p^m}=0\) in \(A/(I_1+I_2)\),
 (3)
\({\mathcal N}(A)^{p^d}=0\).
To prove uniqueness, assume that \(x,x'\in A\) satisfy (a) and (b). In particular, we have \(xx'\in {\mathcal N}(A)\). Write \(x=y^{p^n}\) and \(x'=y'^{p^n}\) for some \(y,y'\in A\). The ring \(A/{\mathcal N}(A)\) has characteristic p, so \(xx'=y^{p^n}y'^{p^n}=(yy')^{p^n}=0\) in \(A/{\mathcal N}(A)\), hence \(yy'\in {\mathcal N}(A)\). We deduce \((yy')^{p^d}=0\) in A, hence also in A / pA. It follows that \(y^{p^d}\equiv y'^{p^d} \pmod {pA}\). Thus \(y^{p^{d+m1}}\equiv y'^{p^{d+m1}} \pmod {p^mA}\) for all \(m\ge 1\). Hence \(y^{p^{u}}=y'^{p^{u}}\) in A for all \(u\ge d+h1\). In particular \(x=x'\). \(\square \)
9.2 Length stratification
Retain the setting of Sect. . In particular, G is a reductive \(\mathbf {F}_p\)group and \(\mu :\mathbf {G}_{m,k}\rightarrow G_k\) is a cocharacter such that the cocharacter datum \((G,\mu )\) arises from a Shimura datum of Hodgetype (Sect. ). By Corollary , there exists an integer \(N\ge 1\) and sections \(h_w\in H^0([E\backslash \overline{G}_w],\omega ^N)\) for each \(w\in {}^I W\), such that the nonvanishing locus of \(h_w\) is exactly the open substack \([E\backslash G_w]\).
Lemma 5.2.1
One has \(\overline{G}_j=\bigcup _{\ell (w) = j}\overline{G}_w\).
Proof
Recall that the underlying topological space of \(\mathop {G\hbox {}{} \mathtt{Zip}}\nolimits ^\mu \) is isomorphic to \({}^I W\) endowed with the topology induced by the partial order \(\preccurlyeq \), which is in general finer than the restriction of the Bruhat order of W [87, Theorem 6.2].
Let \(w\in {\phantom {W}}^{I}{W}\) and let \(w=s_{\alpha _1} \ldots s_{\alpha _r}\) be a reduced expression. Then \(w':=s_{\alpha _1} \ldots s_{\alpha _{r1}}\in {\phantom {W}}^{I}{W}\) and \(\ell (w')=r1\). Furthermore, \(w'\preccurlyeq w\) because \(\preccurlyeq \) is finer than the Bruhat order. Hence any element in \({\phantom {W}}^{I}{W}\) of length \(r\ge 1\) has an element of \({\phantom {W}}^{I}{W}\) of length \(r1\) in its closure. But since \(w\mapsto w_{0,I}ww_0\) is an orderreversing involution, we deduce similarly that any element of length \(\le j\) in \({}^I W\) lies in the closure of an element of \({}^I W\) of length j. \(\square \)
In the next proposition, we prove that the length stratification of G is principally pure, i.e \(G_j\) is the nonvanishing locus of a section over \(\overline{G}_j\).
Proposition 5.2.2
There exists an integer \(N'\ge 1\) and for each \(0\le j \le d\), a section \(h_j \in H^0(\left[ E \backslash \overline{G}_j\right] ,\omega ^{N'})\) such that the nonvanishing locus of \(h_j\) is exactly \(G_j\).
We call the \(h_j\)length Hasse invariants of \(\mathop {G\hbox {}{} \mathtt{Zip}}\nolimits ^\mu \).
Proof
One has the decomposition \( \overline{G}_j=\bigcup _{\ell (w)=j}\overline{G}_w\) and for each w with \(\ell (w)=j\), the section \(h_w\) afforded by Corollary . We may interpret each \(h_w\) as a regular Eeigenfunction on \(\overline{G}_w\) for the character \(N\eta _\omega \). Using Theorem , we obtain a function \(\overline{G}_j\rightarrow \mathbf {A}^1\) which restricts to \(h^{r}_w\) on \(\overline{G}_w\) for some \(r\ge 1\). Hence \(h_j\) is an Eeigenfunction for the character \(Nr\eta _{\omega }\). It follows that \(h_j\) identifies with an element of \(H^0(\left[ E \backslash \overline{G}_j\right] ,\omega ^{N'})\) for \(N'=Nr\), and the result follows. \(\square \)
Proposition 5.2.3
 (a)
The scheme S is equidimensional of dimension d.
 (b)
The stratum \(S_w\) is nonempty for all \(w\in {}^I W\).
 (c)
The stratum \(S_e=S_0\) is zerodimensional.
 (1)
The schemes \(S_j\) and \(S^*_j\) are equidimensional of dimension j,
 (2)
The sections \(h_j\) are injective (Sect. N.2.4); equivalently \(S_j\) is open dense in \(S^*_j\).
 (3)
For \(w\in {}^I W\), \(S_w\) is equidimensional of dimension \(\ell (w)\).
Remark 5.2.4
It is not claimed that \(S_w\) is dense in \(S^*_w\), nor that \(S^*_w\) is equidimensional.
Proof
The section \(f^*(h_j)\) does not restrict to zero on any irreducible component of \(S^*_j\), so \(S_{j}\) is dense in \(S^*_j\) and \(h_j\) is injective, which shows (2).
Finally, we show (3). If \(w\in {}^IW\) satisfies \(\ell (w)\le j\) and \(\dim (S_w)=j\), then \(\ell (w)=j\). Conversely, let \(w\in W\) be an element of length j. Since \(S^*_j\) is pure of dimension j, there exists \(w'\in {}^I W\) of length j satisfying \(\dim (S^*_{w'})=j\) such that \(S_w\) intersects \(S^*_{w'}\). The continuity of f implies that \(w'=w\), so \(\dim (S^*_w)=j\). Hence \(\dim (S_w)=j\) as well. Let Z be an irreducible component of \(S_w\), and assume \(\dim (Z)<j\). Then Z is contained in \(S^*_{w'}\) for an element \(w'\ne w\) of length j, because \(S^*_j\) is pure of dimension j. Again, this contradicts the continuity of f. Hence \(S_w\) is pure of dimension j. \(\square \)
10 Extension of Hasse invariants to compactifications
10.1 Compactifications
We review the results of MadapusiPera [77] that we shall use about integral models of toroidal and minimal compactifications of Hodgetype Shimura varieties. As recalled below, most of these results rely on the corresponding statements in the Siegel case, which are due to Chai–Faltings [21]. Inbetween the works of Chai–Faltings and MadapusiPera, generalizations to the intermediate PEL case were given in a series of works by Lan [65, 66, 71, 72]. We shall often refer to Lan for precise references in the Siegel case.
10.1.1 Toroidal compactifications
10.1.2 Compactification of the universal semiabelian scheme
Let \(\tilde{{\mathcal A}}_g\) be the universal semiabelian scheme over \({\mathscr {S}}_{g, {\tilde{{\mathcal K}}}}^{{\tilde{\Sigma }}}\). Let \(\beta :\overline{{\mathcal A}}_g \rightarrow {\mathscr {S}}_{g, {\tilde{{\mathcal K}}}}^{\tilde{\Sigma }}\) be one of the compactifications of \(\tilde{{\mathcal A}_g}\) constructed in [21, §6.1] and [65, §2.B]. By [65, Proposition 3.19] (see also [21, Chapter VI, §1, Remark 1.4]), after possibly refining \(\tilde{\Sigma }\) (and a corresponding combinatorial datum used in the definition of \(\overline{{\mathcal A}}_g)\) one can arrange that the structure map \(\beta \) is log integral (by [65, Prop 3.18] this is equivalent to \(\beta \) being equidimensional). In particular, by refining \(\Sigma \) one may ensure that \(\Sigma \) is a refinement of \(\varphi ^*{\tilde{\Sigma }}\) for some \({\tilde{\Sigma }}\) which admits a \(\beta \) which is log integral. The log integrality of \(\beta \) will be used to ensure that the Frobenius is wellbehaved on the log de Rham cohomology of \({\mathscr {S}}_{g, \tilde{{\mathcal K}}}^{\tilde{\Sigma }}\), see the proof of Lemma .
10.1.3 Extension of de Rham and Hodge bundles
By [21, Chapter 6, Theorem 4.2 and preceding examples] and [65, Theorem 2.15(d)], \(H^1_{\mathop {\text {logdR} }\nolimits }(\overline{{\mathcal A}}_g/{\mathscr {S}}_{g, {\tilde{{\mathcal K}}}}^{\tilde{\Sigma }})\) is a rank 2g, locally free extension of \(H^1_{\mathrm{dR}}({\mathcal A}_g/{\mathscr {S}}_{g, {\mathcal K}})\). Let \(H_{\mathrm{dR}}^{1,\mathrm{can}}({\mathcal A}_g/{\mathscr {S}}_{g,{\tilde{{\mathcal K}}}})\) be the canonical extension of \(H_{\mathrm{dR}}^{1}({\mathcal A}_g/{\mathscr {S}}_{g,{\tilde{{\mathcal K}}}})\) to \({\mathscr {S}}_{g, {\tilde{{\mathcal K}}}}^{\tilde{\Sigma }}\) . By loc. cit., \(H_{\mathrm{dR}}^{1,\mathrm{can}}({\mathcal A}_g/{\mathscr {S}}_{g,{\tilde{{\mathcal K}}}})=H^1_{\mathop {\text {logdR} }\nolimits }(\overline{{\mathcal A}}_g/{\mathscr {S}}_{g, {\tilde{{\mathcal K}}}}^{{\tilde{\Sigma }}})\).
By [65, Proposition 6.9(2)] the natural pairing on \(H_{\mathrm{dR}}^{1}({\mathcal A}_g/{\mathscr {S}}_{g,{\tilde{{\mathcal K}}}})\) extends (uniquely) to a perfect pairing on \(H_{\mathrm{dR}}^{1,\mathrm{can}}({\mathcal A}_g/{\mathscr {S}}_{g,{\tilde{{\mathcal K}}}})\). The canonical extension \(\Omega ^{\mathrm{can}}_g\subset H_{\mathrm{dR}}^{1, \mathrm{can}}({\mathcal A}_g/{\mathscr {S}}_{g,{\tilde{{\mathcal K}}}})\) is a maximal, totally isotropic, locally direct factor; it is also the pullback along the identity section of \(\Omega ^1_{\tilde{{\mathcal A}_g}/{\mathscr {S}}_{g,{\tilde{{\mathcal K}}}}}\).
In the general case, pull back along \(\varphi ^{\Sigma /\tilde{\Sigma }}\): Put \(H^1_{\mathop {\text {logdR} }\nolimits }(\overline{{\mathcal A}}/{\mathscr {S}}_{{\mathcal K}}^{\Sigma }):=\varphi ^{\Sigma /\tilde{\Sigma },*}H^1_{\mathop {\text {logdR} }\nolimits }(\overline{{\mathcal A}}_g/{\mathscr {S}}_{g, \tilde{\mathcal K}}^{{\tilde{\Sigma }}})\) and \(\Omega ^{\mathrm{can}}:=\varphi ^{\Sigma /\tilde{\Sigma },*}\Omega _g^{\mathrm{can}}\); these give locally free extensions of \(H^1_{\mathrm{dR}}({\mathcal A}_g/{\mathscr {S}}_{{\mathcal K}})\) and \(\Omega \) respectively, see also [77, 5.1.1]. Since the Hodge line bundle will be used so frequently, we abuse notation and continue to write \(\omega \) for \(\det \Omega ^{\mathrm{can}}\) and \(\omega _g\) for \(\det \Omega ^{\mathrm{can}}_g\).
10.1.4 Degeneration of the Hodge–de Rham spectral sequence and base change: the Siegel case
Lemma 6.1.1
For all \(n \ge 0\), formation of \(H^n_{\mathop {\text {logdR} }\nolimits }(\overline{{\mathcal A}}_g/{\mathscr {S}}_{g, {\tilde{{\mathcal K}}}}^{{\tilde{\Sigma }}})\) commutes with arbitrary base change.
10.1.5 Extension of torsors and bundles
By [77, Proposition 5.3.2], the sections \((s_{\alpha , \mathrm{dR}})\) of Sect. extend to \(H^1_{\mathop {\text {logdR} }\nolimits }({\mathcal A}/{\mathscr {S}}_{{\mathcal K}}^{\Sigma })\). Repeating the definition of \(I_{{\mathcal G}},I_{{\mathcal P}}\) (Sect. ) with \({\mathscr {S}}_{{\mathcal K}}^{\Sigma }\), the extended sections and \(H^1_{\mathop {\text {logdR} }\nolimits }({\mathcal A}/{\mathscr {S}}_{{\mathcal K}}^{\Sigma })\) give extensions to torsors \(I_{{\mathcal G}}^{\Sigma },I_{{\mathcal P}}^{\Sigma }\) on \({\mathscr {S}}_{{\mathcal K}}^{\Sigma }\). Again \(I_{{\mathcal L}}^{\Sigma }:=I_{{\mathcal P}}^{\Sigma }/R_u({\mathcal P})\) gives an \({\mathcal L}\)torsor on \({\mathscr {S}}_{{\mathcal K}}^{\Sigma }\). Repeating the construction of Sect. with \(I_{{\mathcal L}}^{\Sigma }\) in place of \(I_{{\mathcal L}}\) associates a vector bundle \({\mathscr {V}}^{\mathrm{can}}(\eta )\) on \({\mathscr {S}}_{{\mathcal K}}^{\Sigma }\) with every \(\eta \in X^*_{+,\mathbf {L}}(\mathbf {T})\). Let \(D=D_{{\mathcal K}}^{\Sigma }\) be the boundary divisor of \({\mathscr {S}}_{{\mathcal K}}^{\Sigma }\) relative \({\mathscr {S}}_{{\mathcal K}}\). Set \({\mathscr {V}}^{\mathrm{sub}}(\eta ):={\mathscr {V}}^{\mathrm{can}}(\eta )(D)\). Since D is a relative, effective Cartier divisor in \({\mathscr {S}}_{{\mathcal K}}^{\Sigma }/{\mathcal O}_{E,{\mathfrak p}}\), the sheaf \({\mathscr {V}}^{\mathrm{sub}}(\eta )\) is again locally free.
10.1.6 The minimal compactification
Let \({\mathscr {S}}_{g, {\tilde{{\mathcal K}}}}^{\min }\) (resp. \({\mathscr {S}}_{{\mathcal K}}^{\min } \)) be the minimal compactification of \({\mathscr {S}}_{g,{\tilde{{\mathcal K}}}}\) (resp. \({\mathscr {S}}_{{\mathcal K}}\)) constructed in [21, Chap. 5, Theorem 2.3] (resp. [77, 5.2.1]). By construction, there are proper maps \(\pi :{\mathscr {S}}_{{\mathcal K}}^{\Sigma } \rightarrow {\mathscr {S}}_{{\mathcal K}}^{\min } \) and \(\pi _g:{\mathscr {S}}_{g, {\tilde{{\mathcal K}}}}^{{\tilde{\Sigma }}} \rightarrow {\mathscr {S}}_{g,{\tilde{{\mathcal K}}}}^{\min }\) satisfying \(\pi _* {\mathcal O}_{{\mathscr {S}}_{{\mathcal K}}^{\Sigma }}={\mathcal O}_{{\mathscr {S}}_{{\mathcal K}}^{\min } }\) and \(\pi _* {\mathcal O}_{{\mathscr {S}}_{g, {\tilde{{\mathcal K}}}}^{{\tilde{\Sigma }}}}={\mathcal O}_{{\mathscr {S}}_{g,{\tilde{{\mathcal K}}}}^{\min }}\). It is expected that the higher direct images of \(\pi , \pi _g\) vanish in general; this is known in the PEL case, see Condition and Remark . This issue will play an important role in Sects. 7–10.
