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The universal p-adic Gross–Zagier formula

Abstract

Let \(\mathrm {G}\) be the group \( (\mathrm {GL}_{2}\times \mathrm{GU}(1))/\mathrm {GL}_{1}\) over a totally real field F, and let \(\mathscr {X}\) be a Hida family for \(\mathrm {G}\). Revisiting a construction of Howard and Fouquet, we construct an explicit section \(\mathscr {P}\) of a sheaf of Selmer groups over \(\mathscr {X}\). We show, answering a question of Howard, that \(\mathscr {P}\) is a universal Heegner class, in the sense that it interpolates geometrically defined Heegner classes at all the relevant classical points of \(\mathscr {X}\). We also propose a ‘Bertolini–Darmon’ conjecture for the leading term of \(\mathscr {P}\) at classical points. We then prove that the p-adic height of \(\mathscr {P}\) is given by the cyclotomic derivative of a p-adic L-function. This formula over \(\mathscr {X}\) (which is an identity of functionals on some universal ordinary automorphic representations) specialises at classical points to all the Gross–Zagier formulas for \(\mathrm {G}\) that may be expected from representation-theoretic considerations. Combined with a result of Fouquet, the formula implies the p-adic analogue of the Beilinson–Bloch–Kato conjecture in analytic rank one, for the selfdual motives attached to Hilbert modular forms and their twists by CM Hecke characters. It also implies one half of the first example of a non-abelian Iwasawa main conjecture for derivatives, in \(2[F:\mathbf {Q}]\) variables. Other applications include two different generic non-vanishing results for Heegner classes and p-adic heights.

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Fig. 1

Notes

  1. Two other recent ideas that our work does not touch upon are nevertheless too important to be ignored: the conjecture of Darmon and Guitart–Madseu–Şengün that there should exist Heegner points attached to any quadratic extension of number fields (see [27, 48]), and the formulas for the p-adic logarithms of Heegner points of [8, 68].

  2. N.B.: the notation \(\mathrm {G}\) used in the informal abstract differs from the notation of the paper.

  3. See however the very recent [87]. (Note added during revision.)

  4. Provided V contains no copies of the trivial representation. Of course in general the meromorphic continuation of L(Vs) is itself conjectural. Note that when V is self-dual, or E is a CM field and V is conjugate-self-dual, we have \(L(V,s)=L(V^{*}(1), s)\).

  5. Other authors consider p-adic L-functions of a variable \(s'\in \mathbf {Z}_{p}\). In our language this corresponds to restricting \(\mathscr {L}_{p}(V, s)\) along the embedding \(\mathbf {Z}_{p}=\mathrm {Spec}\,\mathbf {Z}_{p}\llbracket \mathbf {Z}_{p}\rrbracket _{\mathbf {Q}_{p}}(\mathbf {Q}_{p})\rightarrow \mathscr {E}_{\mathrm{Z}/L}(\mathbf {Q}_{p})\), \(s'\mapsto \chi _{\mathrm{cyc}, F}^{s'}\) where \(\chi _{\mathrm{cyc}, F}=\) (1.8.1) is the cyclotomic character of F.

  6. In a related context, see also the very recent breakthrough of Li–Liu [66]. (Note added during revision.)

  7. In fact, the (incoherent) group that truly underlies our constructions is \((\mathrm {G}\times _\mathrm{Z}\mathrm {H})' = \{(g, h)\, | \, \nu _{\mathrm {G}}(g)=\nu _{\mathrm {H}}(h)\}\) (where \(\nu _{?}:?\rightarrow \mathrm{Z}\) arises from the reduced norm map of \(\mathbf {B}\) (for \(?=\mathrm {G}\)) or from the norm of E/F (for \(?=\mathrm {H}\)). That is, the universal Heegner class and the other associated objects described below descend to the ordinary eigenvariety for \((\mathrm {G}\times _\mathrm{Z}\mathrm {H})'\) (a quotient of the one for \((\mathrm {G}\times \mathrm {H})'\)). Nevertheless, for the sake of simplicity we will content ourselves with working with \((\mathrm {G}\times \mathrm {H})'\).

  8. See Definition 2.4.1: the W/L of interest to us are in bijection with (finite) \(G_{L}\)-orbits of cohomological ‘numerical’ weights as defined above. From now on all numerical or representation-theoretic weights will be tacitly understood to be cohomological.

  9. This approach is inspired by the work of Emerton [37].

  10. By [34, Corollary 5.3.3].

  11. In the sense that for each \(z\in \mathscr {X}^{\mathrm {cl}}\), \(v\in \Sigma \), the finite-dimensional constituent \(\Pi _{z,v}\) of \(\Pi _{z}\) is fixed by \(K_{v}\).

  12. The question in [58] was phrased in terms of the Abel–Jacobi classes of Heegner cycles in a suitable Chow group, defined in that case in [78]; these classes are identical to the \(P_{\Pi }(f)\) from (1.2.4): see [78, §  I.2].

  13. The normalisations of L- and \(\varepsilon \)-factors are as in [98].

  14. Cf. the work [67] discussed in Sect. 1.7 below.

  15. These notions depend of course on \(L\hookrightarrow L'\); nevertheless they will only be used to impose conditions on the weights that are invariant under the Galois group of L.

  16. This notion agrees with the notion of \(\pi \) being nearly ordinary as defined in the work of Hida (e.g. [50]).

  17. If V is a Hodge–Tate representation of \(G_{F_{v}}\) over \(\overline{L}\) and \(\sigma :F_{v}\hookrightarrow \overline{L}\), the \(\sigma \)-Hodge–Tate weights of V are the degrees in which the graded module

    $$\begin{aligned} ( \oplus _{n} \mathbf {C}_{v}(n)\otimes _{\overline{F}_{v},\sigma } V)^{G_{F_{v}}} \end{aligned}$$

    is nonzero; here \(\mathbf {C}_{v}\) is a completion of \(\overline{F}_{v}\) and, in the tensor product, \(\sigma \) is extended to an isomorphism \(\overline{F}_{v}\rightarrow \overline{L}\). In particular our convention is that the Hodge–Tate weight of the cyclotomic character of \(\mathbf {Q}_{p}\) is \(-1\).

  18. We would have \(\mathrm{d}((\mathrm {G}\times _\mathrm{Z}\mathrm {H})'=2[F:\mathbf {Q}]\) for the group of the footnote after (1.2.1), which explains the way the number of variables of Theorem E is counted in the abstract.

  19. I am grateful to David Loeffler and Sarah Zerbes for explaining to me this point of view on the Control Theorem.

  20. The action of \(\mathbf {B}_{p}^{\times }\) has already been traded for an action of the torus, subsumed into the \(\mathscr {E}_{\mathrm {G}}\)-module structure.

  21. There is an essential reason for this, namely the possible presence of Schur indices in representations of those groups.

  22. With respect to the model in (A.4.2), we have \(|W|=\mathrm{ord}_{p}\left( {k-2\atopwithdelims ()(k-2+l)/2}\right) \) for the representation (A.2.2) of \(\mathrm {G}\).

  23. This factor accounts for the ‘\(K_{0}(p^{r})\)’-part of the level.

  24. The same statements hold with some extra notational burden if \(f_{1}, f_{2}\) are only defined over an open subset of \(\mathscr {X}^{(5)}\).

  25. Note that despite the similar notation, the character \(\mu _{v}\) is defined using the Weil–Deligne representations rather than the continuous Galois representations.

  26. In loc. cit., the ring denoted by R is our \(R^{\circ }\), but as our X is open in \(\mathrm {Spec}\,R^{\circ }\), the \(\mathscr {O}_{X}\)-modules \(P^{n}\) are also flat as \(R^{\circ }\)-modules and the cited result applies.

  27. In locc. citt., some restrictive assumptions are made (in particular that E is not CM), but the conjectures make sense even without those and indeed closely related conjectures appear in [39] without those assumptions. Moreover, our statement slightly differs from the ones of [39], which instead of postulating that \((x_{1}, \ldots , x_{r})\) is a basis, postulates that \(\hat{\phi }(Q_{A})\ne 0 \) under the extra assumption that \(\mathrm {Res}_{F/\mathbf {Q}}A\) is simple (equivalently L(As) is primitive). Our slight reformulation appears more uniform and still addresses [39, Remark 1.1] (cf. the comment following [39, Conjecture 1.5]).