10.1.7 \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariance for toroidal compactifications
The notions of \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant sheaf, torsor, subscheme, section and GZip from Sects. , generalize to the (double) tower \(({\mathscr {S}}_{{\mathcal K}}^{\Sigma })_{{\mathcal K}^p, \Sigma }\) of toroidal compactifications. Later we shall consider subtowers consisting of all \({\mathscr {S}}^{\Sigma }_{{\mathcal K}}\) such that \(\Sigma \) is required to satisfy some property P which is attainable by refinement (and no restriction is imposed on \({\mathcal K}^p\)); the examples of P which will occur are: “\(\Sigma \) smooth”, “\(\Sigma \) a refinement of \(\varphi ^*{{\tilde{\Sigma }}}\) for some \({\tilde{\Sigma }}\) which admits \(\beta \) logintegral” (Sect. ) and their common intersection. Then \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariance is defined in the same way, except that we restrict to the morphisms \(\pi ^{\Sigma '/\Sigma }_{{\mathcal K}'/{\mathcal K}}:{\mathscr {S}}_{{\mathcal K}'}^{\Sigma '} \rightarrow {\mathscr {S}}_{{\mathcal K}}^{\Sigma }\) and \(g^{1}_{\Sigma ^g/\Sigma }:{\mathscr {S}}_{g^{1} {\mathcal K}g}^{\Sigma ^g}\rightarrow {\mathscr {S}}_{{\mathcal K}}^{\Sigma }\), where \({\Sigma }\) has property P and \(\Sigma '\) (resp. \(\Sigma ^g\)) is a refinement of \(\pi _{{\mathcal K}'/{\mathcal K}}^*\Sigma \) (resp. \((g^{1})^* \Sigma )\)) having property P.
The system of boundary subschemes \((D^{\Sigma }_{{\mathcal K}})\) is \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant. The torsors \(I_{{\mathcal G}}^{\Sigma },I_{{\mathcal P}}^{\Sigma },I_{{\mathcal L}}^{\Sigma }\) and the automorphic vector bundles \({\mathscr {V}}^{\mathrm{can}}(\eta ),{\mathscr {V}}^{\mathrm{sub}}(\eta )\) are all \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant.
10.2 Extension of the universal \(\mathop {G\hbox {}{} \mathtt{Zip}}\nolimits \)
For the rest of Sect. 6, we assume that the rpcd \(\Sigma \) for \((\mathbf {G}, \mathbf {X}, {\mathcal K})\) is a refinement of \(\varphi ^* {\tilde{\Sigma }}\), where the rpcd \(\tilde{\Sigma }\) for \((GSp(2g), \mathbf {X}_g, \tilde{{\mathcal K}})\) has been chosen so that \(\beta \) is log integral (Sect. ). By contrast, the rpcd \(\Sigma \) for \((\mathbf {G}, \mathbf {X}, {\mathcal K})\) is not assumed to be smooth.
In this section, we prove:
Theorem 6.2.1
Define \(S_w^{\Sigma }:=(\zeta _{{\mathcal K}}^{\Sigma })^{1}([E\backslash G_w])\) and \(S_w^{\Sigma ,*}:=(\zeta _{{\mathcal K}}^{\Sigma })^{1}([E\backslash \overline{G}_w])\) for all \(w\in {}^I W\). Before embarking on the proof, note the following immediate, important corollary:
Corollary 6.2.2
The Hasse invariant \(h_w\in H^0(\overline{S}_w, \omega ^{N_w})\) of Corollary extends to a \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant section \(h_w^{\Sigma } \in H^0(S_w^{\Sigma ,*}, \omega ^{N_w})\) with nonvanishing locus precisely \(S_w^{\Sigma }\).
Let \(A_g/S_{g, {\tilde{{\mathcal K}}}}\) (resp. \(\tilde{A}_g/S_{g, \tilde{\mathcal K}}^{{\tilde{\Sigma }}}, \overline{A}_g/S_{g, {\tilde{{\mathcal K}}}}^{\tilde{\Sigma }}\)) be the special fiber of \({\mathcal A}_g/_{{\mathscr {S}}_{g, \tilde{\mathcal K}}}\) (resp. \(\tilde{{\mathcal A}}_g/{\mathscr {S}}_{g, {\tilde{{\mathcal K}}}}^{\tilde{\Sigma }}, \overline{{\mathcal A}}_g/{\mathscr {S}}_{g, {\tilde{{\mathcal K}}}}^{\tilde{\Sigma }}\)) and similarly for \(H^{1,\mathrm{can}}_{\mathrm{dR}}(A_g/S_{g, \tilde{\mathcal K}})\) etc. (Sect. ).
The Frobenius F and Verschiebung V of \(A_g/S_{g, {\tilde{{\mathcal K}}}}\) induce maps of \({\mathcal O}_{S_{g, {\tilde{{\mathcal K}}}}}\)modules \(F:(H_{\mathrm{dR}}^{1}(A_g/S_{g,{\tilde{{\mathcal K}}}}))^{(p)} \rightarrow H_{\mathrm{dR}}^{1}(A_g/S_{g,{\tilde{{\mathcal K}}}})\) and \(V:H_{\mathrm{dR}}^{1}(A_g/S_{g,{\tilde{{\mathcal K}}}}) \rightarrow H_{\mathrm{dR}}^{1}(A_g/S_{g,{\tilde{{\mathcal K}}}})^{(p)}\). These satisfy the usual conditions \({{\,\mathrm{Ker}\,}}(F)={{\,\mathrm{Im}\,}}(V)\), \({{\,\mathrm{Ker}\,}}(V)={{\,\mathrm{Im}\,}}(F)\). The polarization of \(A_g\) induces perfect pairings on \(H_{\mathrm{dR}}^{1}(A_g/S_{g,{\tilde{{\mathcal K}}}})\) and \(H_{\mathrm{dR}}^{1}(A_g/S_{g,{\tilde{{\mathcal K}}}})^{(p)}\) under which F, V are transpose to each other.
Lemma 6.2.3
 (a)
F, V extend the Frobenius and Verschiebung maps on \(H^1_{\mathrm{dR}}(A_g/S_{g,{\tilde{{\mathcal K}}}})\).
 (b)
\({{\,\mathrm{Ker}\,}}(F)={{\,\mathrm{Im}\,}}(V)\) and \({{\,\mathrm{Ker}\,}}(V)={{\,\mathrm{Im}\,}}(F)\).
 (c)
\({{\,\mathrm{Ker}\,}}(V)\) and \(\mathrm{Im}(V)\) are rank g locally free and locally direct summands of \(H_{\mathrm{dR}}^{1,\mathrm{can}}(A_g/S_{g,{\tilde{{\mathcal K}}}})\).
Proof
The Frobenius \(F:\overline{A}_g \rightarrow \overline{A}_g^{(p)}\) over \(S_{g, {\tilde{{\mathcal K}}}}^{{\tilde{\Sigma }}}\) induces \(F:H_{\mathop {\text {logdR} }\nolimits }^{1}( \overline{A}_g^{(p)}/ S_{g, {\tilde{{\mathcal K}}}}^{{\tilde{\Sigma }}}) \rightarrow H_{\mathop {\text {logdR} }\nolimits }^{1}( \overline{A}_g/ S_{g, {\tilde{{\mathcal K}}}}^{\tilde{\Sigma }})\). Applying Lemma to F gives \(H_{\mathop {\text {logdR} }\nolimits }^{1}( \overline{A}_g^{(p)}/ S_{g, {\tilde{{\mathcal K}}}}^{\tilde{\Sigma }}) \cong H_{\mathop {\text {logdR} }\nolimits }^{1}( \overline{A}/ S_{g, \tilde{\mathcal K}}^{{\tilde{\Sigma }}})^{(p)}\). Together this yields a map as in (6.2.3).
To simplify notation, write \(\overline{{\mathcal M}}=H_{\mathop {\text {logdR} }\nolimits }^{1}(\overline{A}_g/S_{g, \tilde{{\mathcal K}}}^{\tilde{\Sigma }} )\). We have constructed an extension \(F:\overline{{\mathcal M}}^{(p)} \rightarrow \overline{{\mathcal M}}\) over \(S_{g, \tilde{\mathcal K}}^{{\tilde{\Sigma }}}\). Define \(V:\overline{{\mathcal M}} \rightarrow \overline{{\mathcal M}}^{(p)}\) as its transpose with respect to the perfect pairings on \(\overline{{\mathcal M}}\) and \(\overline{{\mathcal M}}^{(p)}\). Clearly, V extends the Verschiebung map on \(S_{g, {\tilde{{\mathcal K}}}}\). It remains to check that F, V satisfy conditions (b) and (c) of the lemma.
Since the relations \(FV=0\) and \(VF=0\) hold over \(S_{g,{\tilde{{\mathcal K}}}}\), which is open dense in \(S_{g, {\tilde{{\mathcal K}}}}^{{\tilde{\Sigma }}}\), they continue to hold over \(S_{g, {\tilde{{\mathcal K}}}}^{{\tilde{\Sigma }}}\). Hence \({{\,\mathrm{Im}\,}}(F)\subset {{\,\mathrm{Ker}\,}}(V)\) and \({{\,\mathrm{Im}\,}}(V)\subset {{\,\mathrm{Ker}\,}}(F)\). On the other hand, we claim that \((\Omega ^{\mathrm{can}}_g)^{(p)} \subset {{\,\mathrm{Ker}\,}}(F)\). Indeed, F induces an injective morphism \((\Omega ^{\mathrm{can}}_g)^{(p)} / ((\Omega ^{\mathrm{can}}_g)^{(p)} \cap {{\,\mathrm{Ker}\,}}(F)) \rightarrow \overline{{\mathcal M}}\). Since we know the result over the open subscheme \(S_{g,{\tilde{{\mathcal K}}}}\subset S_{g, {\tilde{{\mathcal K}}}}^{{\tilde{\Sigma }}}\), the sheaf \({\mathcal N}:=(\Omega ^{\mathrm{can}}_g)^{(p)} / ((\Omega ^{\mathrm{can}}_g)^{(p)} \cap {{\,\mathrm{Ker}\,}}(F))\) restricts to zero on \(S_{g,{\tilde{{\mathcal K}}}}\). But \({\mathcal N}\) is a subsheaf of a locally free sheaf, and since \(S_{g, {\tilde{{\mathcal K}}}}^{\tilde{\Sigma }}\) is reduced and \(S_{g,{\tilde{{\mathcal K}}}}\) is open dense, we deduce \({\mathcal N}=0\). Thus \((\Omega ^{\mathrm{can}}_g)^{(p)} \subset {{\,\mathrm{Ker}\,}}(F)\).
Next, we claim that \({{\,\mathrm{Im}\,}}(F)\) is a locally free sheaf on \(S_{g, {\tilde{{\mathcal K}}}}^{{\tilde{\Sigma }}}\). It suffices to show \(\dim _{k(x)}({{\,\mathrm{Im}\,}}(F)\otimes k(x))=g\) for all \(x\in S_{g, \tilde{\mathcal K}}^{{\tilde{\Sigma }}}\). Since \((\Omega ^{\mathrm{can}}_g)^{(p)}\) is locally a direct summand of \({\mathcal M}^{(p)}\), we have an injection \((\Omega ^{\mathrm{can}}_g)^{(p)} \otimes k(x) \rightarrow {\mathcal M}^{(p)}\otimes k(x)\), which shows \(\dim _{k(x)} ({{\,\mathrm{Im}\,}}(F)\otimes k(x)) \le g\). The converse inequality follows simply from Nakayama’s lemma because this dimension is g generically on \(S_{g, {\tilde{{\mathcal K}}}}^{\tilde{\Sigma }}\). This proves that \({{\,\mathrm{Im}\,}}(F)\) is locally free of rank g.
For all \(x\in S_{g, {\tilde{{\mathcal K}}}}^{{\tilde{\Sigma }}}\), there is a natural surjection \({{\,\mathrm{Im}\,}}(F)\otimes k(x) \rightarrow {{\,\mathrm{Im}\,}}(F_x)\). In particular, \({{\,\mathrm{rk}\,}}(F_x)\le g\). We claim that \({{\,\mathrm{rk}\,}}(F_x)=g\). To see this, consider \({\mathcal M}:=H^1_{\mathop {\text {logcrys} }\nolimits }(\overline{A}_g / S^{\tilde{\Sigma }}_{g, {\tilde{{\mathcal K}}}})\), endowed with its crystalline Frobenius \(\Phi \). One has \({\mathcal M}\otimes _{W(k)} k \simeq \overline{{\mathcal M}}\) and \(\Phi \otimes _{W(k)}k=F\) [70, Remark 1.5]. Recall that \(\tilde{\Sigma }\) was chosen so that \(\beta :\overline{{\mathcal A}}_g \rightarrow {\mathscr {S}}_{g, \tilde{{\mathcal K}}}^{\tilde{\Sigma }}\) is log integral. Using the log integrality of \(\beta \) it is checked in [70] that \({\mathscr {S}}_{g, {\tilde{{\mathcal K}}}}^{\tilde{\Sigma }}\) satisfies the two assumptions of Proposition 1.7 of loc. cit. (see esp. Proposition 1.13, Lemma 7.22 and its proof). In particular, the first implies that the top exterior power \((\wedge ^{2g} {\mathcal M}, \wedge ^{2g} \Phi )\) is the \(\Phi \)span \({\mathcal O}_{S_{g, {\tilde{{\mathcal K}}}}^{\tilde{\Sigma }}/W(k)}(g)\) as defined by Ogus^{6} [84, Definition 5.2.1]. This means that the map \(\wedge ^{2g}\Phi :\wedge ^{2g}{\mathcal M}^{(p)} \rightarrow \wedge ^{2g} {\mathcal M}\) is Zariski locally on \(S_{g, \tilde{\mathcal K}}^{{\tilde{\Sigma }}}\) equal to \(p^g\) times an isomorphism.
In particular, at the point x, the linear map \(\Phi _x: {\mathcal M}_x^{(p)} \rightarrow {\mathcal M}_x\) satisfies that \(\wedge ^{2g} \Phi _x\) is \(p^g\) times an isomorphism. By the theory of invariant factors, there exist suitable bases of \({\mathcal M}_x\) in which \(\Phi _x\) is given by the diagonal matrix \(\mathrm{diag}(p^{a_1}, \ldots , p^{a_{2g}})\) for some integers \(a_i\) satisfying \(a_1 \ge \cdots \ge a_{2g} \ge 0\). Since the unique invariant factor of \(\wedge ^{2g} \Phi _x\) is \(p^g\), the only possibility is that \(a_1=\cdots =a_g=1\) and \(a_{g+1}=\cdots =a_{2g}=0\). Since the reduction modulo p of \(({\mathcal M}_x, \Phi _x)\) is \(({\mathcal M}\otimes k(x), F_x)\), one has \({{\,\mathrm{rk}\,}}(F_x)=g\) as claimed.
Since \(F_x\) and \(V_x\) are transpose to each other, they have the same rank, so we deduce \({{\,\mathrm{Ker}\,}}(F_x)={{\,\mathrm{Im}\,}}(V_x)\) and \({{\,\mathrm{Ker}\,}}(V_x)={{\,\mathrm{Im}\,}}(F_x)\). Since \({{\,\mathrm{Im}\,}}(F)\) is locally free, \({{\,\mathrm{Ker}\,}}(F)\) is locally a direct summand of \(\overline{{\mathcal M}}^{(p)}\). Hence \({{\,\mathrm{Ker}\,}}(F)\otimes k(x)\simeq {{\,\mathrm{Ker}\,}}(F_x)={{\,\mathrm{Im}\,}}(V_x)\) for all \(x\in S_{g, {\tilde{{\mathcal K}}}}^{{\tilde{\Sigma }}}\). In particular, the inclusion \({{\,\mathrm{Im}\,}}(V)\subset {{\,\mathrm{Ker}\,}}(F)\) induces a surjective map \({{\,\mathrm{Im}\,}}(V)\otimes k(x)\rightarrow {{\,\mathrm{Ker}\,}}(F)\otimes k(x)\). So \(({{\,\mathrm{Ker}\,}}(F)/{{\,\mathrm{Im}\,}}(V))\otimes k(x)=0\). Hence \({{\,\mathrm{Ker}\,}}(F)={{\,\mathrm{Im}\,}}(V)\) by Nakayama’s lemma. Thus \({{\,\mathrm{Ker}\,}}(V)\) is locally a direct factor of \({\mathcal M}\). By similar arguments, \({{\,\mathrm{Ker}\,}}(V)={{\,\mathrm{Im}\,}}(F)\). This terminates the proof. \(\square \)
Put \({{\,\mathrm{Fil}\,}}^{{\tilde{\Sigma }}}_{g, \mathrm conj}:={{\,\mathrm{Ker}\,}}(V)\). By Lemma , \(0 \subset {{\,\mathrm{Fil}\,}}_{g, \mathrm conj}^{{\tilde{\Sigma }}} \subset H^{1,\mathrm{can}}_{\mathrm{dR}}(A_g/S_{g, {\tilde{{\mathcal K}}}}^{{\tilde{\Sigma }}})\) extends the conjugate filtration (Sect. ) to \(S_{g, \tilde{\mathcal K}}^{{\tilde{\Sigma }}}\). Set \({{\,\mathrm{Fil}\,}}^{\Sigma }_\mathrm{conj}:=\varphi ^{\Sigma /{\tilde{\Sigma }},*}\text {Fil}_{g, \mathrm conj}\). Then \({{\,\mathrm{Fil}\,}}^{\Sigma }_\mathrm{conj}\) provides an extension of the conjugate filtration on \(H_{\mathrm{dR}}^{1,\mathrm{can}}(A/S_{{\mathcal K}}^{\Sigma })\) which is locally free and locally a direct summand. Replacing \(\text {Fil}_\mathrm{conj}\) with \({{\,\mathrm{Fil}\,}}^{\Sigma }_\mathrm{conj}\) in (4.2.4) gives an étale sheaf \(I_Q^{\Sigma }\) on \(S_{{\mathcal K}}^{\Sigma } \). Furthermore, \(I_Q^{\Sigma }\) is endowed with a natural Qaction.