  28. All the equalities to follow ignore signs and in fact, by our coarse definitions, only make sense at best up to \(\mathbf {Q}^{\times }\).

  29. A less interesting variant of it was sketched in [35].

  30. In the works of Bertolini–Darmon, an explicit test vector f is chosen; cf. [33] for more details on bridging the setups.

  31. Note that \(\pi \) admits a refinement if and only if it is neither supercuspidal nor 1-dimensional.

  32. To compare with (1.2.7), we have \(\zeta _{\mathbf {R}}(2)/L(1, \eta _{\mathbf {C}/\mathbf {R}})=1.\)

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Acknowledgements

I would like to thank Joël Bellaiche, Ashay Burungale, Gaëtan Chenevier, Olivier Fouquet, Ming-Lun Hsieh, David Loeffler, Michele Fornea, Jonathan Pottharst, Ye Tian, Rodolfo Venerucci, and Sarah Zerbes for useful conversations or correspondence or mathoverflow answers. I am also grateful to the referee for coaxing me to write Sect. 7.3, and to Simone Dell’Ariccia for Fig. 1. Finally, I would like to thank Shouwu Zhang for a vague question he asked me in 2010; this paper may be a partial answer.

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This work was supported by ISF grant 1963/20 and BSF grant 2018250. During the preparation of a first draft of this paper, the author was supported by a public grant of the Fondation Mathématique Jacques Hadamard.

Appendix A. p-adic semilocal constructions

Appendix A. p-adic semilocal constructions

Preliminaries

Throughout this appendix, unless otherwise noted L denotes a field of characteristic zero (admitting embeddings into \(\mathbf {C}\)).

Admissible and coadmissible representations

Let \(\mathrm {G}\) be a reductive group over \(\mathbf {Q}_{p}\). We denote

$$\begin{aligned} G_{p}&:=\mathrm {G}(\mathbf {Q}_{p}), \quad G_{\infty }:= \mathrm {G}(\mathbf {Q}_{p}), \quad G=G_{p\infty }:=G_{p}\times G_{\infty },\nonumber \\ G_{\Delta }&:= \Delta (\mathrm {G}(\mathbf {Q}_{p}))\subset G, \end{aligned}$$
(A.1.1)

where \(G_{p}\) and \(G_{\Delta }\) have the p-adic topology, \(G_{\infty }\) has the Zariski topology, and \(\Delta \) is the (continuous) diagonal embedding. The difference between \(G_{p}\), \(G_{\infty }\), \(G_{\Delta }\) will be in the category of modules we choose to consider. Namely, we consider the categories of smooth admissible representations of \(G_{p}\) over L, of algebraic representations of \(G_{\infty }\) over L, and the products of such for G; we call the latter locally algebraic representations of G over L.

Definition A.1.1

Suppose that L is a finite extension of \(\mathbf {Q}_{p}\). A p-adic locally algebraic admissible representation \(\Pi \) of G over L is one such that for each compact open subgroup \(K\subset G_{\Delta }\), there exists a family of \(\mathscr {O}_{L}\)-lattices \(\Pi ^{K, \circ }\subset \Pi ^{K}\), for \(K\subset G_{\Delta }\), with the property that \(\Pi ^{K', \circ }\cap \Pi ^{K}= \Pi ^{K, \circ }\) for all \(K'\subset K\).

The typical example of a p-adic locally algebraic admissible representation is \(\varinjlim _{K_{p}\subset G_{p}} H^{i}(Y_{K^{p}K_{p}}, \mathscr {W})\otimes W^{\vee }\), where \(Y_{K}\) is the system of locally symmetric spaces attached to a model \(\mathrm {G}_{\mathbf {Q}}\) of \(\mathrm {G}\) over \(\mathbf {Q}\), and \(\mathscr {W}\) is the automorphic local system attached to the algebraic representation W of \(G_{\infty }\).

There is a dual notion, introduced in [93, p. 152], see also [94]. Assume that L is endowed with a discrete valuation (possibly trivial), giving it a norm \(|\cdot |\). Let \(G'\) be one of the groups (A.1.1) or an open subgroup. For \(K\subset G'\) a compact open subgroup, let \(\mathscr {D}_{G',K}=\mathscr {H}_{G', K}:=C^{\infty }_{c}(K\backslash G'/K,L)\) and \(\mathscr {D}_{G'}=\varprojlim \mathscr {D}_{G',K}\) be the Hecke algebras of distributions; they are endowed with a natural topology as L-vector space, respectively as the inverse limit. A coadmissible \(G'\)-representation M over \((L, |\cdot |)\) is a topological right \(\mathscr {D}_{G'}\)-module such that, for any compact subgroup \(G^{\circ }\subset G'\), the \(\mathscr {D}_{G^{\circ }}\)-module M admits a presentation of the following form: there exists a system of topological \(\mathscr {D}_{G^{\circ }, K}\)-modules \(M_{K}\) and isomorphisms \(M_{K}\cong M_{K'}\otimes _{\mathscr {D}_{G^{\circ }, K'}} \mathscr {D}_{G^{\circ }, K}\) for \(K'\subset K\subset G^{\circ }\), such that \(M\cong \varprojlim _{K} M_{K}\).

Considering first a field L as endowed with a trivial valuation, we shall consider coadmissible representations M of \(G_{p}\) over L that are smooth in the sense the Lie algebra \(\mathfrak {g}\) of \(G_{p}\) acts trivially; coadmissible representations W of \(G_{\infty }\) that are algebraic (those are just algebraic representations); and the products of such as representations of G, which we call locally algebraic coadmissible representations of G.

Definition A.1.2

Suppose that L is a finite extension of \(\mathbf {Q}_{p}\); denote by \(|\cdot |\) the p-adic norm and by \(|\cdot |_\mathrm{triv}\) the trivial norm on L. A p-adic locally algebraic coadmissible representation M of G over L is one as above for \((L, |\cdot |_\mathrm{triv})\), whose restriction to \(G_{\Delta }\) is coadmissible for \((L, |\cdot |)\).

The typical example of a p-adic locally algebraic coadmissible representation is \(\varprojlim _{K_{p}} H_{i}(Y_{K^{p}K_{p}}, \mathscr {W})\otimes W^{\vee }\), where the notation is as after Definition A.1.1.

Notation

Consider the groups (1.2.1). For a place \(v\vert p\) of F, we let

$$\begin{aligned} G_{v}:=\mathbf {B}_{v}^{\times },\quad H_{v}:=E_{v}^{\times },\quad H_{v}':=E_{v}^{\times }/F_{v}^{\times },\quad (G\times H)'_{v}:= (G_{v}\times H_{v})/F_{v}^{\times } \end{aligned}$$

as topological groups. We use the parallel notation \(G_{*, v, \infty }\) for \(G_{*,v}\) viewed as the group of points of an algebraic group over \(F_{v}\).

We assume from now on that \(\mathbf {B}_{p}\) is split and fix an isomorphism \(\mathrm {G}_{\mathbf {Q}_{p}}\cong \mathrm {Res}_{F_{p}/\mathbf {Q}_{p}}\mathrm {GL}_{2}\), giving a model of \(\mathrm {G}_{*}\) over \(\mathbf {Z}_{p}\). We define involutions

$$\begin{aligned} g^{\iota }:= g^{\mathrm{T}, -1} \quad \text {on}\; \mathrm {G}(\mathbf {Q}_{p}), \quad h^{\iota }:=h^{c, -1} \quad \text {on}\; \mathrm {H}(\mathbf {Q}_{p}), \end{aligned}$$

that induce involutions \(\iota \) on all our groups. The embedding \(\mathrm {H}'\hookrightarrow (\mathrm {G}\times \mathrm {H})'\) is compatible with the involutions.

For \(t\in T_{\mathrm {G}_{*}, p}\), let \(\mathrm {U}_{t}:= K_{p, r} t K_{p,r}\in \mathscr {H}^{K_{p,r}}_{G_{*, p}}\) for any \(r\ge 1\), and

$$\begin{aligned} \mathrm {U}_{t,p\infty }:=\mathrm {U}_{t}\otimes t_{\infty }^{}. \end{aligned}$$

When \(x\in F_{p}^{\times }\), we abuse notation by writing \(\mathrm {U}_{x}= \mathrm {U}_{\left( {\begin{matrix}x&{}\\ &{}1\end{matrix}}\right) }\); we also write

$$\begin{aligned} \mathrm {U}_{p\infty }:= \mathrm {U}_{\left( {\begin{matrix}p&{}\\ &{}1\end{matrix}}\right) , p\infty } \end{aligned}$$

for short.

Ordinary parts of admissible or coadmissible \(G_{*}\)-modules

Let L be a finite extension of \(\mathbf {Q}_{p}\). Let \(\Pi =\Pi _{p}\otimes W\) be a p-adic locally algebraic admissible representation of \(G_{*}\) Let us write

$$\begin{aligned} \Pi ^{N_{0, (r)}}:= \Pi ^{N_{0, (r)}}\otimes W^{N} , \end{aligned}$$

where \(N_{0, r}:=K_{p, r}\). Choose \(\mathscr {O}_{L}\)-lattices \(W^{\circ }\subset W\), \(\Pi _{p}^{\circ , K}\subset \Pi _{p}^{K}\), stable under the Hecke action, and compatibly with the transition maps associated with \(K'\subset K\). Then \(\Pi ^{\circ , N_{0}}:=\Pi _{p}^{\circ , N_{0}} \otimes W^{\circ , N}= \varinjlim _{r} \Pi _{p}^{\circ , K_{p,r}} \otimes W^{\circ , N}\) is stable under the action of \(\mathrm {U}_{p\infty }\). As shown by Hida, the idempotent

$$\begin{aligned} e^{\mathrm {ord}}: = \lim _{n}\mathrm {U}_{p\infty }^{n!} :\Pi ^{\circ , N_{0}}\rightarrow \Pi ^{\circ , N_{0}} \end{aligned}$$

is then well-defined and its image is denoted by \(\Pi ^{\circ , \mathrm {ord}}\). The space \(\Pi ^{\circ , \mathrm {ord}}\) is the maximal split \(\mathscr {O}_{L}\)-submodule of \(\Pi ^{\circ , N_{0}}\) over which \(\mathrm {U}_{p\infty }\) acts invertibly. We also write \(e^{\mathrm {ord}}\) for \(e^{\mathrm {ord}}\otimes 1:\Pi ^{N_{0}}=\Pi ^{\circ , N_{0}}\otimes L\rightarrow \Pi ^{N_{0}}\), and we let \(\Pi ^{\mathrm {ord}}=e^{\mathrm {ord}}\Pi ^{N_{0}}\) be its image. If \(\Pi _{p}\) and W are irreducible, then \(\Pi ^{\mathrm {ord}}\) has dimension either 0 or 1; in the latter case we say that \(\Pi \) is ordinary. (This notion is independent of the choice of lattices.)

Let \(\mathrm{M}=\mathrm{M}_{p}\otimes W^{\vee }\) be a p-adic locally algebraic coadmissible right module for \(G_{*}\) over L. By definition of coadmissibility, the system \((\mathrm{M}_{p,K})_{K\subset G_{*,p}}\) is endowed with a compatible system \(\mathscr {H}_{\mathrm {G}_{*}(\mathbf {Z}_{p}), K}\)-stable lattices \(\mathrm{M}_{p,K}^{\circ }\), so that for some \(\mathrm {G}_{*}(\mathbf {Z}_{p})\)-stable lattice \(W^{\vee , \circ }\), \(\mathrm{M}^{\circ }_{N_{0}}:= \varprojlim \mathrm{M}^{\circ }_{p, K_{p, r}}\otimes W^{\vee , \circ }_{N_{0}}\) is stable under \(\mathrm {U}_{p\infty }\). Then we can again define \(e^{\mathrm {ord}}:\mathrm{M}^{(\circ )}_{N_{0}} \rightarrow \mathrm{M}^{(\circ )}_{N_{0}}\). Its image

$$\begin{aligned} \mathrm{M}^{(\circ ), \mathrm {ord}}:= \mathrm{M}^{(\circ )}_{N_{0}} e^{\mathrm {ord}} \end{aligned}$$

is called the ordinary part of \(\mathrm{M}^{(\circ )}_{N_{0}} \).

The ordinary parts \(\Pi ^{\mathrm {ord}}\), \(\mathrm{M}^{\mathrm {ord}}\) retain an action of the operators \(\mathrm {U}_{t, p\infty }\).

Special group elements, and further notation

The following notation will be in use throughout this appendix. Let \(v\vert p\) be a place of F. We denote by \(e_{v}\) be the ramification degree of \(E_{v}/F_{v}\), and fix a uniformiser \(\varpi _{v}\in F_{v}\) chosen so that \(\prod _{v\vert p}\varpi _{v}^{e_{v}}=p\). Let \(\mathrm{Tr}_{v}=\mathrm {Tr}_{E_{v}/F_{v}}\) and \(\mathrm{Nm}_{v}:=\mathrm{Nm}_{E_{v}/F_{v}}\) be the trace and norm. Fix an isomorphism \(\mathscr {O}_{E, v}=\mathscr {O}_{F, v}\times \mathscr {O}_{F, v}\) if v is split. If v is nonsplit, let c be the Galois conjugation of \(E_{v}/F_{v}\), and fix an element \(\theta _{v}\in \mathscr {O}_{E,v}\) such that \(\mathscr {O}_{E, v}=\mathscr {O}_{F,v}[\theta _{v}]\) (thus \(\theta _{v}\) is a unit if v is inert and a uniformiser if v is ramified). We define a purely imaginary \(\mathrm{j}_{v}\in E_{v}^{\times }\) to be

$$\begin{aligned} \mathrm{j}_{v}:= {\left\{ \begin{array}{ll} (-1_{w},1_{w^{c}}) &{} \text {if}\; E_{v}=E_{w}^{\times }\times E_{w^{c}}^{\times }, \\ \theta _{v}^{c}-\theta _{v} &{} \text {if}\; E_{v}\; \text {is a field}. \end{array}\right. } \end{aligned}$$
(A.1.2)

We assume that \(E_{v}\) embeds in \(\mathbf {B}_{v}\) and fix the embedding \(E_{v}\rightarrow \mathbf {B}_{v}\) to be

$$\begin{aligned} t=(t_{w},t_{w^{c}}) \mapsto \left( \begin{array}{cc}t_{w}&{}\\ &{}t_{w^{c}}\end{array}\right) \quad \text {if}\ E_{v}=E_{w}^{\times } \times E_{w^{c}}^{\times }, \\ \end{aligned}$$
$$\begin{aligned} t=a+\theta b\mapsto \left( \begin{array}{cc}a+b\mathrm {Tr}_{v}\theta _{v}&{}b\mathrm{Nm}_{v}\theta _{v}\\ -b&{}a\end{array}\right) \quad \text {if}\; E_{v} \text { is a field}. \end{aligned}$$

For \(r\ge 0\), let

$$\begin{aligned} w_{r, v}&:=\left( \begin{array}{cc}&{}1\\ -p^{r}&{}\end{array}\right) \in \mathrm {GL}_{2}(F_{v}),\\ \gamma _{r,v}&:= {\left\{ \begin{array}{ll} {\left( \begin{array}{cc}p^{r}&{}1\\ &{}1\end{array}\right) } &{}\text {if}\; v\;\text {splits}\\ {\left( \begin{array}{cc}p^{r}\mathrm{Nm}_{v}(\theta _{v})&{}\\ &{}1\end{array}\right) }&\text {if}\; v\;\text {is nonsplit} \end{array}\right. } \in (G\times H)_{v}' \end{aligned}$$

and

$$\begin{aligned} w_{r}:=\prod _{v\vert p}w_{r, v}\in \mathrm {G}(\mathbf {Q}_{p}), \quad \gamma _{r}:= \prod _{v\vert p}\gamma _{r,v}\in (\mathrm {G}\times \mathrm {H})'(\mathbf {Q}_{p}). \end{aligned}$$

Toric, ordinary, and anti-ordinary parts

Let L be a finite extension of \(\mathbf {Q}_{p}\). We perform some twists.

Ordinary and anti-ordinary parts

Let \(w:=\left( {\begin{matrix}&{}1\\ -1&{}\end{matrix}}\right) \in G_{*,\Delta }\) and let \(\pi ^{w}\) be the representation on the same space as \(\pi \) but with G-action given by \(\pi ^{w}(g)v:=\pi (w^{-1}gw)v\). Let \(N^{-}:= w^{-1}Nw\), and \(U_{p\infty }^{-}:= U_{w^{-1}\left( {\begin{matrix}p&{}\\ &{}1\end{matrix}}\right) w, p\infty }\).