Lemma 6.2.4
The étale sheaf \(I_Q^{\Sigma }\) is a Qtorsor on \(S_{{\mathcal K}}^{\Sigma } \)
Proof
We thank Torsten Wedhorn for his help with this. Choose an étale cover \(U\rightarrow S_{{\mathcal K}}^{\Sigma } \) which trivializes the Gtorsor \(I^{\Sigma }\). Hence we have an isomorphism \(I^{\Sigma }_U\simeq G\times U\). The stabilizer of \(\text {Fil}_\mathrm{conj}^{\Sigma }\) is a parabolic subgroup \({\mathcal Q}\) of \(G\times U\), and we need to show that it is étale locally on U isomorphic to Q. To check this, it suffices to show that \({\mathcal Q}\otimes k(x)\simeq Q\otimes k(x)\) at every point x. This follows from the fact that the type of the parabolic \({\mathcal Q}\otimes k(x)\) is a locally constant function, and that the result holds generically (on the open subset \(S_{{\mathcal K}}\)). \(\square \)
Proof of Theorem 6.2.1
By Sect. and Lemma , we have torsors \(I_{G}^{\Sigma },I_P^{\Sigma },I_Q^{\Sigma }\) for G, P, Q extending \(I_G, I_P,I_Q\) respectively. It remains to extend the isomorphisms \(\iota _0\), \(\iota _1\) to \(S_{{\mathcal K}}^{\Sigma } \) (Sect. ). On \(S_{g,{\tilde{{\mathcal K}}}}\), these isomorphisms are naturally induced by the maps F, V. Since these maps extend to \(S_{g, {\tilde{{\mathcal K}}}}^{{\tilde{\Sigma }}}\) by Lemma , so do \(\iota _0\) and \(\iota _1\). One obtains similar isomorphisms on \(S_{{\mathcal K}}^{\Sigma } \) by pullback from \(S_{g, {\tilde{{\mathcal K}}}}^{{\tilde{\Sigma }}}\). This completes the construction of the Gzip \(\underline{I}^{\Sigma }\) over \(S_{{\mathcal K}}^{\Sigma } \). \(\square \)
10.3 Affineness in the minimal compactification
Proposition 6.3.1
 (a)
The \(S^{\min }_w\) are pairwise disjoint and locally closed.
 (b)
The Stein factorization of \(\pi :\overline{S}^{\Sigma }_w\rightarrow \overline{S}^{\min }_w\) is given by \(\pi =f\circ g\) where \(g:\overline{S}^{\Sigma }_w\rightarrow T\) is proper with connected fibers such that \(g_{*}{\mathcal O}_{\overline{S}^{\Sigma }_w}={\mathcal O}_T\), and \(f:T\rightarrow \overline{S}^{\min }_w\) is a finite morphism which is a universal homeomorphism.
 (c)
The stratum \(S^{\min }_w\) is affine.
Remark 6.3.2
10.3.1 Comparison with Ekedahl–van der Geer [28]
We shall first prove Proposition (a) in the Siegel case and then reduce the general case to the Siegel one. For the Siegel case, we need to compare the extension of the EO stratification to \(S_{g, {\tilde{{\mathcal K}}}}^{{\tilde{\Sigma }}}\) afforded by Theorem with the extension defined in [28, §5].
Lemma 6.3.3
(Comparison) Let \(S_{g,w}^{{\tilde{\Sigma }}}\) be the EO strata defined by pulling back the stratification of \(\mathop {GSp(2g)\hbox {}{} \mathtt{Zip}}\nolimits ^{\mu _g}\) along \(\zeta _{\tilde{g}, {\mathcal K}}^{{\tilde{\Sigma }}}:S_{g, {\tilde{{\mathcal K}}}}^{{\tilde{\Sigma }}} \rightarrow \mathop {GSp(2g)\hbox {}{} \mathtt{Zip}}\nolimits ^{\mu _g}\). Then \('S_{g,w}^{{\tilde{\Sigma }}}= S_{g,w}^{\tilde{\Sigma }}\).
Proof
It suffices to prove that the two extensions of the EO stratification agree one boundary stratum at a time, i.e., to show that \('S_{g,w}^{{\tilde{\Sigma }}} \cap S_{g , {\tilde{{\mathcal K}}}}^{\langle i \rangle } =S_{g,w}^{{\tilde{\Sigma }}} \cap S_{g ,{\tilde{{\mathcal K}}}}^{\langle i \rangle }\) for all i. As in [77, 4.3.1], there is a universal, polarized 1motive over \(S_{g ,\tilde{\mathcal K}}^{\langle i \rangle }\); let \(H_{\mathrm{dR}}^{1, \langle i \rangle }\) denote its (contravariant) de Rham realization (loc. cit., 1.1.3 and [3, 4.3]). It is a locally free sheaf of rank 2g equipped with a nondegenerate symplectic pairing, induced from the polarization of the 1motive. By the characterization of \(H_{\mathrm{dR}}^{1, \langle i \rangle }\) as a canonical extension in [77, 4.3.1] and the characterization of \(H_{\mathrm{dR}}^{1,\mathrm{can}}(A_g/S_{g,{\tilde{{\mathcal K}}}})=H^1_{\mathop {\text {logdR} }\nolimits }(\overline{A}_g/S_{g, {\tilde{{\mathcal K}}}}^{{\tilde{\Sigma }}})\) in [65, Theorem 2.15(3)(d)], the restriction of \(H^1_{\mathop {\text {logdR} }\nolimits }(\overline{A}_g/S_{g, {\tilde{{\mathcal K}}}}^{{\tilde{\Sigma }}})\) to \(S_{g ,{\tilde{{\mathcal K}}}}^{\langle i \rangle }\) is \(H_{\mathrm{dR}}^{1, \langle i \rangle }\).
Consider the fiber of (6.3.5) over a kpoint \(x \in S_{g, {\tilde{{\mathcal K}}}}^{\langle i \rangle }\) with underlying semiabelian scheme \(\tilde{A}\) (6.3.4). We get a filtration of kvector spaces stable by F and V, viewed as \(\sigma \)linear and \(\sigma ^{1}\)linear maps respectively. We claim that there is a (noncanonical) splitting which is stable under both F and V. Set \(H^{1, \langle i \rangle }_{\mathrm{dR},x}:=H^{1, \langle i \rangle }_{\mathrm{dR}} \otimes k(x)\), \(W_{j,x}:=W_jH^{1, \langle i \rangle }_{\mathrm{dR}} \otimes k(x)\) and \({{\,\mathrm{Gr}\,}}^W_{j,x}:={{\,\mathrm{Gr}\,}}^W_j \otimes k(x)\). The action of F (resp. V) on \({{\,\mathrm{Gr}\,}}^W_{0,x}\) (resp. \({{\,\mathrm{Gr}\,}}^W_{2,x}\)) is invertible (cf. [3, 4.3]).
Recall the following elementary fact of \(\sigma \)linear algebra: Assume N is a kvector space, \(f:N \rightarrow N\) is a \(\sigma \)linear map and \(N'\) is a subspace stable by f. If f is invertible on either \(N'\) or the quotient \(N/N'\), then there is a (noncanonical) fstable splitting \(N= N' \oplus N/N'\).
Remark 6.3.4
The proof of Lemma is similar to [68, Lemmas 3.2.6, 3.4.3] which uses the language of Raynaud extensions instead of 1motives. Lemma is also implicit in [28, §5] but it seems to us that the arguments in [28, §5] may be incomplete. We thank the referee for suggesting to us that there might be an issue with [28, §5].
Proof of Proposition 6.3.1
We prove that the \(S_w^{\min }\) are disjoint and locally closed by first explaining why it is true in the Siegel case and then reducing to that case. Since \( \pi \) maps a semiabelian kscheme \(\tilde{A}\) to its abelian part A, in view of the definition of the \('S_w^{{\tilde{\Sigma }}}\) recalled above, the disjointness of the images \(\pi ('S_w^{{\tilde{\Sigma }}})\) is just the injectivity of \(\rho _i:W_{gi} \rightarrow W_g\). Thus the disjointness of the \(S_w^{\min }\) in the Siegel case follows from the comparison Lemma .
Next we show that the \(S^{\min }_w\) are locally closed. Since \(\zeta ^{\Sigma }_{{\mathcal K}}\) is continuous, \(S_w^{\Sigma , *}\) is closed in \(S_{{\mathcal K}}^{\Sigma }\). The complement of \(S_w^{\Sigma }\) in \(S_w^{\Sigma ,*}\) is also closed, as it is the finite union of the \(S_{w'}^{\Sigma ,*}\) for all \(w'\preccurlyeq w\) in \({}^IW\). Since \(\pi \) is proper, the images \(\pi (S_w^{\Sigma , *})\) and \(\pi (S_w^{\Sigma ,*}{\setminus } S_w^{\Sigma })\) are both closed in \(S_{{\mathcal K}}^{\min }\). Since the \(S_w^{\min }\) are disjoint, \(\pi (S_w^{\Sigma })=S_w^{\min }\) is the complement of \(\pi (S_w^{\Sigma ,*}{\setminus } S_w^{\Sigma })\) in \(\pi (S_w^{\Sigma ,*})\). Hence \(\pi (S_w^{\Sigma })=S_w^{\min }\) is locally closed.
The Stein factorization of the map \(\pi :\overline{S}^{\Sigma }_w\rightarrow \overline{S}^{\min }_w\) gives a scheme T with a finite map \(f:T\rightarrow \overline{S}^{\min }_w\), such that \(\pi \) factors through a proper map \(g:\overline{S}^{\Sigma }_w\rightarrow T\) with connected fibers, satisfying \(g_{*}{\mathcal O}_{\overline{S}^{\Sigma }_w}={\mathcal O}_T\). By (6.3.9), we see that \(\pi \) has connected fibers, hence f has connected fibers. It follows that f is a universal homeomorphism. This proves (b).
Finally, we prove (c). Denote again by \(\omega \) the line bundle \(f^*\omega \) on T. Note that \(\omega \) is again ample on T. Since \(g_{*}{\mathcal O}_{\overline{S}^{\Sigma }_w}={\mathcal O}_T\), it follows from the projection formula that \(g_{*}\omega =\omega \). In particular, the section \(h_w^\Sigma \) descends to a section \(h_T\in H^0(T,\omega ^{N_w})\). The nonvanishing locus \(U\subset T\) of \(h_T\) satisfies \(g^{1}(U)=S^{\Sigma }_w\). Using again (6.3.9), we deduce \(U=f^{1}(S^{\min }_w)\). Since \(\omega \) is ample on T, the open U is affine, and hence \(S^{\min }_w=f(U)\) is affine by a theorem of Chevalley. \(\square \)
10.4 The length stratification of Hodgetype Shimura varieties
We apply Sect. to \(\zeta _{{\mathcal K}}^{\Sigma } :S_{{\mathcal K}}^{\Sigma }\rightarrow \mathop {G\hbox {}{} \mathtt{Zip}}\nolimits ^\mu \). Write \(S^{\Sigma }_j\) and \(S^{\Sigma ,*}_j\) for the length strata in \(S_{{\mathcal K}}^{\Sigma }\).
Assumption (a) of Proposition follows from [77, Theorem 2]. Assumption (b) was proved in the PEL case in [104, Theorem 2], and has recently been generalized to general Hodgetype Shimura varieties, see [57, 76, 111]. This also follows from [103] using a different language. As for Assumption (c):
Lemma 6.4.1
Suppose \(\varphi \) (Sect. ) is a PEL embedding . Then \(S_e^{\Sigma }\) intersects the boundary trivially, i.e., \(S_e=S_e^{\Sigma }\). In particular, \(S_e^{\Sigma }\) is zerodimensional.
Proof
 (a)
For every boundary point x of \(S_{g, {\tilde{{\mathcal K}}}}^{{\tilde{\Sigma }}} {\setminus } S_{g, {\tilde{{\mathcal K}}}}\), the underlying semiabelian variety \(\tilde{A}\) (6.3.4) has \({{\,\mathrm{mult}\,}}{{\,\mathrm{rk}\,}}\tilde{A}[p]>0\).
 (b)
The multiplicative rank is constant along the strata \(S_{g,w}^{{\tilde{\Sigma }}}\) for \(w \in W_g\).
 (c)
The stratum \(S_{g,w}^{{\tilde{\Sigma }}}\) containing \(S_e^{\Sigma }\) has multiplicative rank 0.
Now consider (b). It follows from [83, §15] that \({{\,\mathrm{mult}\,}}{{\,\mathrm{rk}\,}}\tilde{A}\) is the semisimple rank of \(V: H^0(\tilde{A}, \Omega ^1_{\tilde{A}}) \rightarrow H^0(\tilde{A}, \Omega ^1_{\tilde{A}_x})^{(p)}\), since V induces an isomorphism on the differentials of the torus part T. In view of the transpose relation between F and V, s is also the semisimple rank of \(F:(\ker V)^{(p)}\rightarrow \ker V\). Since the semisimple rank is equal to the stable rank (i.e., the rank of all sufficiently large powers of F), one sees that s is an invariant of the GSp(2g)Zip \(I^{{\tilde{\Sigma }}}_x\) associated to x by Theorem .
To prove (c), by (b) it is enough to show that the multiplicative rank of the underlying abelian variety of a point of \(S_e\) is zero, under the assumption that \(\mathbf {G}^{{{\,\mathrm{ad}\,}}}(\mathbf {R})\) has no compact factors. This can be checked casebycase using Moonen’s “standard objects” [81, §§4.9,5.8]. (Given \(w \in {\phantom {W}}^{I}{W}\), the associated standard object is an explicit representative of the isomorphism class of the GZips of type w.) \(\square \)
It is expected that Lemma remains true for an arbitrary Shimura variety of Hodge type. For now we single this out as a condition; it will play an important role in Sects. 8 and 10.
Condition 6.4.2
The minimal EO stratum \(S_e^{\Sigma }\) has dimension zero.
Remark 6.4.3
As in Lemma , Condition holds when \(S_e^{\Sigma }\) does not meet the boundary, i.e., when \(S_e=S_e^\Sigma \).
Putting together Propositions , and Lemma gives the following result on length strata of Hodgetype Shimura varieties and their Hasse invariants:
Corollary 6.4.4
 (a)
The length strata \(S^{\Sigma }_j\) and \(S^{\Sigma ,*}_j\) are equidimensional of dimension j, and \(S^{\Sigma }_j\) is open dense in \(S^{\Sigma ,*}_j\).
 (b)
For \(w\in {}^I W\), \(S^{\Sigma }_w\) is equidimensional of dimension \(\ell (w)\).
 (c)
For every j, there exists \(N_j \ge 1\) and a \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant \(h_j \in H^0(S_j^{\Sigma ,*}, \omega ^{N_j})\) satisfying \({{\,\mathrm{nonvanish}\,}}(h_j)=S_{j}^{\Sigma }\).
We call the \(h_j\) the length Hasse invariants of \(S^{\Sigma }_{{\mathcal K}}\).
11 Geometry of Hasse regular sequences
11.1 Cohomological vanishing on \({\mathscr {S}}_{{\mathcal K}}^{\Sigma ,n} \)
The following variant of “cohomology and base change” will be used to reduce vanishing statements to the special fiber.