Let \(\pi =\pi _{p}\otimes W\) be a p-adic admissible locally algebraic representation of G over L. The anti-ordinary part of \(\pi \) is the space

$$\begin{aligned} \pi ^\mathrm{a}:=\pi _{p}^\mathrm{a}\otimes W^{N^{-}}\subset \pi \end{aligned}$$

of ‘ordinary’ elements with respect to \(N^{-}\) and \(U_{p\infty }^{-}\). Because \(\pi ^{w}\) is isomorphic to \(\pi \), the spaces \(\pi ^\mathrm{a}\) and \(\pi ^{\mathrm {ord}}\) have the same dimension.

Let \(M=M_{p}\otimes W\) be a p-adic coadmissible locally algebraic representation of G over L. The anti-ordinary part of M is the quotient

$$\begin{aligned} M^\mathrm{a}:= M_{p}^\mathrm{a}\otimes W_{N^{-}} \end{aligned}$$

of M that is its ‘ordinary’ part with respect to \(N_{0}^{-}\) and and \(U_{p\infty }^{-}\).

Proposition A.2.1

Let W be an algebraic representation of \(G_{\infty }\).

  1. 1.

    Let \(\pi \) be a p-adic locally algebraic admissible representation of G. There is an isomorphism

    $$\begin{aligned} w^{\mathrm {ord}}_\mathrm{a}:\pi ^{\mathrm {ord}}&\rightarrow \pi ^\mathrm{a}\\ f&\mapsto \lim _{r\rightarrow \infty } p^{r[F:\mathbf {Q}]} w_{r,p}w_{0, \infty }^{\iota }\mathrm {U}_{p}^{-r}f, \end{aligned}$$

    where the sequence stabilises as soon as \(r\ge 1 \) is such that \(f_{p}\in \pi _{p}^{U_{1}^{1}(p^{r})}\).

  2. 2.

    Let M be a p-adic locally algebraic coadmissible representation of G. There is a map

    $$\begin{aligned} w^{\mathrm {ord}}_\mathrm{a}:M^{\mathrm {ord}}&\rightarrow M^\mathrm{a}\\ m=m_{p}\otimes m_{\infty }&\mapsto \lim _{r\rightarrow \infty } p^{r[F:\mathbf {Q}]} [m (U_{p}^{})^{-r} w_{r,p}]_{N_{0}^{-}} \otimes [m_{\infty }w_{0, \infty }^{\iota }]_{N^{-}}, \end{aligned}$$

    where before applying \(w_{*}\), we take arbitrary lifts from \(N_{0}^{}\)-coinvariants to the module M.

Proof

That the maps are well-defined is a standard result left to the reader. At least for admissible representations, the map is an isomorphism (equivalently, nonzero) because of Lemma A.3.3 below. \(\square \)

Let \(\pi _{v}^{\mathrm {ord}}\) (respectively \(\pi _{v}^\mathrm{a}\)) denote the preimage of \(\pi ^{\mathrm {ord}}\) (respectively \(\pi ^\mathrm{a}\)) in \(\pi _{v}\), and let \(W_{v}\) be the \(G_{v, \infty }\)-component of W. The following local components of the above map are similarly well-defined:

$$\begin{aligned} w_{\mathrm{a}, v}^{\mathrm {ord}}:\pi _{v}^{\mathrm {ord}}&\rightarrow \pi _{v }^\mathrm{a},&\quad w_{\mathrm{a}, v, \infty }^{\mathrm {ord}}:W_{v}^{N_{v}}&\rightarrow W_{v}^{N_{v}^{-}}\nonumber \\ f_{v}&\mapsto \lim _{r\rightarrow \infty } p^{r[F_{v}:\mathbf {Q}_{p}]} w_{r , v} \mathrm {U}_{p,v}^{-r}f_{v},&f_{v, \infty }&\mapsto w_{0,v, \infty }^{\iota }f_{v, \infty }. \end{aligned}$$
(A.2.1)

Lemma A.2.2

Let \(\pi \) be an ordinary representation of G. If \((\ , \ ):\pi \otimes \pi ^{\vee }\rightarrow L\) is a nondegenerate G-invariant pairing, then the pairing

$$\begin{aligned} (\ , \ )^{\mathrm {ord}} :\pi ^{\mathrm {ord}}\otimes \pi ^{\vee , \mathrm {ord}}&\rightarrow L \\ (f_{1}, f_{2})^{\mathrm {ord}}&:= (w_\mathrm{a}^{\mathrm {ord}} f_{1}, f_{2}) \end{aligned}$$

is a nondegenerate pairing.

Proof

It suffices to see this for a specific pairing \((\ , \ )\): we may take the product of the pairings (A.3.2) below, that are known to be nondegenerate, and any nondegenerate pairing on \(W\otimes W^{\vee }\). Then the result follows from Lemmas A.3.3 and A.4.1 below. \(\square \)

Ordinary and toric parts

We construct a map from the ordinary part of a representation of \((G\times H)'\) to its toric coinvariants, as well as a dual map in the opposite direction for coadmissibe modules. These map are the key to the interpolation of toric periods.

Suppose that \(W_{(v)}\) (respectively \(W=\bigotimes _{v\vert p}W_{v}\) is an algebraic representation of \((G\times H)_{(v) , \infty }\) (respectively \((G\times H)'_{\infty }\)) over L such that, for a field extension \(L'/L\) splitting E, \(W_{(v), \infty }\otimes _{L} L'=\bigotimes _{\sigma :F_{(v)}\hookrightarrow L} W_{\sigma }\) with

$$\begin{aligned} W_{\sigma }=W_{\sigma , w, k, l}:= \mathrm{Sym}^{k_{\sigma }-2}\mathrm{Std} \cdot {\det }^{w-k_{\sigma }+2\over 2} \otimes \sigma ^{{l_{\sigma }-w}\over 2}{(\sigma ^{c})}^{-l_{\sigma }-w\over 2} \end{aligned}$$
(A.2.2)

for some integers \(k_{\sigma }\ge 2, |l_{\sigma }|<k_{\sigma }, w\) of the same parity. (Here we have chosen, for each \(\sigma :F\hookrightarrow L\), an extension \(\sigma :E\hookrightarrow L\).) Then we define a constant

$$\begin{aligned} c(W_{\sigma })&:= \mathrm{j}_{v}^{-w-k_{\sigma }+2 }\cdot {k_{\sigma }-2 \atopwithdelims ()(k_{\sigma }-2-l_{\sigma })/2}\nonumber \\&\quad \cdot {\left\{ \begin{array}{ll} 1&{} \text {if}\; v\; \text {splits in}\; E\\ \theta ^{c,(k-2-l)/2} \theta ^{(k-2+l)/2} &{} \text {if}\; v\; \text {does not split in}\; E, \end{array}\right. }\nonumber \\ c(W_{(v)})&:=\prod _{\sigma :F_{(v)}\hookrightarrow L} c(W_{\sigma }). \end{aligned}$$
(A.2.3)

(Note that \(\mathrm{j}_{v}^{-w-k_{\sigma }+2 }=1\) if v splits in E, as \(w+k_{\sigma }-2\) is even.)

Lemma A.2.3

Recall the congruence subgroups \(V_{v,r}'\), \(K_{v,r}\) defined in Sect. 2.1.5. For all \(r\ge 1\), we have the identity of Hecke operators in the Hecke algebra for \((G\times H)'_{v}\):

$$\begin{aligned} V_{v, r+1}'\left( \sum _{t\in V_{v,r}'/V_{v,r+1}'} t \right) \cdot \gamma _{r+1,v} K_{v,r}= V_{v, r+1}' \gamma _{r,v}\cdot \mathrm {U}_{\varpi _{v}} K_{v, r}. \end{aligned}$$

Proof

This is a consequence of the following matrix identity.

Let \(v\vert p\) be a prime of F. For \(r\in \mathbf {Z}_{\ge 1}\), \(j\in \mathscr {O}_{F,v}\) let \(b_{j,v}:= \left( \begin{array}{cc}\varpi &{}j\\ &{}1\end{array}\right) \). In the split case, let

$$\begin{aligned} t_{j,r, v}= k_{j,r,v}:= \left( \begin{array}{cc}1+j\varpi ^{r}&{}\\ &{}1\end{array}\right) \in E_{v}^{\times } \end{aligned}$$

In the nonsplit case, let

$$\begin{aligned} t_{j,r, v}=1+\theta _{v}\varpi _{v}^{r}, \quad k_{j, r, v}=\left( \begin{array}{cc}1+j{\mathrm {Tr}}_{v}(\theta _{v}) \varpi _{v}^{r}&{}\mathrm{Tr}_{v}(\theta _{v})-j\varpi _{v}^{r}\\ -j\mathrm{N}_{v}(\theta _{v}) \varpi _{v}^{2r}&{} 1+j^{2}\mathrm{N}_{v}(\theta _{v} )\varpi _{v}^{r}\end{array}\right) . \end{aligned}$$

Then

$$\begin{aligned} t_{j,r, v}\gamma _{r+1,v}=\gamma _{r,v} b_{j,v} k_{j,r, v} \end{aligned}$$

in \(\mathrm {GL}_{2}(F_{v})\). \(\square \)

Proposition A.2.4

Let W be an algebraic representation of \((G\times H)'_{\infty }\).

  1. 1.

    Let \(\Pi =\Pi _{p}\otimes W\) be a p-adic locally algebraic admissible representation of \((G\times H)'\). There is a map

    $$\begin{aligned} \gamma _{H'}^{\mathrm {ord}} :\Pi ^{\mathrm {ord}}&{\rightarrow } \Pi _{H'}\nonumber \\ f&\mapsto \lim _{r} \ {H'_{}}[ p^{r[F:\mathbf {Q}]} \cdot c(W)^{-1}\cdot \gamma _{r, p\infty } \mathrm {U}_{p\infty }^{-r} f], \end{aligned}$$
    (A.2.4)

    where \({H'_{}}[-]:\Pi \rightarrow \Pi _{H'}\) is the natural projection. The sequence in the right hand side of (A.2.4) stabilises as soon as \(f_{p}\in \Pi ^{K_{p, r}}\), where \(K_{p, r}\subset (G\times H)_{p}'\) is defined at the end of Sect. 2.1.

  2. 2.

    Let \(\mathrm{M}:= \mathrm{M}_{p}\otimes W^{\vee }\) be a p-adic locally algebraic coadmissible representation of \((G\times H)'\). There is a map

    $$\begin{aligned} \gamma _{H'}^{\mathrm {ord}} :\mathrm{M}^{H'}&\rightarrow \mathrm{M}^{\mathrm {ord}}\\ m&\mapsto \lim _{r}\ [ p^{r[F:\mathbf {Q}]} \cdot c(W)^{-1}\cdot m \gamma _{r,p\infty }] {N_{0,r}} e^{\mathrm {ord}} \mathrm {U}_{p\infty }^{-r}, \end{aligned}$$

    where \([ -]{N_{0,r}}:\mathrm{M} \rightarrow \mathrm{M}_{N_{0}}\) is the natural projection.

The constant c(W) is justified by Lemma A.4.2 below.

Proof

For part 1, let \(f\in \Pi _{p}^{K_{p, r}}\). Then it follows from Lemma A.2.3 that, denoting by \([f_{r}]_{H'}\) the sequence in the right hand side of (A.2.4), we have

$$\begin{aligned} {1\over p^{[F:\mathbf {Q}]}} \sum _{t\in V_{p,r}'/V_{p,r+1}'} \Pi (t) f_{r+1}=f_{r}, \end{aligned}$$

hence \([f_{r+1}-f_{r}]_{H'}=0 \) and the sequence stabilises.

For part 2, Lemma A.2.3 similarly implies (the boundedness and) the convergence of the sequence in \(\varprojlim _{r} \mathrm{M}_{N_{0, r}}^{\mathrm {ord}}\). \(\square \)

Let \(\Pi _{v}^{\mathrm {ord}}\) denote the preimage of \(\Pi ^{\mathrm {ord}}\) in \(\Pi _{v}\), and let \(W_{v}\) be the \((G\times H)'_{v, \infty }\)-component of W. The following local components of the above maps are similarly well-defined:

$$\begin{aligned} \gamma _{H',v}^{\mathrm {ord}}:\Pi _{v}^{\mathrm {ord}}&\rightarrow \Pi _{v, H'_{v}}\nonumber \\ \gamma _{H',v, \infty }^{\mathrm {ord}}:W_{v}^{N}&\rightarrow W_{v, H'_{v}}\nonumber \\ f_{v}&\mapsto \lim [p^{r[F_{v}:\mathbf {Q}_{p}]} \gamma _{r , v} \mathrm {U}_{\varpi _{v}}^{-r}f_{v}]_{H'_{v}},\nonumber \\ f_{v, \infty }&\mapsto c(W_{v})^{-1} \cdot \gamma _{0,v, \infty }^{\iota }f_{v, \infty }. \end{aligned}$$
(A.2.5)

Exceptional representations and vanishing of \(P^{\mathrm {ord}}\)

We show that \(\gamma _{H'}^{\mathrm {ord}} \) acts by zero precisely on those representations that are exceptional.

Lemma A.2.5

Let \(\Pi =\pi \otimes \chi \) be an ordinary, distinguished, irreducible representation of \((G\times H)'\). The following are equivalent:

  1. 1.

    \(\Pi \) is exceptional;

  2. 2.

    \(e_{v}(V_{(\pi , \chi )}) = 0\);

  3. 3.

    there exists \(P\in \Pi ^{*, H'} -\{ 0\}\) such that \(P^{\mathrm {ord}}:=P \gamma _{H'}^{\mathrm {ord}} =0\);

  4. 4.

    for all \(P\in \Pi ^{*, H'}\), we have \(P^{\mathrm {ord}}=0\);

Proof

The equivalence of 1. and 2. is a reminder from Lemma 6.4.6. The equivalence of 3. and 4. is a consequence of multiplicity-one. Consider 3. Let \(P\in \Pi ^{*, H'}\). Identify \(\Pi ^{\vee }=\Pi ^{\iota }\) (the representation on the same space as \(\Pi \), with group action twisted by the involution \(\iota \)). Then the identity map on spaces yields isomorphisms \(\Pi ^{*, H'}\cong \Pi ^{\vee , *, H'}\) and \(\Pi ^{\mathrm {ord}, *}\cong \Pi ^{\vee , \mathrm {ord}, *}\), and it follows from the explicit description of \(\gamma _{H'}^{\mathrm {ord}} \) that if \(P^{\vee }\) denotes the image of P, then the image of \(P^{\vee }\) is \(P^{\vee ,\mathrm {ord}}\). Hence, \(P^{\mathrm {ord}}\) is zero if and only if so is \(P^{\vee , \mathrm {ord}}\), if and only if so is \(P\otimes P^{\vee }\circ \gamma _{H'}^{\mathrm {ord}} \otimes \gamma _{H'}^{\mathrm {ord}} \). Now by the theory recalled in Sect. 1.2.6 , \(P \otimes P^{\vee }\) is necessarily a multiple of the explicit functional \(Q_{dt, (, )}\) defined there. Therefore it suffices to show that \(Q_{dt, (, )}\) vanishes on the line \(\gamma _{H'}^{\mathrm {ord}} \Pi ^{\mathrm {ord}}\otimes \gamma _{H'}^{\mathrm {ord}} \Pi ^{\vee , \mathrm {ord}}\) if and only if \(e_{v}(V_{(\pi , \chi )}) = 0\). This follows from the explicit computations of Propositions A.3.4 and A.4.3 below, cf. also Proposition 4.3.4. \(\square \)

Pairings at p

The goal of this subsection is to relate the p-components of the toric terms Q and their ordinary variants \(Q^{\mathrm {ord}}\), as defined in Sects. 4.24.3.

Let \(v \vert p\) be a place of F.