Lemma 7.1.1
Let \((R,{\mathfrak m})\) be a Noetherian local ring, \(f:X\rightarrow {{\,\mathrm{Spec}\,}}R\) a proper morphism, and \({\mathcal F}\) a coherent \({\mathcal O}_X\)module, flat over R. Denote by x the point \({\mathfrak m}\in {{\,\mathrm{Spec}\,}}R\). Assume that \(H^i(X_x,{\mathcal F}_x)=0\) for \(i\ge 0\). Then \(H^i (X,{\mathcal F})=0\).
In particular, \(H^i (X_n,{\mathcal F}_n)=0\) for all \(n\ge 1\), where \(X_n=X\times _R R/{\mathfrak m}^n\) and \({\mathcal F}_n\) is the pullback of \({\mathcal F}\) via \(X_n\rightarrow X\).
Proof
The base change map \(\varphi ^i(x):R^i f_*({\mathcal F})_x \otimes k(x) \longrightarrow H^i(X_x, {\mathcal F}_x)\) is surjective, because its target is zero by assumption. By “cohomology and base change” (see [51, Theorem 12.11(a)] in the projective case and [46, §7] in general), the map \(\varphi ^i(x)\) is an isomorphism. Hence \(R^i f_*({\mathcal F})_x \otimes k(x)=0\). Since \({{\,\mathrm{Spec}\,}}R\) is affine, \(R^i f_*({\mathcal F})=H^i(X,{\mathcal F})^\sim \), and \(H^i(X,{\mathcal F})\) is a finite type Rmodule since f is proper. So the first statement follows from Nakayama’s lemma.
The second part of the lemma follows from the first one applied to the base change \(X_n \rightarrow {{\,\mathrm{Spec}\,}}R/{\mathfrak m}^n\). \(\square \)
The next condition on the vanishing of higher direct images is key to studying cohomological vanishing on \({\mathscr {S}}_{{\mathcal K}}^{\Sigma ,n} \) and its subschemes. Recall that \(S_{{\mathcal K}}^{\Sigma }:={\mathscr {S}}_{{\mathcal K}}^{\Sigma ,1}\). Denote by \(\pi :S_{{\mathcal K}}^{\Sigma } \rightarrow S_{{\mathcal K}}^{\min } \) the natural map.
Condition 7.1.2
Let \(\eta \in X^*_{+,\mathbf {L}}(\mathbf {T})\). The pair \(({\mathscr {S}}_{{\mathcal K}}^{\Sigma },\eta )\) satisfies \(R^i\pi _* {\mathscr {V}}^{\mathrm{sub}}(\eta )=0\) for all \(i>0\).
Remark 7.1.3
In the PEL case, Condition is a theorem of Lan [72, Theorem 8.2.1.2]. See also [67] for a simpler argument under a mild additional hypothesis on p. In the general Hodge case, Condition is a theorem of Stroh^{7} in [99] when \(\eta \in {{\mathbf {Q}}} \eta _{\omega }\).
Recall the convention regarding notation established in Sect. N.2.3: \({\mathscr {S}}_{{\mathcal K}}^{\Sigma , +}:={\mathscr {S}}_{{\mathcal K}}^{\Sigma }\) and \({\mathscr {S}}_{{\mathcal K}}^{\Sigma , 0}\) is its generic fiber.
Lemma 7.1.4
Proof
Corollary 7.1.5
Proof
11.2 Hasseregular sequences
The next definition singles out those \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant subschemes of a Shimura variety (modulo a prime power) which are obtained as successive zero schemes of length Hasse invariants (Sects. , ). Let \(({\mathscr {S}}_{{\mathcal K}}^{\Sigma ,n})_{{\mathcal K}^p, \Sigma }\) be the reduction modulo \({\mathfrak p}^n\) of the tower introduced in Sect. .
Definition 7.2.1
 (a)
One has \(Z_0={\mathscr {S}}_{{\mathcal K}}^{\Sigma , n}\) and \((a_j)_{j=0}^r\) is a sequence of positive integers independent of \({\mathcal K}^p,\Sigma \).
 (b)
For all \(j>0\), \(Z_{j} \subset Z_{j1}\) is a closed subscheme with \((Z_{j})_{\mathrm{red}}=S_{dj}^{\Sigma ,*}\), the \((dj)\)th length stratum (Sect. ).
 (c)
For all j, \(f_j\in H^0(Z_j,\omega ^{a_j})\) is a section constructed by applying Theorem to the length Hasse invariant \(h_{dj}\) (Corollary ).
 (d)
For all \(j<r\), the zero scheme of \(f_j\) is \(Z_{j+1}\).
Remark 7.2.2
 (a)
Given \(0 \le s \le r\), the truncation \((Z_j, a_j, f_j)_{j=0}^s\) is a Hasse regular sequence of length s.
 (b)
Since the length Hasse invariants \(h_{dj}\) are \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant, it follows inductively from the uniqueness in Theorem (a) that the systems of \(Z_j\) and \(f_j\) in a Hasseregular sequence are \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant and compatible with refinement of rpcd \(\Sigma \).
 (c)
By Corollary , one has \(\dim Z_j=dj\) for all j.
 (d)
In Sect. , it is shown that the coherent cohomology of \(Z_j\) admits an action of the unramified Hecke algebra \({\mathcal H}\) (away from p), compatible with the inclusions \(Z_j \hookrightarrow Z_{j1}\).
 (d)
It is possible (and indeed happens in our arguments) that the last section \(f_r\) is nowhere vanishing.
Definition 7.2.3
A system of subschemes \(Z \subset {\mathscr {S}}_{{\mathcal K}}^{\Sigma ,n}\) is Hasse regular if \(Z=Z_r\) for some Hasse regular sequence \((Z_j, a_j, f_j)_{j=0}^r\).
Lemma 7.2.4
Let \((Z_j, a_j, f_j)_{j=0}^r\) be a Hasseregular sequence in \({\mathscr {S}}_{{\mathcal K}}^{\Sigma ,n} \). Then \(Z_j\) is Cohen–Macaulay for all j.
Proof
First note that the scheme \({\mathscr {S}}_{{\mathcal K}}^{\Sigma ,n} \) is Cohen–Macaulay by [79, Theorem 23.9], since the ring \(\mathcal O_E/{\mathfrak p}^n\) is Cohen–Macaulay and \({\mathscr {S}}_{{\mathcal K}}^{\Sigma ,n} \) is smooth over \(\mathcal O_E/{\mathfrak p}^n\). Then the result follows from [79, Theorem 17.3], which implies that an effective Cartier divisor in a Cohen–Macaulay scheme is again Cohen–Macaulay. \(\square \)
Corollary 7.2.5
Let \((Z_j, a_j, f_j)_{j=0}^r\) be a Hasseregular sequence in \({\mathscr {S}}_{{\mathcal K}}^{\Sigma ,n} \). Then \(f_j\) is injective for all j.
Proof
By Macaulay’s “Unmixedness Theorem”, a Cohen–Macaulay scheme has no embedded components, cf. [27, Corollary 18.14]. In view of Lemma , \(Z_j\) has no embedded components. Then the result follows from Theorem (b) and Corollary . \(\square \)
11.3 Cohomological vanishing on Hasseregular subschemes of \({\mathscr {S}}_{{\mathcal K}}^{\Sigma ,n} \)
Lemma 7.3.1
Proof
12 Part 3. Hecke algebras and Galois representations
Part 3 contains our applications of grouptheoretical Hasse invariants to the action of the Hecke algebra on coherent cohomology and the existence of ‘automorphic’ Galois representations. Our most fundamental application to coherent cohomology Hecke algebras is the factorization theorem (Theorem , compare Theorem I.3.1). Section 8 is concerned with the statement and the majority of the proof of this result. The flag space of a Hodgetype Shimura variety is introduced in Sect. . It is used to complete the proof of Theorem in Sect. . Our results on Galois representations are contained in Sect. 10. The application to ‘Serre’s letter to Tate’ constitutes Sect. 11.
13 Construction of Hecke factorizations
13.1 Hecke algebras
13.1.1 The abstract Hecke algebra
13.1.2 Systems of Hecke eigenvalues
Suppose \(\kappa \) is a field and M is a finite dimensional \(\kappa \)vector space which is also an \({\mathcal H}\)module. Recall that a system of Hecke eigenvalues \((b_T)_{T\in {\mathcal H}}\) appears in M if there exists a finite extension \(\kappa '/\kappa \) and \(m \in M\otimes \kappa '\) such that \(Tm=b_Tm\) in \(M \otimes \kappa '\) for all \(T \in {\mathcal H}\).
13.1.3 Coherent cohomology Hecke algebras
Let \(({\mathbf {G}}, {\mathbf {X}})\) be a Shimura datum of Hodge type. Recall the associated structure theory (Sect. ) and the symplectic embedding of toroidal compactifications \(\varphi ^{\Sigma /{\tilde{\Sigma }}}:{\mathscr {S}}_{{\mathcal K}}^{\Sigma } \rightarrow {\mathscr {S}}_{g, {\tilde{{\mathcal K}}}}^{{\tilde{\Sigma }}}\) (6.1.1). We choose \({\tilde{\Sigma }}\) and \(\Sigma \) smooth, so that \({\mathscr {S}}_{{\mathcal K}}^{\Sigma }\) is smooth. Let \(\eta \in X^*_{+,\mathbf {L}}(\mathbf {T})\). By Sects. , , one has the coherent cohomology modules \(H^i({\mathscr {S}}_{{\mathcal K}}^n, {\mathscr {V}}({\eta }))\), \(H^i({\mathscr {S}}_{{\mathcal K}}^{\Sigma ,n}, {\mathscr {V}}^{\mathrm{can}}(\eta ))\) and \(H^i({\mathscr {S}}_{{\mathcal K}}^{\Sigma ,n}, {\mathscr {V}}^{\mathrm{sub}}(\eta ))\) for every \(n \in \mathbf {Z}_{\ge 0}\cup \{+\}\) and every \(i \ge 0\) (notation as in Sect. N.2.3). We explain how the Hecke algebra \({\mathcal H}\) acts on each of these modules. This is standard for \(H^i({\mathscr {S}}_{{\mathcal K}}^n, {\mathscr {V}}({\eta }))\), but for \(H^i({\mathscr {S}}_{{\mathcal K}}^{\Sigma ,n}, {\mathscr {V}}^{\mathrm{can}}(\eta ))\) and \(H^i({\mathscr {S}}_{{\mathcal K}}^{\Sigma ,n}, {\mathscr {V}}^{\mathrm{sub}}(\eta ))\) the definition of the trace maps is less obvious. For \({\mathscr {S}}_{{\mathcal K}}^n\), we also describe an action of \({\mathcal H}\) on \(H^i({\mathscr {S}}_{{\mathcal K}}^n, {\mathscr {F}})\) where \({\mathscr {F}}\) is an arbitrary (not necessarily locally free) \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant coherent sheaf. This will be used in Sect. 11 with \({\mathscr {F}}\) the ideal sheaf of \(S_e\) in \({\mathscr {S}}^1_{{\mathcal K}}=S_{{\mathcal K}}\).
Let \(g \in {\mathbf {G}}(\mathbf {Q}_v)\). For each of the spaces above, we define below an associated Heckeoperator \(T_g\). Since \({\mathcal H}_v\) is generated by the characteristic functions of double cosets represented by \(g \in {\mathbf {G}}(\mathbf {Q}_v)\), one deduces an action of all of \({\mathcal H}\).
13.1.4 Hecke action I: \({\mathscr {S}}_{{\mathcal K}}^n\)
13.1.5 Hecke action II: \({\mathscr {S}}_{{\mathcal K}}^{\Sigma ,n}\)
The \(\pi _j^0\) are finite and we claim they are also flat (we learned this fact from Tilouine [101, §8.1.2, p. 1409] and we thank B. Stroh and D. Rydh for discussions about why it is true). It suffices to check the flatness over \({\mathcal O}_{E,{\mathfrak p}}\); the corresponding statement over \({\mathcal O}_{E,{\mathfrak p}}/{\mathfrak p}^n\) then follows by base change.
Observe that all of the toroidal compactifications considered here are Cohen–Macaulay over \({\mathcal O}_{E,{\mathfrak p}}\); specifically we need to justify why the \({\mathscr {S}}_{{\mathcal K}_g}^{\Sigma _g^j}\), which are associated to potentially nonsmooth rpcd, are Cohen–Macaulay. Recall that, given any finite rpcd \(\Sigma '\) and any base scheme S, one has an associated torus embedding \(X_{\Sigma '}(S)\) over S, functorial in \(\Sigma '\), as described in [21, Chapter IV, Theorem 2.5]. By construction, the toroidal compactifications \({\mathscr {S}}_{{\mathcal K}_g}^{\Sigma _g^j}\) are étale locally torus embeddings of this type over \({\mathcal O}_{E,{\mathfrak p}}\). Thus we are reduced to checking that a torus embedding \(X_{\Sigma '}(R)\) over a DVR R is Cohen–Macaulay (the argument which follows works equally well over regular local R). By [56, Chapter 1, §3, Theorem 14] a torus embedding over a field is Cohen–Macaulay. Since a torus embedding over R is flat over R, the result follows from [79, Theorem 23.9]. Thus \({\mathscr {S}}_{{\mathcal K}_g}^{\Sigma _g^j}\) is Cohen–Macaulay.
Since the target \({\mathscr {S}}_{{\mathcal K}}^{\Sigma }\) of \(\pi _j^0\) is regular (recall that \(\Sigma \) was assumed smooth), we conclude that \(\pi ^0_j\) is flat by [79, Theorem 23.1], which implies that a quasifinite morphism between equidimensional, locally Noetherian schemes of the same dimension is flat if the domain is Cohen–Macaulay and the target is regular.
Finally, note that the definition of \(T_g^{\Sigma }\) reduces to that of \(T_g\) away from the boundary. The inclusion \({\mathscr {S}}_{{\mathcal K}}^n \rightarrow {\mathscr {S}}_{{\mathcal K}}^{\Sigma , n}\) is \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant; it induces an \({\mathcal H}\)equivariant restriction map \(H^0({\mathscr {S}}_{{\mathcal K}}^{\Sigma , n}, {\mathscr {V}}^{\mathrm{can}}(\eta )) \rightarrow H^0({\mathscr {S}}_{{\mathcal K}}^{n}, {\mathscr {V}}({\eta }))\). The latter is an isomorphism when \(({\mathbf {G}}, {\mathbf {X}})\) satisfies the Koecher principle.
13.1.6 Hecke action III: Hasseregular subschemes
For \(j \in \{1,2\}\), let \(r^j\) and \(\pi ^0_j\) be as in Sect. . Let \((Z^0_{t}, a_t, f_{t,0})_{t=0}^r\) (resp. \((Z_{t}, a_t, f_{t})_{t=0}^r\)) denote the component of a Hasse regular sequence for \(({\mathcal K}, \Sigma )\) (resp. \(({\mathcal K}^g, \Sigma _g)\)). Even though the \(\Sigma _g^j\) may not be smooth, we may still define a component \((Z^j_{t}, a_t, f_{t,j})_{t=0}^r\) of the Hasse regular sequence for \(({\mathcal K}_g, \Sigma _g^j)\) by pullback: Put \(Z^j_{t}:=(\pi _j^0)^{1} (Z^0_{t})\) and \(f_{t,j}:=(\pi ^{0}_j)^* f_{t,0}\). Then \((r^j)^{1}(Z^j_t)=Z_t\) and \((r^{j})^* f_{t,j}=f_t\) are the components for \(\Sigma _g\). Let \(y^{t}:Z_{t} \rightarrow {\mathscr {S}}_{{\mathcal K}_g}^{\Sigma _g}\), \(y^{t,0}:Z_t^0 \rightarrow {\mathscr {S}}_{{\mathcal K}}^{\Sigma }\) and \(y^{t,j}:Z_{t}^{j} \rightarrow {\mathscr {S}}_{{\mathcal K}_g}^{\Sigma ^j_g}\) denote the inclusions.
Lemma 8.1.1
 (a)
For all \(i>0\), \(R^i r^j_* (y^t_* {\mathcal O}_{Z_t})=0\) and \( r^j_* (y^t_* {\mathcal O}_{Z_t})=y^{t,j}_* {\mathcal O}_{Z_t^j}\).