Integrals and gamma factors

If \(\pi \) (respectively \(\chi \)) is an irreducible representation of \(G_{v}\) over L, we denote by \(V_{\pi }\) (respectively \(V_{\chi })\) the associated 2- (respectively 1-) dimensional Frobenius-semisimple representation of \(\mathrm{WD}_{F_{v}}\) (respectively of \(\mathrm{WD}_{E_{v}}:=\prod _{w\vert v}\mathrm{WD}_{E_{w}}\); we choose the “Hecke” normalisation, so that \(\det V_{\pi }\) is the cyclotomic character if \(\pi \) is self-dual. If \(\Pi =\pi \otimes \chi \) is an irreducible representation of \((G\times H)'_{v}\), we denote by \(V_{\Pi }= V_{\pi |\mathrm{WD}_{E_{v}}}\otimes V_{\chi }\) the associated 2-dimensional representation of \(\mathrm{WD}_{E_{v}}\). If \(E_{*} \) is F or E, \(w\vert p\) is a prime of \(E_{*}\) and V is any representation of \(\mathrm{WD}_{E_{*,v}}\) as above, we let \(V_{w}:= V_{|\mathrm{WD}_{E_{*, w}}}\).

If \(\psi :F_{v}\rightarrow \mathbf {C}^{\times }\) is a nontrivial character, we denote by \(d_{\psi } y\) the selfdual Haar measure on \(F_{v}\) and \(d^{\times }_{\psi }y:= d_{\psi }y/|y|\). The level of \(\psi \) is the largest n such that \(\psi _{|\varpi ^{-n}\mathscr {O}_{F, v}}=1\). We recall that if \(\psi \) has level 0, then \(\mathrm {vol}(\mathscr {O}_{F, v},d_{\psi }y)=1\).

Recall the Deligne–Langlands \(\gamma \)-factor of (1.4.3).

Lemma A.3.1

[35, Lemma A.1.1]. Let \(\mu :F_{v}^{\times }\rightarrow \mathbf {C}^{\times }\) and \(\psi :F_{v}\rightarrow \mathbf {C}^{\times }\) be characters, with \(\psi _{v}\ne 1\). Let \(d^{\times }y\) be a Haar measure on \(F_{v}^{\times }\). Then

$$\begin{aligned} \int _{F_{v}^{\times }} \mu (y) \psi (y) d^{\times }y={d^{\times }y \over d^{\times }_{\psi }y} \cdot \mu (-1)\cdot \gamma (\mu , \psi )^{-1} . \end{aligned}$$

Local pairing

The following isolates those representations that can be components of an ordinary representation.

Definition A.3.2

A refined representation \((\pi , \alpha )\) of \(G_{v}\) over a field L consists of a smooth irreducible infinite-dimensional representation \(\pi \) and a character \(\alpha :F_{v}\rightarrow L^{\times }\), such that \(\pi \) embeds into the un-normalised induction \(\mathrm{Ind}( |\ |\alpha , |\ |^{-1}\omega \alpha ^{-1}))\) for some other character \(\omega :F_{v}^{\times }\rightarrow L^{\times }\).Footnote 31 Sometimes we abusively simply write \(\pi \) instead of \((\pi , \alpha )\). A refined representation \(\Pi =\pi \otimes \chi \) \((G\times H)'_{v}\) is the product of a refined representation \(\pi =(\pi , \alpha )\) of G and a character \(\chi \) of H, such that \(\omega \chi _{|F_{v}^{\times }}=\mathbf {1}\).

If \((\pi , \alpha )\) is a refined representation of \(G_{v}\), we let \(\pi ^{\mathrm {ord}}\subset \pi ^{N_{0}}\) be the unique line on which the operator \(\mathrm {U}_{t}\) acts by \(\alpha (t)\). If \(\Pi =\pi \otimes \chi \) is a refined representation of \((G\times H)'_{v}\), we let \(\Pi ^{\mathrm {ord}}:=\pi ^{\mathrm {ord}}\otimes \chi \). The associated Weil–Deligne representation \(V_{\pi }\) is reducible, and we have a unique filtration

$$\begin{aligned} 0\rightarrow V_{\pi }^{+}\rightarrow V_{\pi } \rightarrow V_{\pi }^{-}\rightarrow 0 \end{aligned}$$

such that \(\mathrm{WD}_{F_{v}}\) acts on \(V_{\pi }^{+}\) through the character \(\alpha |\cdot |\).

Let \(\pi \) be a refined representation of \(G_{v}\) over L, and let \((\ , \ )_{\pi }:\pi \otimes \pi ^{\vee }\rightarrow L\) be a G-invariant pairing. Then we define

$$\begin{aligned} (\ , \ )_{\pi }^{\mathrm {ord}}:\pi ^{\mathrm {ord}}\otimes (\pi ^{\vee })^{\mathrm {ord}}&\rightarrow L \\ f\otimes f^{\vee }&\mapsto (w^{\mathrm {ord}}_\mathrm{a}f, f^{\vee }), \end{aligned}$$

where \(w^{\mathrm {ord}}_\mathrm{a} \) is the operator denoted \(w_{\mathrm{a}, v}^{\mathrm {ord}}\) in (A.2.1). If \(\Pi \) is a refined representation of \((G\times H)_{v}'\) over L and \((\ , \ ) =(\ , \ )_{\pi }(\ ,\ )_{\chi }:\Pi \otimes \Pi ^{\vee } \rightarrow L\) is a pairing, we define \((\ , \ )^{\mathrm {ord}}:= (\ , \ )_{\pi }^{\mathrm {ord}} (\ ,\ )_{\chi }\), a pairing on \(\Pi ^{\mathrm {ord}}\otimes \Pi ^{\vee , \mathrm {ord}}\).

Lemma A.3.3

Let \((\pi , \alpha )\) be a refined representation of \(G_{p}\) over \(\mathbf {C}\), with central character \(\omega \) as in Definition A.3.2. Let \(\alpha ^{\vee }=\alpha \omega ^{-1}\). Let

$$\begin{aligned} \mathrm {ad}(V_{\pi })^{++}(1)= \mathrm {Hom}\,(V_{\pi }^{-}, V_{\pi }^{+})(1). \end{aligned}$$

Fix a character \(\psi :F_{v}\rightarrow \mathbf {C}^{\times }\) of level 0, and Kirillov models of \(\iota \pi _{v}\), \(\pi _{v}^{\vee }\) with respect to \(\psi _{v}\), \(-\psi _{v}\). Let

$$\begin{aligned} f_{v}^{(\vee )}(y):= \mathbf {1}_{\mathscr {O}_{F, v}}(y) \alpha _{v}^{(\vee )} |\ |_{v}(y)\in \pi ^{\mathrm {ord}}_{v}. \end{aligned}$$
(A.3.1)

Suppose that \(( \ , \ )_{\pi ,{v}}\) is, in the Kirillov models, the pairing

$$\begin{aligned} (f, f^{\vee })_{\pi }:= \int _{F^{\times }} f(y)f^{\vee }(y) d^{\times }_{\psi }y . \end{aligned}$$
(A.3.2)

Then

$$\begin{aligned} (f, f^{\vee })^{\mathrm {ord}}_{\pi ,v}= \omega _{v}(-1)\cdot \gamma (\mathrm {ad}(V_{\pi })^{++}(1), \psi )^{-1} . \end{aligned}$$

Proof

We omit all remaining subscripts v and argue similarly to [60, Lemma 2.8]. The inner product \((f, f^{\vee })^{\mathrm {ord}}_{\pi }\) is the value at \(s=0\) of

$$\begin{aligned}&\alpha |\ |(\varpi )^{-r}Z(s+1/2,w_{r}f, \alpha ^{\vee }|\ |), \quad Z(s+1/2, w_{r}f, \alpha ^{\vee }|\ | )\\&\quad := \int _{F^{\times }} w_{r}f(y)\alpha ^{\vee }|\ |(y) |y|^{s}d^{\times }_{\psi }y. \end{aligned}$$

By the functional equation for \(\mathrm {GL}_{2}\), this equals

$$\begin{aligned}&\omega (-1)\cdot \gamma (s+1/2,\pi \otimes \alpha ^{\vee }|\ |, \psi )^{-1} \cdot \int _{p^{-r}\mathscr {O}_{F}-\{0\}} \alpha \alpha ^{\vee , -1}\omega ^{-1}|\ |^{-s}(y) d^{\times }_{\psi }y\\&\quad = \omega (-1)\cdot \gamma (s,\alpha \alpha ^{\vee }|\ |^{2}, \psi )^{-1} \cdot \gamma (s, |\ |, \psi )^{-1} \cdot \zeta _{F}(1)^{-1} \zeta _{F}(-s), \end{aligned}$$

using the fact that the domain of integration can be replaced with \(F^{\times }\), the additivity of gamma factors, and the relation \(\alpha ^{\vee }=\alpha \omega ^{-1}\). Evaluating at \(s=0\) we find \(\gamma (\mathrm {ad}(V_{\pi })^{++}(1), \psi )^{-1}\) as desired. \(\square \)