 (b)
The pullback of \(\pi _j^0\) along the inclusion \(y^{t,0}:Z_t^0 \rightarrow {\mathscr {S}}_{{\mathcal K}}^{\Sigma }\) is a finite flat map \(Z_t^j \rightarrow Z_t^0\).
Proof
13.1.7 Notation for coherent cohomology Hecke algebras
Let \({\mathcal H}^{i,n}(\eta )\) denote the image of \({\mathcal H}\) in \(\mathrm{End}(H^i({\mathscr {S}}_{{\mathcal K}}^{\Sigma ,n} , {\mathscr {V}}^{\mathrm{sub}}(\eta )))\); we omit \({\mathcal K}\) and \(\Sigma \) from the notation. If Z is a Hasse regular subscheme (Definition ) of \({\mathscr {S}}_{{\mathcal K}}^{\Sigma ,n} \), then write \({\mathcal H}^{i,n}_Z(\eta )\) for the image of \({\mathcal H}\) in \(\mathrm{End}(H^i(Z, {\mathscr {V}}^{\mathrm{sub}}(\eta )))\). An arrow \({\mathcal H}\rightarrow {\mathcal H}^{i,n}(\eta )\) or \({\mathcal H}\rightarrow {\mathcal H}_Z^{i,n}(\eta )\) will always signify the defining projection.
13.2 Statement of the factorization theorem
From now on, in addition to assuming \(\Sigma \) to be smooth, we assume \(\Sigma \) is a refinement of \(\varphi ^*{\tilde{\Sigma }}\) for some \({\tilde{\Sigma }}\) which admits \(\beta \) log integral (Sect. ). Recall that this can always be achieved by refining \(\Sigma \). Then we may apply Theorem and the results of Sects. , 7 which depend on it.
Theorem 8.2.1
 (a)
There exists an arithmetic progression A such that \(\eta +a\eta _{\omega } \in F(i, n, \eta )\) for all \(a \in A \cap \mathbf {Z}_{\ge 1}\).
 (b)
Let \({\mathcal C}\subset X^*(\mathbf {T})\) be the “global sections cone” defined in (3.4.3). For all \(\nu \in {\mathcal C}\) and \(\eta _1 \in F(i, n, \eta )\) there exists \(m=m(\nu ,n) \in \mathbf {Z}_{\ge 1}\) such that \(\eta _1+jm\nu \in F(i,n,\eta )\) for all \(j \in \mathbf {Z}_{\ge 1}\).
 (c)
For all \(\delta \in \mathbf {R}_{\ge 0}\), \(F(i,n,\eta )\) contains a \(\delta \)regular character (Definition N.5.5).
Remark 8.2.2
(Concerning our assumptions) We expect it will soon be shown Conditions and hold for all Shimura varieties of Hodge type and all \(\eta \in X^*_{+,L}(\mathbf {T})\). These conditions are vacuous in the compact case and are known to hold in the PEL case and in general when \(\eta \) is a multiple of \(\eta _{\omega }\); see Lemma and Remark .
Remark 8.2.3
Theorem (c) illustrates merely one of many regularity conditions that follow from (b). Following the proof that (b) imples (c) in Sect. , it should be clear to the reader that “\(\delta \)regular” may be replaced by any regularity condition whose singular locus is a finite union of hyperplanes. In particular, when \(\mathbf {G}=GSp(2g)\), this applies to the condition “spinregular” studied in [63].
Remark 8.2.4
For PEL Shimura varieties of type A and C (resp. Scholze’s integral models of general Hodgetype Shimura varieties) Part (a) of Theorem was also obtained simultaneously and independently in [11] (resp. [86]). The assumptions , are not needed in [86]. By contrast, as far as we can tell, parts (b)–(c) do not follow by the methods of [11, 86].
Remark 8.2.5
(Reduction to \(H^0\), modulo p) In the case \(n=1\) (action of the Hecke algebra on the cohomology of the special fiber), a standard argument gives a more elementary statement in terms of systems of Hecke eigenvalues. The set \(F(i,1,\eta )\) consists of all \(\eta '\) such that every system of Hecke eigenvalues that appears in \(H^i(S_{{\mathcal K}}^{\Sigma } , {\mathscr {V}}^{\mathrm{sub}}(\eta ))\) also appears in \(H^0(S_{{\mathcal K}}^{\Sigma } , {\mathscr {V}}^{\mathrm{sub}}(\eta '))\).
13.2.1 Strategy of the proof
The proof of Theorem (a) naturally breaks down into three steps: (i) “Descent” (Lemma ), (ii) “Weight Increase” (Lemma ) and (iii) “Ascent” (Lemma ). “Descent” shows that \({\mathcal H}\rightarrow {\mathcal H}^{i,n}(\eta )\) factors through \({\mathcal H}\rightarrow {\mathcal H}_Z^{0,n}(\eta +b\eta _{\omega })\) for some \(b \in \mathbf {Z}_{\ge 1}\) and some Heckeequivariant, nilpotent thickening Z of a length stratum. “Weight increase” then shows that the latter factors through \({\mathcal H}\rightarrow {\mathcal H}_Z^{0,n}(\eta +s\eta _{\omega })\) for all sufficiently large s in an arithmetic progression containing b. By choosing s large enough, the vanishing lemma applies. Finally, “Ascent” yields the factorization claimed in (a).
The flag space is introduced in Sect. to prove (b)–(c). The latter are proved in Sect. by applying results of Sect. about \(\mathop {G\hbox {}{} \mathtt{ZipFlag}}\nolimits ^{\mu }\) and \({\mathcal C}\).
13.3 Descent, weight increase and going up
Throughout Sect. , adopt the assumptions of Theorem . If \(H^i({\mathscr {S}}^{\Sigma ,n}_{{\mathcal K}}, {\mathscr {V}}^{\mathrm{sub}}(\eta ))=0\), then Theorem is vacuous for \((i,n,\eta )\). So assume for the rest of Sect. 8 that \(H^i({\mathscr {S}}^{\Sigma ,n}_{{\mathcal K}}, {\mathscr {V}}^{\mathrm{sub}}(\eta ))\ne 0\). The following linear algebra lemma will be applied repeatedly:
Lemma 8.3.1
Suppose R is a \({\mathbf {Z}}_p\)algebra and \(\epsilon :M'\rightarrow M\) is a morphism of \(R\mathcal H\)modules. Write \(\mathcal H_M\) (resp. \(\mathcal H_{M'}\)) for the image of \(\mathcal H\) in \(\mathrm{End}_R(M)\) (resp. \(\mathrm{End}_R(M')\)). If \(\epsilon \) is injective (resp. surjective), then the map \({\mathcal H}\twoheadrightarrow {\mathcal H}_{M'}\) (resp. \({\mathcal H}\twoheadrightarrow {\mathcal H}_{M}\)) factors through \({\mathcal H}_{M}\) (resp. \({\mathcal H}_{M'}\)).
Proof
Both factorizations are equivalent to the corresponding inclusions of kernels, which are trivial. \(\square \)
Lemma 8.3.2
Proof
For the rest of Sect. , fix the notation and the Hasseregular sequence of Lemma .
Lemma 8.3.3
Proof
Lemma 8.3.4
Proof
Proof of Theorem 8.2.1(a)
Apply Lemmas , and successively. \(\square \)
14 Increased regularity via the flag space
14.1 The flag space
Remark 9.1.1
Note that the kfibers of \(\pi \) are flag varieties isomorphic to \(L/B_L\). If the character \(\eta \) is not in the cone \(X^*_{+,\mathbf {L}}(\mathbf {T})\), then \(H^0(L/B_L,{\mathscr {L}}({\eta }))=0\), so we deduce \(\pi _*{\mathscr {L}}({\eta })={\mathscr {V}}({\eta })=0\). This implies that if \({\mathscr {L}}({\eta })\) admits a nonzero global section on \( Fl_{{\mathcal K}}^{\Sigma } \), then \(\eta \in X^*_{+,\mathbf {L}}(\mathbf {T})\). In particular, one has \({\mathcal C}\subset X^*_{+,\mathbf {L}}(\mathbf {T})\) (Sect. ).
Recall (Sect. ) that D denotes the boundary divisor of the toroidal compactification \({\mathscr {S}}_{{\mathcal K}}^{\Sigma }\). Put \({\mathscr {L}}^{\mathrm{sub}}({\eta }):={\mathscr {L}}({\eta })\otimes \pi ^*\mathcal O(D)\). Since \(\mathcal O(D)\) is a line bundle, the projection formula and (9.1.2) gives \(\pi _*{\mathscr {L}}^{\mathrm{sub}}({\eta })={\mathscr {V}}^{\mathrm{sub}}(\eta )\).
Remark 9.1.2
(Motivation for the flag space) The \({\mathcal O}_{{\mathfrak p}}\)scheme \( {\mathcal F}l_{{\mathcal K}}^{\Sigma } \) is a simultaneous generalization of aspects of work of Griffiths–Schmid over \(\mathbf {C}\) [44] (see also the works Carayol cited in the introduction) and work of Ekedahl–van der Geer [28] in characteristic \(p>0\). We were inspired by both of these works.
Griffiths–Schmid studied homogenous complex manifolds of the form \(\Gamma \backslash G/T\), where G is a connected, semisimple, real Lie group, T is a compact Cartan subgroup, G / T is endowed with a complex structure and \(\Gamma \) is an arithmetic subgroup. Carayol termed these Griffiths–Schmid manifolds [16].
Griffiths and his school refer to those complex structures on G / T which fiber holomorphically over a Hermitian symmetric domain as the classical case. In the classical case, the Griffiths–Schmid manifolds \(\Gamma \backslash G /T\) are algebraic. By contrast, it has recently been shown that all Griffiths–Schmid manifolds in the nonclassical case are not algebraic [43].
The complex manifolds \( {\mathcal F}l_{{\mathcal K}}^{\Sigma } (\mathbf {C})\) are adelic versions of algebraic Griffiths–Schmid manifolds. They provide a moduli interpretation of every algebraic Griffiths–Schmid manifold which lies over a Shimura variety of Hodgetype.
For Siegel modular varieties, Ekedahl–van der Geer defined a flag space \(Fl_g\) over \({{\mathbf {F}}}_p\) by more elementary means than ours, namely in terms of full symplectic flags refining the Hodge filtration in \(H^1_{\mathrm{dR}}\) of an abelian scheme. They went on to define a stratification of \(Fl_g\) and study its relation to the EO stratification of \({\mathscr {S}}_{g,{\tilde{{\mathcal K}}}}\otimes {{\mathbf {F}}}_p\). The stratification obtained above via (9.1.1) agrees with that of [28] for Siegel varieties.
Remark 9.1.3
14.2 Increased regularity
The flag space affords additional Hecke factorizations as follows:
Proof of Theorem 8.2.1(b)
Proof of Theorem 8.2.1(c)
By Sect. , the cone \({\mathcal C}\) spans \(X^*(\mathbf {T})_{\mathbf {Q}}\). On the other hand, the singular locus in \(X^*(\mathbf {T})_{\mathbf {Q}}\) is a finite union of hyperplanes viz. the root hyperplanes \(\alpha ^{\perp }\) for \(\alpha \in \Phi (\mathbf {G}, \mathbf {T})\) (Sect. N.3.4). Therefore \({\mathcal C}\) contains a regular element \(\nu \in X^*_{+,\mathbf {L}}(\mathbf {T})\).
By Theorem (a), \(F(i,n,\eta )\) is nonempty. So let \(\eta _1 \in F(i,n,\eta )\). By Theorem (b), there exists \(m=m(\nu , n)\) such that \(\eta _1+jm\nu \in F(i,n,\eta )\) for all \(j \in \mathbf {Z}_{\ge 1}\); since \(\nu \) is regular, \(\eta _1+jm\nu \) is \(\delta \)regular for all sufficiently large j. \(\square \)
15 Galois representations
This section is devoted to Theorem I.3.2 on the association of Galois representations (resp. pseudorepresentations) to automorphic representations whose Archimedean component is a nondegenerate limit of discrete series (LDS) (resp. coherent cohomology modulo \(p^n\)). Sections and recall facts and introduce notation regarding Archimedean and nonarchimedean representations respectively. To simplify the ensuing statements, we formalize the existence of a Galois representation associated to an automorphic representation as a condition in Sect. . Our results are then stated as four theorems in Sects. and . These theorems are proved in Sect. . We hope that by carefully stating our hypotheses in a general context, it will be easy to apply them as soon as new results are established for cohomological representations (e.g., see Remark ).
Recall our notation and conventions regarding structure theory of reductive groups, (based) root data and related objects (Sect. N.3) and Shimura varieties of Hodge type Sect. . Fix a Shimura datum of Hodge type \(({\mathbf {G}}, {\mathbf {X}})\) together with an integral symplectic embedding \(\varphi \) (4.1.1). Recall from Sect. that associated to \((\mathbf {G}, \mathbf {X}, \varphi )\) we have: The conjugacy class of cocharacters \([\mu ]_{\overline{E}_{\mathfrak p}}\) over \({\overline{E}_{\mathfrak p}}\), a representative \(\mu \) defined over \(E_{{\mathfrak p}}\), a Borel pair \((\mathbf {B}, \mathbf {T})\) of \(\mathbf {G}_{\mathbf {Q}_p}\), opposite parabolic subgroups \(\mathbf {P}, \mathbf {P}^+\) such that \(\mathbf {B}_{E_p}\subset \mathbf {P}\) and the Levi \(\mathbf {L}=\mathrm{Cent}(\mu )=\mathbf {P}\cap \mathbf {P}^+\) over \(E_{{\mathfrak p}}\) containing \(\mathbf {T}_{E_{{\mathfrak p}}}\).
Throughout Sect. 10, fix an isomorphism (of abstract fields) \(\iota :\overline{\mathbf {Q}}_{p}{\mathop {\longrightarrow }\limits ^{\sim }} {{\mathbf {C}}}\). We choose \(\iota \) compatibly with \(\mathbf {P}, \mathbf {P}^\) and \(\Delta \) as explained in Sect. , so that \(\iota \mathbf {P}_{\overline{\mathbf {Q}}_{p}}\) is the stabilizer of the Hodge filtration for the \(\mathbf {R}\)Hodge structure \({{\,\mathrm{Ad}\,}}\circ h\), where \(h \in \mathbf {X}\) gives rise to \(\iota \mu _{\overline{\mathbf {Q}}_{p}}\) via Sect. N.6.2.
15.1 Archimedean representation theory
We briefly review notation and results about real groups and Lie algebra cohomology needed later in Sect. 10.
15.1.1 Infinitesimal character
Following Buzzard–Gee [14], we say that \(\chi \in X^*(\mathbf {T})_{\mathbf {C}}\) is Lalgebraic (resp. Calgebraic) if \(\chi \in X^*({\mathbf {T}})\) (resp. \(\chi \in X^*({\mathbf {T}})+\rho \)).
Let K be a maximal compact subgroup of \(\mathbf {G}(\mathbf {R})\). Let \(\pi _{\infty }\) be an irreducible HarishChandra module i.e., an irreducible \(({\mathfrak g}, K_{\mathbf {C}})\)module. Write \(\chi _{\infty }\) for the infinitesimal character of \(\pi _{\infty }\), identified with an element of \(X^*(\mathbf {T})_{\mathbf {C}}/W\) via (10.1.1).
Given a property \({\mathcal P}\) of elements of \(X^*(\mathbf {T})_{\mathbf {C}}\) which is stable under the action of W, we say that \(\pi _{\infty }\) has property \(\mathcal P\) if \(\chi _{\infty }\) does. We say that an automorphic representation has property \({\mathcal P}\) if its Archimedean component does.
15.1.2 Limits of discrete series
The parametrization of limits of discrete series (LDS) for \(\mathbf {G}(\mathbf {R})\) is complicated by the fact that \(\mathbf {G}(\mathbf {R})\) is often neither semisimple nor connected in the classical topology. For a semisimple real Lie group which is connected in the classical topology, LDS were parametrized in [59, §1]. In the generality considered here, a full parametrization of LDS was given by the first author in both [35] and [34], based on explanations by Schmid and Vogan, see also [1]. Here we restrict attention to Calgebraic LDS of \(\mathbf {G}(\mathbf {R})\), since they are the only LDS considered in this paper.
A Calgebraic LDS HarishChandra parameter is a pair \((\lambda , {\mathcal C})\), where \({\mathcal C}\) is a Weyl chamber for the root datum \({\mathcal R}{\mathcal D}(\mathbf {G},\mathbf {T})\) (see Sects. N.3.1, N.3.4), \(\lambda \in \partial {\mathcal C}\cap (X^*(\mathbf {T}) +\rho )\) and \(\langle \lambda , \alpha ^{\vee } \rangle \ne 0\) for all \({\mathcal C}\)simple \(\alpha \in \Phi _{\mathbf {L}}\).^{8} For every Calgebraic, LDS HarishChandra parameter \((\lambda ,{\mathcal C})\), there exists a Calgebraic LDS HarishChandra module \(\pi (\lambda , {\mathcal C})\). The LDS \(\pi (\lambda , {\mathcal C})\) may be constructed as the image of a discrete series \(\pi _{\rho }\) by the Zuckerman translation functor \({{\,\mathrm{Transl}\,}}_{W\lambda }\) which goes from HarishChandra modules of infinitesimal character \(W\rho \) to ones of infinitesimal character \(W\lambda \) (cf. the discussion preceding Theorem 2.1 in [97]).
The fundamental dichotomy of nondegenerate versus degenerate was introduced for LDS by Knapp–Zuckerman in their classification of tempered representations [59]. An LDS \(\pi (\lambda , {\mathcal C})\) is degenerate if \(\lambda \) is orthogonal to \(\alpha ^{\vee }\) for some \(\alpha \in \Phi _L\) (which is then necessarily not \({\mathcal C}\)simple); otherwise, it is called nondegenerate.
Following [32, §2.3], we say that an LDS is \(\mathbf {X}\)holomorphic if it is isomorphic to some \(\pi (\lambda , {\mathcal C})\) with \({\mathcal C}\) the \(\Delta \)dominant chamber [here \(\mathbf {X}\) pertains to the Shimura datum \(({\mathbf {G}}, {\mathbf {X}})\)]. As noted in loc. cit., every \(\mathbf {X}\)holomorphic LDS is nondegenerate.
Remark 10.1.1
(Degeneracy via Langlands Parameters) The first author has shown that an equivalent formulation of degeneracy is that the image of the restriction to \({{\mathbf {C}}}^{\times }\) of the Langlands parameter of \(\pi (\lambda , {\mathcal C})\) contains a simple group of rank at least two [34]. In turn, this characterization allows to generalize the nondegenerate/degenerate dichotomy to arbitrary HarishChandra modules [31, 34].
15.1.3 Lie algebra cohomology
Let \({\mathfrak p}\) (resp. \({\mathfrak l}\)) be the complexified Lie algebra of \(\mathbf {P}\) (resp. \(\mathbf {L}\)). For \(\eta \in X^*_{+,\mathbf {L}}(\mathbf {T})\), let \(V_{\eta }\) be the irreducible finitedimensional representation of \(\mathbf {L}_{\mathbf {C}}\) of highest weight \(\eta \). Given a HarishChandra module \(\pi _{\infty }\) for \(\mathbf {G}(\mathbf {R})\), write \(H^i({\mathfrak p}, {\mathfrak l}, \pi _{\infty } \otimes V_{\eta })\) for the relative Lie algebra cohomology of the pair (\({\mathfrak p}\), \({\mathfrak l}\)) with coefficients in \(\pi _{\infty } \otimes V_{\eta }\) (cf. [9, 47, 105], though note that the latter reference’s notation is somewhat different).
Theorem 10.1.2
Remark 10.1.3
Schmid computed the \({\mathfrak n}\)cohomology of discrete series [92]. The computation was generalized to nondegenerate LDS by Williams [108]. Harris translated their results to the setting of \(({\mathfrak p}, {\mathfrak l})\)cohomology [48, Theorem 3.4]. In Theorem 3.4 of loc. cit., Harris makes the additional claim that the \(({\mathfrak p}, {\mathfrak l})\)cohomology of \(\pi (\lambda , {\mathcal C})\) is zero in all degrees other than \({{\,\mathrm{cd}\,}}({\mathcal C})\). This is false already for \(\mathbf {G}=GL(2)\). It is true when \(\mathbf {G}(\mathbf {R})\) is semisimple and connected in the classical topology. We shall only use the part of loc. cit. stated in Theorem , which is correct.
Corollary 10.1.4
Remark 10.1.5
Though it will not be used in this paper, note that Corollary is purely a result over \(\mathbf {C}\) and is valid for any Shimura datum, even if not of Hodge type and not known to admit integral models.
Proof of Corollary 10.1.4
15.2 Applications of the Satake isomorphism
Let v be a nonArchimedean place of \(\mathbf {Q}\). Assume \(v \not \in {{\,\mathrm{Ram}\,}}(\mathbf {G}) \cup \{p\}\) and let \(\mathrm{Frob}_v\) be a geometric Frobenius at v. Recall \(\iota :\overline{\mathbf {Q}}_{p}{\mathop {\rightarrow }\limits ^{\sim }} \mathbf {C}\) and write \({\mathcal H}_{v,\mathbf {C}}:={\mathcal H}_v \otimes _{{\mathbf {Z}}_p, \iota } \mathbf {C}\).
Let \({\phantom {\mathbf {G}}}^{L}{\mathbf {G}}^{\circ }\) be the dual group of \(\mathbf {G}\) over \(\mathbf {C}\), i.e., the connected, reductive \(\mathbf {C}\)group whose root datum is dual to \({\mathcal R}{\mathcal D}(\mathbf {G}, \mathbf {T})\) (Sect. N.3.1). Let \({\phantom {\mathbf {G}}}^{L}{\mathbf {G}}={\phantom {\mathbf {G}}}^{L}{\mathbf {G}}^{\circ }\rtimes \mathrm{Gal}(\overline{{{\mathbf {Q}}}}/ \mathbf {Q})\) (resp. \({\phantom {\mathbf {G}}}^{L}{\mathbf {G}}_v={\phantom {\mathbf {G}}}^{L}{\mathbf {G}}^{\circ }_v\rtimes \mathrm{Gal}(\overline{\mathbf {Q}}_{v}/ \mathbf {Q}_v)\)) denote the Galois form of the Lgroup of \(\mathbf {G}\) (resp. \(\mathbf {G}_v\)), cf. [8, §2]. Similar to [14, §2.1], we view \({\phantom {\mathbf {G}}}^{L}{\mathbf {G}}\) and \({\phantom {\mathbf {G}}}^{L}{\mathbf {G}}_v\) as group schemes over \(\mathbf {C}\) with component groups \(\mathrm{Gal}(\overline{{{\mathbf {Q}}}}/ \mathbf {Q})\) and \(\mathrm{Gal}(\overline{\mathbf {Q}}_{v}/ \mathbf {Q}_v)\), respectively.^{9} A representation of \({\phantom {\mathbf {G}}}^{L}{\mathbf {G}}\) will always mean a morphism of \(\mathbf {C}\)group schemes \(r:{\phantom {\mathbf {G}}}^{L}{\mathbf {G}}\rightarrow GL(m)\) for some \(m\ge 1\), which factors through \({\phantom {\mathbf {G}}}^{L}{\mathbf {G}}^{\circ }\rtimes \mathrm{Gal}(F/\mathbf {Q})\) for some finite, Galois extension F over which \(\mathbf {G}\) splits; representations of \({\phantom {\mathbf {G}}}^{L}{\mathbf {G}}_v\) are defined analogously (cf. [8, 2.6]). Given a representation \(r:{\phantom {\mathbf {G}}}^{L}{\mathbf {G}}\rightarrow GL(m)\) of \({\phantom {\mathbf {G}}}^{L}{\mathbf {G}}\), we denote by \(r_v:{\phantom {\mathbf {G}}}^{L}{\mathbf {G}}_v\rightarrow GL(m)\) the representation of \({\phantom {\mathbf {G}}}^{L}{\mathbf {G}}_v\) obtained by restriction.
First, there is a canonical bijection between the set of (complex) characters of \({\mathcal H}_{v,\mathbf {C}}\) and the set of \({\phantom {\mathbf {G}}}^{L}{\mathbf {G}}^{\circ }_v(\mathbf {C})\)conjugacy classes of semisimple elements in \({\phantom {\mathbf {G}}}^{L}{\mathbf {G}}^{\circ }_v\rtimes \mathrm{Frob}_v\). Since the former set is in bijection with the set of unramified, admissible, complex representations of \(\mathbf {G}(\mathbf {Q}_v)\), so is the latter. If \(\pi _v\) is an unramified, admissible, complex representation of \(\mathbf {G}(\mathbf {Q}_v)\), we denote by \({{\,\mathrm{Class}\,}}(\pi _v)\) the corresponding \({\phantom {\mathbf {G}}}^{L}{\mathbf {G}}^{\circ }_v(\mathbf {C})\)conjugacy class. Recall that the correspondence \(\pi _v \leftrightarrow {{\,\mathrm{Class}\,}}(\pi _v)\) is given as follows: The Hecke algebra \({\mathcal H}_{v,\mathbf {C}}\) acts on the line \(\pi _v^{{\mathcal K}_v}\) by a character \(\chi (\pi _v):{\mathcal H}_{v,\mathbf {C}} \rightarrow \mathbf {C}\). Then \({{\,\mathrm{Class}\,}}(\pi _v)\) is characterized as the unique \({\phantom {\mathbf {G}}}^{L}{\mathbf {G}}^{\circ }_v(\mathbf {C})\)conjugacy class such that the character of \(R_{{{\,\mathrm{fd}\,}}}^{\mathrm{ss}}({\phantom {\mathbf {G}}}^{L}{\mathbf {G}}_v)\) corresponding to \(\chi (\pi _v)\) via (10.2.1) is given by evaluation at \({{\,\mathrm{Class}\,}}(\pi _v)\).
In addition, for \(r:{\phantom {\mathbf {G}}}^{L}{\mathbf {G}}\rightarrow GL(m)\) as above, let \({{\,\mathrm{Class}\,}}(\pi _v, r_v)\) be the conjugacy class in \(GL(m, \mathbf {C})\) generated by the image \(r_v({{\,\mathrm{Class}\,}}(\pi _v))\). Let \({{\,\mathrm{Class}\,}}_{p,\iota }(\pi _v, r_v)\) denote the conjugacy class in \(GL(m, \overline{\mathbf {Q}}_{p})\) obtained from \({{\,\mathrm{Class}\,}}(\pi _v, r_v)\) via \(\iota \).
For all \(j \ge 1\), the function \({{\,\mathrm{tr}\,}}^j(r):{\phantom {\mathbf {G}}}^{L}{\mathbf {G}}_v(\mathbf {C}) \rightarrow \mathbf {C}\) defined by \({\tilde{g}}\mapsto {{\,\mathrm{tr}\,}}(r({\tilde{g}})^j)\) lies in \(R_{{{\,\mathrm{fd}\,}}}^{\mathrm{ss}}({\phantom {\mathbf {G}}}^{L}{\mathbf {G}}_v)\).
Definition 10.2.1
Denote by \(T^{(j)}_v(r)\in {\mathcal H}_v[\sqrt{v}]\) the Hecke operator associated to \({{\,\mathrm{tr}\,}}^j(r)\) by (10.2.1). Write \(T^{(j)}_v(r;i,n,\eta )\) for the image of \(T^{(j)}_v(r)\) in the coherent cohomology Hecke algebra \({\mathcal H}^{i,n}(\eta )\) (Sect. ).
Remark 10.2.2
By definition, if \({\mathcal H}\rightarrow {\mathcal H}^{i,n}(\eta )\) factors through \({\mathcal H}\rightarrow {\mathcal H}^{i',n'}(\eta ')\), then \(T_v^{(j)}(r;i',n',\eta ')\) maps to \(T_v^{(j)}(r;i,n,\eta )\) via the induced map \({\mathcal H}^{i',n'}(\eta ') \rightarrow {\mathcal H}^{i,n}(\eta )\).
15.3 Existence of automorphic Galois representations
Let \(\pi \) be an automorphic representation of \({\mathbf {G}}\). Fix a prime \(p \not \in {{\,\mathrm{Ram}\,}}(\pi )\) (Sect. N.3.6). Let \(r:{\phantom {\mathbf {G}}}^{L}{\mathbf {G}}\longrightarrow GL(m)\) be a representation as defined in Sect. .
It will be convenient to introduce a condition which says that the pair \((\pi , r)\) admits a padic Galois representation with weak localglobal compatibility. If X is a conjugacy class in \(GL(m, \overline{\mathbf {Q}}_{p})\), then write \(X^{\mathrm{ss}}\) for its semisimplification i.e., \(X^{\mathrm{ss}}\) is the unique semisimple conjugacy class with the same characteristic polynomial as X.
Condition 10.3.1
Remark 10.3.2
 (a)
Suppose \(\pi \) is Lalgebraic. Assume that, as predicted by Langlands and Buzzard–Gee (and mentioned in Sect. I.3.1) there exists \(R_{p,\iota }(\pi ):\mathrm{Gal}(\overline{{{\mathbf {Q}}}}/ \mathbf {Q})\rightarrow {\phantom {\mathbf {G}}}^{L}{\mathbf {G}}(\overline{\mathbf {Q}}_{p})\) such that for every \(v \notin {{\,\mathrm{Ram}\,}}(\pi )\cup \{p\}\), one has \(R_{p,\iota }(\pi )(\mathrm{Frob}_v)^{\mathrm{ss}}={{\,\mathrm{Class}\,}}_{p,\iota }(\pi _v)\) as \({\phantom {\mathbf {G}}}^{L}{\mathbf {G}}^{\circ }(\overline{\mathbf {Q}}_{p})\)conjugacy classes in \({\phantom {\mathbf {G}}}^{L}{\mathbf {G}}^{\circ }(\overline{\mathbf {Q}}_{p}) \rtimes \mathrm{Frob}_v\). Then \(R_{p,\iota }(\pi , r)=r \circ R_{p,\iota }(\pi )\).
 (b)
Assume (weak) Langlands functoriality holds for \((\pi ,r)\) as in [14, Conj. 6.1.1], i.e., that there exists an automorphic representation \(r_*\pi \) of GL(m) which is a transfer of \(\pi \) at infinity and at all unramified places. If \((\pi , r)\) satisfies Condition , then \(R_{p,\iota }(\pi ,r)=R_{p, \iota }(r_*\pi )\).
15.4 Galois pseudorepresentations associated to coherent cohomology modulo a prime power
Retain the notation of Sect. for the rest of Sect. 10.
15.4.1 The general Hodgetype case
Let \(({\mathbf {G}}, {\mathbf {X}})\) be a Hodgetype Shimura datum \(({\mathbf {G}}, {\mathbf {X}})\) and \(p \notin {{\,\mathrm{Ram}\,}}(\mathbf {G}) \cup \{2\}\). Fix an integral model \({\mathscr {S}}_{{\mathcal K}}\) over \({\mathcal O}_{E,{\mathfrak p}}\) as in Sect. and a toroidal compactification \({\mathscr {S}}_{{\mathcal K}}^{\Sigma }\) as in Sect. . We assume that \(\Sigma \) (hence also \({\mathscr {S}}_{{\mathcal K}}^{\Sigma }\)) is smooth, so that we have the action of the Hecke algebra \({\mathcal H}^{i,n}(\eta )\) on \(H^i({\mathscr {S}}_{{\mathcal K}}^{\Sigma }, {\mathscr {V}}^{\mathrm{sub}}(\eta ))\) as defined in Sect. . Recall the notion of a \(\delta \)regular character (Definition N.5.5).
Let \(\eta \in X^*_{+,\mathbf {L}}(\mathbf {T})\). If \(({\mathbf {G}}, {\mathbf {X}})\) is neither of PELtype nor of compact type, assume \(S_{{\mathcal K}}^{\Sigma }\) satisfies Condition and \((S_{{\mathcal K}}^{\Sigma }, \eta )\) satisfies Condition .
Theorem 10.4.1
Upon consulting the proof of Theorem , it should be clear to the reader that the proof can easily be adapted to treat several variants.
Remark 10.4.2
(Variant I: Twisting) Under additional assumptions, one can introduce twisting between Lalgebraic and Calgebraic in several ways, e.g., on the “domain” \(\mathbf {G}\) or on the “target” GL(m).