Local toric period

We compute the value of the local toric periods on the lines of interest to us. Let \(\Pi =\pi \otimes \chi \) be a refined representation of \((G\times H)'_{v}\). Let dt be a measure on \(H'_{v}\), and set as in (4.3.1)

$$\begin{aligned} {\mathrm {vol}^{\circ }(H'_{v}, dt_{})} := {\mathrm {vol}(\mathscr {O}_{E, v}^{\times }/\mathscr {O}_{F, v}^{\times }, dt_{})\over e_{v} L(1, \eta _{v})^{-1}}. \end{aligned}$$

Then for all \(f_{1}, f_{3}\in \Pi ^{\mathrm {ord}}\), \(f_{2},f_{4}\in \Pi ^{\vee , \mathrm {ord}}\) with \(f_{3}, f_{4}\ne 0\), we define

$$\begin{aligned} Q_{dt}^{\mathrm {ord}}\left( {f_{1}\otimes f_{2} \over f_{3}\otimes f_{4}}\right) : = \mu ^{+}(\mathrm{j}_{v})^{}\cdot {\mathrm {vol}^{\circ }(H'_{v}, dt_{})} \cdot {f_{1}\otimes f_{2} \over f_{3}\otimes f_{4}}, \end{aligned}$$
(A.3.3)

where \(\mathrm{j}_{v}=\) (A.1.2) and \(\mu ^{+}=\chi _{v}\cdot \alpha |\cdot |\circ N_{E_{v}/F_{v}}\) is the character giving the action of \(E_{v}^{\times }\) on \(V^{+}:=V_{\pi }^{+}\otimes \chi \).

Proposition A.3.4

Let \(\Pi =\pi \otimes \chi \) be a refined representation of \((G\times H)'_{v}\) over L, with associated Weil–Deligne representation \(V=V_{\pi |\mathrm{WD}_{E_{v}}}\otimes \chi \). Let \({\gamma _{H'}^{\mathrm {ord}} }={\gamma _{H'}^{\mathrm {ord}} }_{,v}\) be as defined in (A.2.5). Then

$$\begin{aligned} Q_{dt}^{}\left( { \gamma _{H'}^{\mathrm {ord}} f_{1}\otimes \gamma _{H'}^{\mathrm {ord}} f_{2} \over w_\mathrm{a}^{\mathrm {ord}} f_{3}\otimes f_{4}}\right) = e_{v}(V_{(\pi , \chi )}) \cdot Q_{dt}^{\mathrm {ord}}\left( {f_{1}\otimes f_{2} \over f_{3}\otimes f_{4}}\right) . \end{aligned}$$

Here

$$\begin{aligned} e_{v}(V_{(\pi , \chi )})&= \mathscr {L}(V_{(\pi , \chi )}, 0)^{-1}\cdot \iota ^{-1}\bigg ( |d|_{v}^{-1/2} \gamma (\mathrm{ad}(\iota V_{\pi }^{++})(1), \psi _{v})\\&\quad \cdot \left. \prod _{w\vert v} \gamma (\iota V^{+}_{|\mathrm{WD}_{E_{w}}}, \psi _{E_{w}})^{-1}\right) \end{aligned}$$

is defined independently of any choice of an embedding \(\iota :L\hookrightarrow \mathbf {C}\) and nontrivial character \(\psi :F_{v}\rightarrow \mathbf {C}^{\times }\).

Proof

Identify \(\chi ^{\pm 1}\) with L and assume that \(f_{i}=f_{i, \pi }f_{i, \chi }\) with \(f_{i, \chi }\) identified with 1. Fix \(\iota :L\hookrightarrow \mathbf {C}\) (omitted from the notation) and \(\mathbf {1}\ne \psi :F_{v}\rightarrow \mathbf {C}^{\times }\). Identify \(\pi \), \(\pi ^{\vee }\) with Kirillov models with respect to \(\psi \), \(-\psi \). Let \((\ , \ )=(\ , \ )_{\pi }\cdot (\, \ )_{\chi }\) be the invariant pairing on \(\Pi \otimes \Pi ^{\vee }\) such that \(( \ , \ )_{\pi } =\) (4.2.3) and \((1 \ ,1 \ )_{\chi }=1\). Assume, after a harmless extension of scalars, that \(dt=|D_{v}|^{-1/2} d^{\times }_{\psi _{E}}z/d^{\times }_{\psi }y\), which gives \(\mathrm {vol}^{\circ }(H', dt)=1\). Let \(f_{1}=f_{3}=f_{\pi }\), \(f_{2}=f_{4}=f_{\pi }^{\vee }\) with \(f_{\pi }^{(\vee )}\) as in (A.3.1).

In view of the definitions (4.2.2), (A.3.3) and of Lemma A.3.3, it suffices to show that

$$\begin{aligned} Q^{\sharp }(\gamma _{H'}^{\mathrm {ord}} f, \gamma _{H'}^{\mathrm {ord}} f^{\vee })&:= \int _{H'_{v}}(\pi (t)\gamma _{H'}^{\mathrm {ord}} f, \gamma _{H'}^{\mathrm {ord}} f^{\vee }) \chi (t) \, dt \\&= \omega _{}(-1)\cdot \mu ^{+}(\mathrm{j}_{v}) \cdot \prod _{w\vert v} \gamma (V^{+}_{|\mathrm{WD}_{E_{w}}}, \psi _{E_{w}})^{-1}. \end{aligned}$$

We denote by \(\alpha \) the refinement of \(\pi \), and we fix \(r\ge 1\) to be larger than the valuations of the conductors of \(\pi \) and of the norm of the conductor of \(\chi \).

Split case. Suppose first that \(E_{v}/F_{v}\) is split and identify \(E_{v}^{\times }=F_{v}^{\times }\times F_{v}^{\times }\) as usual. Then as in [32, Lemma 10.12] we find

$$\begin{aligned} Q^{\sharp }(\gamma _{H'}^{\mathrm {ord}} f, \gamma _{H'}^{\mathrm {ord}} f^{\vee })&= \prod _{w\vert v} \int _{E_{w}^{\times }}\alpha \chi _{w}|\ |_{w}(y_{w}) \psi _{w}(y_{w}) d^{\times }y_{w}\\&\quad \cdot \int _{E_{w^{c}}^{\times }}\alpha _{}\chi _{w^{c}}|\ |_{w^{c}}(y_{w^{c}}) \psi _{w^{c}}(-y_{w^{c}}) d^{\times }y_{w^{c}}\\&= \omega _{v}(-1)\cdot \mu ^{+}_{}(\mathrm{j}_{v}) \cdot \gamma (V^{+}_{v}, \psi _{v})^{-1}, \end{aligned}$$

where we have used Lemma A.3.1.

Nonsplit case. Now suppose that \(E_{v}=E_{w}\) is a field and drop all subscripts v, w. We abbreviate \(\mathrm{T}:=\mathrm {Tr}(\theta )\), \(\mathrm {N}:=\mathrm{Nm}(\theta )\).

We have

$$\begin{aligned} Q^{\sharp } (\gamma _{H'}^{\mathrm {ord}} f,\gamma _{H'}^{\mathrm {ord}} f^{\vee }) = \int _{H'} \alpha \alpha ^{\vee }|\ |^{2}(\varpi )^{-r} \cdot (\pi (\gamma _{r}^{-1}t \gamma _{r})f_{\pi } , f_{\pi }^{\vee })\chi (t) \, dt. \end{aligned}$$
(A.3.4)

There is a decomposition

$$\begin{aligned} H'&=H'_{1}\sqcup H'_{2}, \quad H'_{1}=\{ 1+b\theta \ |\ b\in \mathscr {O}_{F}\} , \\ H'_{2}&= \{a\mathrm{N} +\theta \ |\ a\in \mathrm{N}^{-1}\varpi \mathscr {O}_{F}\}, \end{aligned}$$

that is an isometry when \(H'_{1}\), \(H_{2}'\) are endowed with the measures \(d_{\psi }b\), \(d_{\psi }a\).