On the domain: Assume \(\mathbf {G}\) admits a twisting element \(\theta \) as in [14, Definition 5.2.1]; for simplicity assume \(\mathbf {G}\) is \(\mathbf {Q}\)split and that the restriction of r to \({\phantom {\mathbf {G}}}^{L}{\mathbf {G}}^{\circ }\) is irreducible of highest weight \(\chi \in X^*(\mathbf {T})\). Let \(T_v^{(j), C}(r):=v^{\langle \chi , \rho \theta \rangle }T_v^{(j)}(r)\) be the twists of our old Hecke operators (the Tate normalization in [45, §8]). Mimicking the proof of Theorem , one can prove the twisted variant where we assume instead that for all \((\pi , r)\) as in the theorem, \((\pi \otimes \cdot ^{\rho \theta }, r)\) satisfies Condition and in the conclusion we replace the Hecke operators \(T_v^{(j)(r)}\) with the twists \(T_v^{(j),C}(r)\).
On the target: Assume a weak transfer \(r_*\pi \) exists for all Calgebraic, discrete series and nondegenerate LDS [Remark (b)], and suppose that \(r_*\) maps Calgebraic to Calgebraic. Then one may again adapt the proof of Theorem to show that, if for all \((\pi , r)\) as in the theorem Condition holds for \((r_*\pi \otimes \cdot ^{(1m)/2}, \mathrm{Id})\), then the conclusion holds where we now replace \(T_v^{(j)}(r)\) with \(v^{(1m)/2}T_v^{(j)}(r)\).
In terms of twisting, the optimal generalization of Theorem may be one formulated in terms of a zextension of \(\mathbf {G}\) which admits a twisting element, following [14, §5] (and what Buzzard–Gee call the “Cgroup” of \(\mathbf {G}\)). However, it seems this may require extending some of our results on Shimura varieties to zextension of \(\mathbf {G}\) and we have not attempted to carry this out.
Remark 10.4.3
(Variant II: \(\mathrm{Gal}(\overline{F}/F)\)) Another notable variant is to replace \(\mathrm{Gal}(\overline{{{\mathbf {Q}}}}/ \mathbf {Q})\) by \(\mathrm{Gal}(\overline{F}/F)\) for some number field F which plays a special role for the Shimura variety \({{\,\mathrm{Sh}\,}}({\mathbf {G}}, {\mathbf {X}})\). An example which incorporates both variants for unitary similitude groups is given by Theorem below. In the Hilbert case, taking F to be the totally real field recovers the main result of [29].
Remark 10.4.4
In the special case where \(\mathbf {G}=GSp(4)\) and r is the ‘standard’ fourdimensional representation (which coincides with the spin representation since \(g=2\)), both twisted variants of Theorem apply unconditionally (a twisting element exists and Calgebraic is mapped to Calgebraic). The hypothesis holds with \(\delta =0\). This can be deduced in two ways: By using the work of Arthur [4] to transfer to GL(4) and then applying Shin’s result [95], or by the work of Weissauer [107] and Laumon [73, 74]. (The Conditions , hold because \(({\mathbf {G}}, {\mathbf {X}})\) is of PELtype.)
15.4.2 Unitary similitude groups
Let \(({\mathbf {G}}, {\mathbf {X}})\) be an arbitrary Shimura datum of PEL type A. Then \(\mathbf {G}\) is an inner form of the quasisplit unitary group associated to a quadratic extension of an imaginary CM field F over its totally real subfield \(F^+\).
Given \(v \not \in {{\,\mathrm{Ram}\,}}(\mathbf {G}) \cup \{p\}\), a prime w of F above v and \(j \ge 1\), let \(T_w^{(j), U}\) be the Hecke operator defined between Lemmas 6.1 and 6.2 of [50] and denoted \(T_v^{(i)}\) there. Let \(T_w^{(j), U}(i,n, \eta )\) be the image of \(T_w^{(j), U}\) in \({\mathcal H}^{i,n}(\eta )\).
Theorem 10.4.5
15.5 Galois representations associated to nondegenerate LDS
15.5.1 The Hodgetype case
Return to the setting of Sect. . If \(({\mathbf {G}}, {\mathbf {X}})\) is neither of PELtype nor of compact type, assume \(S_{{\mathcal K}}^{\Sigma }\) satisfies Condition for all sufficiently small \({\mathcal K}^p\).
Theorem 10.5.1
(LDS, Hodge case) Suppose there exists \(\delta \in \mathbf {R}_{\ge 0}\) and \(r: {\phantom {\mathbf {G}}}^{L}{\mathbf {G}}\rightarrow GL(m)\) such that, for every \(\delta \)regular, cuspidal, Calgebraic automorphic representation \(\pi '\) with \(\pi '_{\infty }\) discrete series, the pair \((\pi ', r)\) satisfies Condition (GalRepp). Let \(\pi \) be a cuspidal automorphic representation of \(\mathbf {G}\) with \(\pi _{\infty }=\pi (\lambda , {\mathcal C})\) a Calgebraic, nondegenerate LDS. If \(p \notin {{\,\mathrm{Ram}\,}}(\pi )\) and \((S_{{\mathcal K}}^{\Sigma }, w_{0, \mathbf {L}}\lambda \rho )\) satisfies Condition , then the pair \((\pi , r)\) satisfies (GalRepp).
As in Remarks –, one has variants with twisting between L and Calgebraic, replacing \(\mathrm{Gal}(\overline{{{\mathbf {Q}}}}/ \mathbf {Q})\) with \(\mathrm{Gal}(\overline{F}/F)\) for suitable F related to \(({\mathbf {G}}, {\mathbf {X}})\) and one deduces an unconditional result for \(\mathbf {G}=GSp(4)\), r the ‘standard’ fourdimensional representation.
Remark 10.5.2
It is interesting whether Theorem may be used to extend [63] to produce \(GSpin(2g+1)\)valued Galois representations for nondegenerate LDS.
15.5.2 Unitary similitude groups
Return to the setting of Sect. : \(({\mathbf {G}}, {\mathbf {X}})\) is of PELtype A associated to a CM extension \(F/F^+\).
We have found useful the description of local base change from unitary similitude groups to GL(m) at unramified primes in [50, §1.3].^{11} Let \(\pi \) be a cuspidal automorphic representation of \({\mathbf {G}}\). For every \(v \not \in {{\,\mathrm{Ram}\,}}(\pi )\) and w a place of F above v, define the base change of \(\pi _v\), denoted \({{\,\mathrm{BC}\,}}(\pi _v)\), and its wpart \({{\,\mathrm{BC}\,}}(\pi _v)_w\) as in [50, §1.3]. Write \({{\,\mathrm{rec}\,}}_{F_w}\) for the local Langlands correspondence, normalized as in [49]. Let \(W_{F_{w}}\) be the Weil group of \(F_{w}\). A superscript \(()^\mathrm{ss}\) will denote semisimplification.
Theorem 10.5.3
15.5.3 Remarks about Hodge–Tate weights
Let \(\pi \) be as in Theorem or . Assume \(\pi \) is Lalgebraic. The Hodge–Tate cocharacter of \(R_{p,\iota }(\pi )\) is conjectured to be the infinitesimal character \(\chi _{\infty }\) of \(\pi _{\infty }\), when viewed as a cocharacter of \({\phantom {\mathbf {G}}}^{L}{\mathbf {G}}^{\circ }\), [14, Remark 3.2.3]. When \(\pi \) is Calgebraic but not Lalgebraic, after possibly passing to a zextension of \( \mathbf {G}\), one must apply twisting to reduce to the Lalgebraic case, as explained in loc. cit.
By the first author’s work [34], the condition “\(\pi _{\infty }\) is a nondegenerate LDS” is equivalent to the following condition on the Hodge–Tate cocharacter: The adjoint group of \(\mathrm{Cent}_{{\phantom {\mathbf {G}}}^{L}{\mathbf {G}}^{\circ }}(\chi _{\infty })\) is a product of copies of PGL(2).
Concretely, assume that \(\mathbf {G}\) is a form of GL(n) (resp. SO(2n), \(SO(2n+1)\), Sp(2n)) and that r is the ‘standard’ representation of \({\phantom {\mathbf {G}}}^{L}{\mathbf {G}}\) of dimension n (resp. 2n, 2n, \(2n+1\)). Then “\(\pi _{\infty }\) nondegenerate LDS” conjecturally means that every Hodge–Tate weight of \(R_{p,\iota }(\pi , r)\) has multiplicity \(\le 2\), except that for \(\mathbf {G}_{\mathbf {C}}=Sp(2n)\) (resp. \(\mathbf {G}_{\mathbf {C}}=SO(2n)\)) the weight 0 may have multiplicity 3 (resp. 4). Recall that, by the standard dictionary between Hodge–Tate weights and Hodge numbers, this means that if \(R_{p,\iota }(\pi , r)\) is the padic realization of a Motive M, then the Hodge numbers \(h^{p,q}\) of the Betti realization of M are all \(\le 2\) (with the same exceptions for Sp(2n) and SO(2n) as before).
By contrast, in the same examples, the condition “\(\pi _{\infty }\) is a holomorphic LDS” conjecturally corresponds to the stricter restriction that all Hodge–Tate weights have multiplicity 1, except possibly for one weight of multiplicity 2 if \(\mathbf {G}_{\mathbf {C}}=GL(n)\), the weight 0 with multiplicity 3 if \(\mathbf {G}_{\mathbf {C}}=Sp(2n)\) and two weights of multiplicity 2 if \(\mathbf {G}_{\mathbf {C}}=SO(2n+1)\) or \(\mathbf {G}_{\mathbf {C}}=SO(2n)\). In the other direction, “\(\pi _{\infty }\) an arbitrary LDS” (conjecturally) imposes no restriction on the multiplicities of the Hodge–Tate weights.
15.6 Proof of Theorems and
The results of Sects. – will be deduced from our factorization theorem (Theorem ) coupled with the previously known Corollary on which automorphic representations appear in the coherent cohomology of Shimura varieties. The proofs in the case of general groups and in that of unitary groups are almost identical. The only difference is that in the former case we assume that some very regular \(\pi \) satisfy Condition , while in the latter case this is given to us by Corollary 1.3 of [50]. Moreover, the argument we use is analogous to the one introduced by Taylor [100], and then applied in [32, 39, 50, 54]. For these reasons, we only treat the case of general groups to avoid repetition.
Remark 10.6.1
Note that [50, Corollary 1.3] is a concise combination of Shin’s results [95, Theorem 1.2] and [96, Theorem A.1]. These build on the work of many people, see also Remark .
Consider first the case of torsion.
Proof of Theorem 10.4.1
 (i)
\(\nu +ak\eta _{\omega } \in F(i,n,\eta )\),
 (ii)
\(w_{0,\mathbf {L}}(\nu +\rho +ak\eta _{\omega })\) is \(\max \{\delta ,1\}\)regular, and
 (iii)
the map (10.6.1) is surjective.
The case of LDS will now be deduced from that of torsion.
Proof of Theorem 10.5.1
 (a)
We may apply Sect. to the integral model \({\mathscr {S}}_{{\mathcal K}}\),
 (b)
For all \(v \not \in {{\,\mathrm{Ram}\,}}(\pi ) \cup \{p\}\), one has \({\mathcal K}_v' \subset {\mathcal K}_v\).
First we explain how to pass from the “interior cohomology” (10.1.4) to that of the subcanonical extension. Let \({\bar{{\mathcal H}}}^{i,0}(\eta )\) be the image of the Hecke algebra \({\mathcal H}\) in \(\mathrm{End}(\bar{H}^i({\mathscr {S}}_{{\mathcal K}}^{\Sigma } \otimes \overline{\mathbf {Q}}_{p}, {\mathscr {V}}({\eta })))\) By definition, one has the surjection \(H^i({\mathscr {S}}_{{\mathcal K}}^{\Sigma } \otimes \overline{\mathbf {Q}}_{p}, {\mathscr {V}}^{\mathrm{sub}}(\eta )) \twoheadrightarrow \bar{H}^i({\mathscr {S}}_{{\mathcal K}}^{\Sigma } \otimes \overline{\mathbf {Q}}_{p}, {\mathscr {V}}({\eta }))\). Thus \({\mathcal H}\rightarrow {\bar{{\mathcal H}}}^{i,0}(\eta )\) factors through \({\mathcal H}^{i,0}(\eta )\). Therefore there exists an eigenclass \(\gamma _{\pi } \in H^i({\mathscr {S}}_{{\mathcal K}}^{\Sigma } \otimes \overline{\mathbf {Q}}_{p}, {\mathscr {V}}^{\mathrm{sub}}(\eta ))\) with the same eigenvalues as \(\gamma _{\pi }'\). After replacing \({\mathcal O}_{{\mathfrak p}}\) by a finite extension, we may assume that \(\gamma _{\pi }\) lies in \(H^i({\mathscr {S}}_{{\mathcal K}}^{\Sigma } , {\mathscr {V}}^{\mathrm{sub}}(\eta ))\).
Remark 10.6.2
In view of the “change of weight” afforded by Theorem (c), in the case of unitary similitude groups we only use a weak version of [50, Corollary 1.3], where the Archimedean component is \(\delta \)regular. By applying Theorem (c) with \(\delta =1\), it suffices for us to combine Shin’s earlier work [95, Theorem 1.2] with Labesse’s restricted base change [64]. In particular our results use neither Shin’s extended base change [96, Theorem A.1], nor the work of Chenevier–Harris [22].^{12}
16 Systems of Hecke eigenvalues on the generalized superspecial locus
This section is concerned with the generalization of Serre’s letter to Tate, Theorem I.4.1.
16.1 Statement of the result and a corollary
Theorem 11.1.1
 (\(\star \))

There exists a \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant Cartier divisor \(D'\) such that \(D'_{\mathrm{red}}=D\) and \(\omega ^k(D')\) is ample on \({\mathscr {S}}_{{\mathcal K}}^{\Sigma }\) for all \(k \gg 0\).
Then each of the \({\mathcal H}\)modules \(M^0, M^0_{\mathrm{sub}}, M^0_{\mathrm{can}}\) and \(M_e\) admits precisely the same systems of Hecke eigenvalues.
Remark 11.1.2
 (a)
If \(({\mathbf {G}}, {\mathbf {X}})\) is of compact type, then \((\star )\) is trivially satisfied with \(D=D'=0\), since then \(\omega \) is ample on \({\mathscr {S}}_{{\mathcal K}}\).
 (b)
If \(({\mathbf {G}}, {\mathbf {X}})\) is of noncompact, PEL type, then [66, Theorem 7.3.3.4] (explained further in [69, (2.1)]) produces a \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant Cartier divisor \(D'\) satisfying (\(\star \)) for \(\Sigma \) as above which is also projective [69, Definition 7.3.1.1]. The construction of \(D'\) should apply in the general Hodge case to the compactifications of [77] (still associated to projective \(\Sigma \)). Since we were unable to find a precise reference to this effect, the existence of such \(D'\) appears as hypothesis (\(\star \)) in Theorem .
 (c)
Note that Theorem assumes neither Koecher’s Principle nor Conditions , for \(S_{{\mathcal K}}\).
An immediate consequence of Remark and Theorem is:
Corollary 11.1.3
Remark 11.1.4
Using Lan’s Higher Koecher Principle [71] one obtains results analogous to Corollary for \(M^j\), \(j>0\), under additional hypotheses about the codimension of the boundary of \(S_{{\mathcal K}}^{\min } \).
16.2 Proof of Theorem and Corollary
Remark 11.2.1
The argument given in [30, Proposition 24] and cited in [90, Proposition 5.17] for Lemma seems to contain a serious error: If \(j:S_{{\mathcal K}} \rightarrow {{\mathbf {P}}}^n\) is the immersion associated to a very ample power \(\omega ^m\) of \(\omega \), then it is claimed in loc. cit. that every \({\mathscr {V}}(\lambda )\) admits a locally free extension to \({{\mathbf {P}}}^n\). Already for the Siegel scheme \(S_{g, {\mathcal K}}\) with \(g \ge 2\) this cannot be: It would contradict the wellknown fact that most \({\mathscr {V}}(\lambda )\) fail to admit a locally free extension to \(S_{{\mathcal K}}^{\min }\).
Lemma 11.2.2
One has \(H^0(S_{{\mathcal K}}^{\Sigma }, {\mathscr {V}}^{\mathrm{sub}}(\eta )) \rightsquigarrow _{{\mathcal H}} H^0(S_{{\mathcal K}}^{\Sigma }, {\mathscr {V}}^{\mathrm{can}}(\eta ))\) and \(H^0(S_{{\mathcal K}}^{\Sigma }, {\mathscr {V}}^{\mathrm{can}}(\eta )) \rightsquigarrow _{{\mathcal H}} H^0(S_{{\mathcal K}}, {\mathscr {V}}({\eta }))\).
Proof
To show that \(M^0 \rightsquigarrow _{{\mathcal H}} M_e\), we shall need an auxiliary lemma about “good filtrations” of Lmodules. Let V be an Lmodule (not necessarily finitedimensional). Recall that an ascending filtration \(0=V_0 \subset V_1 \subset \cdots \subset V_{i} \subset \cdots \subset V\) of V by Lsubmodules \(V_i\) is called a good filtration if \(\bigcup _i V_i =V\) and every graded piece \(V_{i}/V_{i1} \cong V_{\eta _i}\) for some \(\eta _i \in X^*_{+,L}(T)\), [53, II, 4.1.6]. Recall that the assumption \(p>2\) is still in force.
Lemma 11.2.3
For every \(n \ge 1\) and \(\eta \in X^*_{+,L}(T)\), the Lmodule \({{\,\mathrm{Sym}\,}}^n(({{\,\mathrm{Lie}\,}}(G)/{{\,\mathrm{Lie}\,}}(P))^{\vee }) \otimes V_{\eta }\) admits a good filtration.
Proof
The proof amounts to combining a result of Andersen–Janzten [2] on good filtrations for \({{\,\mathrm{Sym}\,}}^n {{\,\mathrm{Lie}\,}}(G)\) with closure properties for good filtrations given by Donkin [26].
Since \(({\mathbf {G}}, {\mathbf {X}})\) is of Hodge type, \(\mathbf {G}\) (and hence also G and L) are of classical type (i.e., have no factors of exceptional type) [24, 1.3.10]. Therefore every prime \(p>2\) is good for G in the sense of Steinberg [2, §4.4]: The \(\Delta \)dominant Tweights \(\chi \) of \({{\,\mathrm{Lie}\,}}(G)\) all satisfy \(0 \le \langle \chi , \alpha ^{\vee } \rangle \le p1\) for all \(\alpha \in \Delta \). Let \({\tilde{G}}\) be the simplyconnected cover of the derived subgroup of G (in the sense of algebraic groups, as in [53, II, 1.6]). Since p is good for G, the \(\tilde{G}\)module \({{\,\mathrm{Sym}\,}}^n {{\,\mathrm{Lie}\,}}({\tilde{G}})\) admits a good filtration for all \(n \ge 1\) by [2, Proof of Proposition 4.4].
Let \(V',V''\) be two Lmodules. By [2, 4.1(1)], the sum \(V \oplus V'\) admits a good filtration if and only if both summands \(V,V'\) admit good filtrations. Further, if \(V'\) and \(V''\) both admit good filtrations, so does \(V' \otimes V''\), [26, Theorem 4.3.1] (see also [2, 4.1.4(3)]).^{13} It follows as in [2, 4.1.4] that \({{\,\mathrm{Sym}\,}}^n(V' \oplus V'')\) admits a good filtration for all \(n \ge 1\) if and only if \({{\,\mathrm{Sym}\,}}^n(V')\) and \({{\,\mathrm{Sym}\,}}^n(V'')\) both admit good filtrations for all \(n \ge 1\).
Let \(U^+\) be the unipotent radical of \(P^+\). One has \({{\,\mathrm{Lie}\,}}(G)={{\,\mathrm{Lie}\,}}(P) \oplus {{\,\mathrm{Lie}\,}}(U^+)\) and \({{\,\mathrm{Lie}\,}}(P)^{\vee }={{\,\mathrm{Lie}\,}}(P^+)\) as Lmodules. Also \({{\,\mathrm{Lie}\,}}(G) \simeq {{\,\mathrm{Lie}\,}}(G)^{\vee }\) as Gmodules. Thus \(({{\,\mathrm{Lie}\,}}(G)/{{\,\mathrm{Lie}\,}}(P))^{\vee }={{\,\mathrm{Lie}\,}}(U)\) as Lmodules. Since \({{\,\mathrm{Lie}\,}}(G)={{\,\mathrm{Lie}\,}}(P^+) \oplus {{\,\mathrm{Lie}\,}}(U)\) as Lmodules, we are reduced to showing that \({{\,\mathrm{Sym}\,}}^n {{\,\mathrm{Lie}\,}}(G)\) admits a good filtration as Lmodule for all \(n \ge 1\). By reductions of Donkin, [26, Propositions 3.2.7, 3.4.3], since \({{\,\mathrm{Sym}\,}}^n {{\,\mathrm{Lie}\,}}(\tilde{G})\) admits a good filtration as a \(\tilde{G}\)module, it also admits a good filtration as Gmodule. Since \({{\,\mathrm{Lie}\,}}(G)\) is the direct sum of its center and \({{\,\mathrm{Lie}\,}}(\tilde{G})\), the result used above about symmetric powers of a sum implies that \({{\,\mathrm{Sym}\,}}^n {{\,\mathrm{Lie}\,}}(G)\) admits a good filtration as a Gmodule for all \(n \ge 1\). By [26, Theorem 4.3.1], the property of admitting a good filtration is stable under restriction to a Levi subgroup. Thus \({{\,\mathrm{Sym}\,}}^n {{\,\mathrm{Lie}\,}}(G)\) admits a good filtration as Lmodule for all \(n \ge 1\). \(\square \)
Lemma 11.2.4
One has \(M^0 \rightsquigarrow _{{\mathcal H}} M_e\).
Proof
The automorphic vector bundle \(\Omega ^1_{S_{{\mathcal K}}}\) is associated to the Lmodule \(({{\,\mathrm{Lie}\,}}(G)/{{\,\mathrm{Lie}\,}}(P))^{\vee }={{\,\mathrm{Lie}\,}}(U)\) via the construction of Sects. , . Indeed, it is sufficient to check this for the cotangent bundle over \({\mathscr {S}}_{{\mathcal K}, \mathbf {C}}\), where it follows trivially from the complex uniformization of \({\mathscr {S}}_{{\mathcal K}}(\mathbf {C})\) and the Borel embedding \(\mathbf {X}\hookrightarrow (\mathbf {G}/\mathbf {P})(\mathbf {C})\). Since the associated bundle construction is an exact tensor functor, \({{\,\mathrm{Sym}\,}}^j(\Omega ^1_{S_{{\mathcal K}}})\otimes {\mathscr {V}}({\eta })\) is associated to the Lmodule \({{\,\mathrm{Sym}\,}}^j(({{\,\mathrm{Lie}\,}}(G)/{{\,\mathrm{Lie}\,}}(P))^{\vee }) \otimes V_{\eta }\). By Lemma , this Lmodule admits a good filtration. Hence the system of Hecke eigenvalues of f appears in \(H^0(S_e, {\mathscr {V}}(\eta '))\) for some \(\eta ' \in X^*_{+,\mathbf {L}}(\mathbf {T})\) and some graded piece \(V_{\eta '}\) of a good filtration for \({{\,\mathrm{Sym}\,}}^j(({{\,\mathrm{Lie}\,}}(G)/{{\,\mathrm{Lie}\,}}(P))^{\vee }) \otimes V_{\eta }\) . \(\square \)
By Theorem (with \(w=e\)), there exists a nowhere vanishing, Heckeequivariant section \(h_e \in H^0(S_e, \omega ^{N_e})\) for some \(N_e \ge 1\). The following is then immediate:
Lemma 11.2.5
The last step, of going up from \(S_e\) to \(S_{{\mathcal K}}\), is given by:
Lemma 11.2.6
For every \(\eta \in X^*_{+,\mathbf {L}}(\mathbf {T})\) and every system of Hecke eigenvalues \((b_T)_{T \in {\mathcal H}}\) appearing in \(H^0(S_e, {\mathscr {V}}(\eta ))\), there exists \(m \ge 1\) such that \((b_T)_{T \in {\mathcal H}}\) also appears in \(H^0(S_{{\mathcal K}}, {\mathscr {V}}({\eta })\otimes \omega ^m)\).
Proof
Let \(f \in H^0(S_e, {\mathscr {V}}({\eta }))\) be an eigenform with eigenvalues \((b_T)\). By Lemma , \(fh_e^{a} \in H^0(S_e, {\mathscr {V}}({\eta })\otimes \omega ^{aN_e})\) is an eigenform with the same eigenvalues \((b_T)\) for all \(a\ge 1\).
Therefore \((b_T)\) appears in \(H^0(S_{{\mathcal K}}^{\Sigma } , {\mathscr {V}}^{\mathrm{can}}(\eta )\otimes \omega ^{kr}(rD'))\). Since the latter is Heckeequivariantly embedded in \(H^0(S_{{\mathcal K}}^{\Sigma } , {\mathscr {V}}^{\mathrm{can}}(\eta )\otimes \omega ^{kr}(D))\) and \({\mathscr {V}}^{\mathrm{can}}(\eta )\otimes \omega ^{kr}(D)={\mathscr {V}}^{\mathrm{sub}}(\eta ) \otimes \omega ^{kr}\), we conclude that \((b_T)\) also appears in \(H^0(S_{{\mathcal K}}^{\Sigma } , {\mathscr {V}}^{\mathrm{sub}}(\eta )\otimes \omega ^{kr})\). \(\square \)
Proof of Theorem 11.1.1
Apply Lemmas , , and consecutively. \(\square \)
Remark 11.2.7
Suppose that the theory of integral models and their compactifications applies to \(({\mathbf {G}}, {\mathbf {X}})\) also for \(p=2\). e.g., this is the case if \(({\mathbf {G}}, {\mathbf {X}})\) is of PELtype with no factors of type D. Then it is clear that the proof of Lemma works for \(p=2\). However, even in the Siegel case, we don’t know whether Lemma remains valid for \(p=2\). According to [53, p. 241], it seems likely that Lemma is false when \(p=2\).
Proof of Corollary 11.1.3
Let \(d=\dim S_{{\mathcal K}}^{\Sigma } \). The canonical bundle of \(S_{{\mathcal K}}^{\Sigma } \) is isomorphic to \(\omega ^m(D)\) for some \(m \ge 1\), [77, 5.3.11]. By Serre duality, \(H^i(S_{{\mathcal K}}^{\Sigma } ,{\mathscr {V}}^{\mathrm{can}}(\eta )) \cong H^{di}(S_{{\mathcal K}}^{\Sigma } , {\mathscr {V}}^{\mathrm{sub}}(m\eta _{\omega }\eta ))\). By Theorem , \(F(i,1,\eta ) \ne \emptyset \) for all \(\eta \). Hence Remark and Theorem imply that every system system of Hecke eigenvalues that appears in (11.1.1) also appears in \(M_e\) (recall that \({\mathscr {V}}({\eta })\), \({\mathscr {V}}^{\mathrm{can}}(\eta )\) and \({\mathscr {V}}^{\mathrm{sub}}(\eta )\) all have the same restriction to \(S_e\)). The finiteness then follows by the same argument as that given in [30, 90, 94]. \(\square \)
Remark 11.2.8
The finiteness of Hecke eigensystems appearing in (11.1.1) can also be deduced as follows: Recall that the (noncompactified) flag space \(Fl_{\mathcal K}\) introduced in (9.1.1) admits a smooth map \(Fl_{\mathcal K}\rightarrow \mathop {G\hbox {}{} \mathtt{ZipFlag}}\nolimits ^\mu \). By pullback, the stratification defined in Sect. induces a stratification of \(Fl_{\mathcal K}\). There is a unique zerodimensional stratum \(Fl_e\), and the projection \(\pi :Fl_{\mathcal K}\rightarrow S_{{\mathcal K}}\) induces a \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant map \(Fl_e\rightarrow S_e\). Hence, we obtain a Heckeequivariant injection \(H^0(S_e,{\mathscr {V}}(\eta ))\rightarrow H^0(Fl_e,\pi ^*{\mathscr {V}}(\eta ))\). Furthermore, each \(\pi ^*{\mathscr {V}}(\eta )\) admits a filtration with graded pieces of the form \({\mathscr {L}}(\lambda )\). For each character \(\chi \in X^*(T)\), there exists an integer \(N\ge 1\) such that \({\mathscr {L}}(\chi )^N\) admits a nonvanishing \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant section (obtained by pullback from the corresponding stratum in \(\mathop {G\hbox {}{} \mathtt{ZipFlag}}\nolimits ^\mu \)). In particular, the lattice of weights of \({\mathbf {G}}({{\mathbf {A}}}_f^p)\)equivariant nonvanishing sections on \(Fl_e\) has finite covolume in \(X^*(T)\). The finiteness of Heckeeigensystems follows immediately.
Footnotes
 1.
Unless otherwise stated, the Zariski closure is always equipped with the reduced scheme structure.
 2.
To avoid confusion, we specify that for us the classical superspecial locus means those points whose underlying abelian scheme is isomorphic to a product of supersingular elliptic curves. Some refer to this locus simply as the “superspecial locus”, while others reserve the same term for the zerodimensional EO stratum. The latter two conventions are jointly incompatible, already in the PEL case.
 3.
Such a section is sometimes referred to as regular (cf. [98, Definition 11.17]), but we find the term injective less confusing.
 4.
Note that contrary to loc. cit., we use the convention \(B\subset P\). This choice seems more natural from the point of view of Hodge theory and applications to characteristic 0. It slightly modifies the parametrization of the Eorbits, and other results are changed accordingly.
 5.
Of course \({{\,\mathrm{Sh}\,}}({\mathbf {G}}, {\mathbf {X}})_{{\mathcal K}}\) will only be a stack for certain \({\mathcal K}\); this does not matter for our purposes.
 6.
Ogus calls his spans Fspans since he uses F for the crystalline Frobenius.
 7.
 8.
Our definition implicitly exploits that \(\mathbf{G}(\mathbf{R})\) arises from a Shimura datum \((\mathbf{G}, \mathbf{X})\) of Hodge type. In particular, it implies that the Lpacket of a Calgebraic LDS of \(\mathbf{G}(\mathbf{R})\) is uniquely determined by its infinitesimal character. In greater generality, one should define HarishChandra parameters differently in terms of roots of \(K_C\) and genuine characters as defined in [1].
 9.
The group denoted \({\phantom {\mathbf {G}}}^{L}{\mathbf {G}}\) in [8] is the topological group we denote \({\phantom {\mathbf {G}}}^{L}{\mathbf {G}}(\mathbf {C})\).
 10.
The isomorphism from loc. cit. is with \(\mathbf {C}\)coefficients, i.e., \({\mathcal H}_{v,\mathbf {C}} {\mathop {\rightarrow }\limits ^{\sim }} R_{{{\,\mathrm{fd}\,}}}^{\mathrm{ss}}({\phantom {\mathbf {G}}}^{L}{\mathbf {G}}_v)_{\mathbf {C}}\). However the integral version (10.2.1) holds because the Satake transform is defined over \({\mathbf {Z}}_p[\sqrt{v}]\), cf. [93, Proof of Lemma V.1.6].
 11.
This seems to be one of many things that has been potentially “wellknown to experts” for a long time, but difficult to extract from the literature prior to [50].
 12.
To be completely precise, Labesse’s result has the disadvantage of being stated for unitary groups rather than unitary similitude groups, and to assume \(F^+ \ne {{\mathbf {Q}}}\), so in that respect we do use Shin’s [96, Theorem A.1], but not concerning the regularity of the Archimedean component.
 13.
The theorem of Donkin cited above assumed that \(p>2\) when G has factors of type \(E_7\) or \(E_8\). Mathieu later showed [78] that the existence of good filtrations is closed under tensor product without any restriction on p or G (still assumed connected, reductive).
Notes
Acknowledgements
We thank our respective collaborators MarcHubert Nicole and Torsten Wedhorn: The joint projects [39, 62] were invaluable in leading us to the results of the current paper. We are grateful to David Geraghty for the many important ideas that he contributed during the initial part of this project. In addition, we thank Joseph Ayoub, Laurent Clozel, Pierre Deligne, Phillip Griffiths, David Helm, Matt Kerr, Arno Kret, Ben Moonen, MarcHubert Nicole, Stefan Patrikis, Jonathan Pottharst, Sug Woo Shin, Benoit Stroh, Jacques Tilouine, Adrian Vasiu and Torsten Wedhorn for helpful conversations and correspondence. Finally, we thank the referee for his/her very thorough work, detailed comments and valuable suggestions, which we believe have led to considerable improvements in the paper.
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