Let \(r':= r+e-1\) and let us redefine, for the purposes of this proof, \(w_{r'}:= \left( {\begin{matrix}&{}1\\ -\mathrm {N}^{-1}\varpi ^{-r}&{}\end{matrix}}\right) \). Let \(\sim _{r'} \) denote the relation in \(\mathrm {GL}(2, F)\) of equality up to right multiplication by an element of \(U_{1}^{1}(\varpi ^{r'})\), and let \(t^{(r)}:= \gamma _{r}^{-1}t\gamma _{r}\).

Contribution from \(H_{1}'\). For \(t=1+b\theta \in H_{1}'\), we have

$$\begin{aligned} t^{(r)}= \left( \begin{array}{cc}1+b\mathrm{T}&{}b\varpi ^{-r}\\ -b\mathrm {N}\varpi ^{r}&{}1\end{array}\right) \sim _{r'} \left( \begin{array}{cc}1+b\mathrm{T}+b^{2}\mathrm {N}&{}b\varpi ^{-r}\\ &{}1\end{array}\right) . \end{aligned}$$

Hence the integral over \(H_{1}'\) equals

$$\begin{aligned}&\omega ^{-1}\alpha ^{2}|\ |^{2}(\varpi )^{-r} \int _{\mathscr {O}_{F}} \int _{\mathscr {O}_{F}-\{0\}} \psi (by\varpi ^{-r}) \alpha |\ |(\mathrm{Nm}(1+b\theta )y) \alpha \omega ^{-1}|\ |(y) \\&\qquad \cdot \chi (1+b\theta ) \,d^{\times }_{\psi }y \, d_{\psi }b \\&\quad = \int _{\mathscr {O}_{F}} \int _{ \varpi ^{-r}\mathscr {O}_{F}-\{0\}} \chi \cdot \alpha |\ |\circ \mathrm{Nm} ((1+b\theta )y)\cdot \psi (by) \, d_{\psi }^{\times }y \, d_{\psi }b. \end{aligned}$$

We show that the domain of integration in y can be harmlessly extended to \(F^{\times }\), i.e. that

$$\begin{aligned} \int _{\mathscr {O}_{F}} \int _{v(y)\le -r-1} \mu ^{+}((1+b\theta ) y)\psi (by) \, d^{\times }_{\psi }y \, db \end{aligned}$$

vanishes. Consider first the contribution from \(v(b)\ge r\). On this domain, \(\mu ^{+}(1+b\theta )=1\) and integration in db yields \(\int _{\varpi ^{r}\mathscr {O}_{F}}\psi (by)\, db= \mathbf {1}_{\varpi ^{-r}\mathscr {O}_{F}}(y)\), that vanishes on \(v(y)\le -r-1\). Consider now the contribution from \(v(b)\le r-1\)

$$\begin{aligned} \int _{0\le v(b) \le r-1} \mu ^{+}(1+b\theta ) \int _{v(y)\le -r-1} \mu ^{+}(y)\psi (by) \, d^{\times }_{\psi }y db. \end{aligned}$$
(A.3.5)

Let n be the conductor of \(\mu ^{+}_{|F^{\times }}\). Then (A.3.5) vanishes if \(n=0\); otherwise only the annulus \(v(y)=-n-1\) contributes, and after a change of variable \(y'=by\) we obtain

$$\begin{aligned} \varepsilon (\mu ^{+}_{|F^{\times }}, \psi )^{-1}\cdot \int _{r- n \le v(b) \le r-1} \mu ^{+}(1+b\theta )\mu ^{+}(b)^{-1} db. \end{aligned}$$

On our domain \(\mu ^{+}(1+b\theta )=1\), and \(\int \mu ^{+}(b)^{-1}=0\) as \(\mu ^{+}_{|F^{\times }} \) is ramified.

We conclude that the contribution from \(H_{1}'\) is

$$\begin{aligned}&\int _{\mathscr {O}_{F}} \int _{F^{\times }} \mu ^{+}((1+b\theta ) y)\psi (by) \, d^{\times }_{\psi }y \, db \int _{H_{1}'} \int _{ F^{\times }} \mu ^{+}(ty)\\&\quad \cdot \psi _{E}(ty/(\theta -\theta ^{c})) \, d_{\psi }^{\times }y \, dt. \end{aligned}$$

Contribution from \(H_{2}'\). For \(t=a\mathrm{N}+\theta \in H_{2}'\), we have

$$\begin{aligned} t^{(r)} = \left( \begin{array}{cc}a\mathrm {N}+\mathrm{T}&{}\varpi ^{-r}\\ -\mathrm {N}\varpi ^{r}&{}a\mathrm {N}\end{array}\right)&= w_{r}' \left( \begin{array}{cc}1&{}-a\varpi ^{-r}\\ a\mathrm {N}+\mathrm{T}&{}\varpi ^{-r}\end{array}\right) \\&\sim _{r} w_{r'}\left( \begin{array}{cc}1+a\mathrm{T}+a^{2}\mathrm {N}&{}-a\varpi ^{-r}\\ &{}\varpi ^{-r}\end{array}\right) . \end{aligned}$$

Then the integral over \(H_{2}'\) is

where we have observed that \(w_{r'}f^{\vee }\) vanishes outside \(\mathscr {O}_{F}\), and that \(\psi (-ay)=1\) for \(y\in \mathscr {O}_{F}\). Applying first the same argument as in the proof of Lemma A.3.3, then Lemma A.3.1, this equals

$$\begin{aligned}&\gamma (\mathrm {ad}(V_{\pi })^{++}(1),- \psi )^{-1} \cdot \int _{\mathrm {N}^{-1}\varpi \mathscr {O}_{F}} \alpha |\ | \circ \mathrm{Nm} \cdot \chi (a\mathrm {N}+\theta ) d_{\psi }a \\&\quad = \int _{F^{\times }}\mu ^{+}(y) \psi (y) \, d^{\times }_{\psi }y \cdot \int _{\mathrm {N}^{-1}\varpi \mathscr {O}_{F}} \mu ^{+}(a\mathrm {N}+\theta ) d_{\psi }a\\&\quad = \int _{H_{2}'} \int _{F^{\times }} \mu ^{+}(ty) \psi _{E}(ty/(\theta -\theta ^{c})) \, d_{\psi }^{\times }y. \end{aligned}$$

Conclusion. Summing up the two contributions to (A.3.4) yields

$$\begin{aligned} \mu ^{+}(\theta ^{c}-\theta ) \cdot \int _{H'} \int _{F^{\times }} \mu ^{+}(u) \psi _{E}(u) \, d^{\times }u = \omega _{}(-1)\cdot \mu ^{+}(\mathrm{j})\cdot \gamma (\mu ^{+}, \psi _{E})^{-1}, \end{aligned}$$

as desired. \(\square \)

Pairings at infinity

Fix a place \(v\vert p\) of F.

Models for algebraic representations and pairings

Suppose that W is the representation (A.2.2) of \((G\times H)_{v, \infty }'\) over \(L{\mathop {\hookleftarrow }\limits ^{\sigma }} E\). We identify W with the space of homogeneous polynomials p(xy) of degree \(k-2\) in L[xy], where x and y are considered as the components of a column (respectively row) vector if W is viewed as a right (respectively left) representation. In those two cases, the action is respectively:

$$\begin{aligned} p. (g, h)(x, y)&= \det (g)^{w-k+2\over 2}\sigma (h)^{l-w\over 2}\sigma ^{c}(h)^{-l-w \over 2}\cdot p(g(x, y)^\mathrm{T}) \nonumber \\ (g, h).p(x, y)&= \det (g)^{w-k+2\over 2}\sigma (h)^{l-w\over 2}\sigma ^{c}(h)^{-l-w \over 2}\cdot p((x, y)g) . \end{aligned}$$
(A.4.1)

In either case, we fix the invariant pairing

$$\begin{aligned} (x^{k-2-a}y^{a}, x^{a'}y^{k-2-a'}) = (-1)^{a} {k-2 \atopwithdelims ()a}^{-1}\delta _{a, a'}. \end{aligned}$$
(A.4.2)

Lemma A.4.1

Let \(W=\) (A.2.2), viewed as a left representation of \(G_{v, \infty }\) only. Let \(w_\mathrm{a}^{\mathrm {ord}}:W^{N}\rightarrow W_{N}\) be the map denoted by \(w_{\mathrm{a}, v, \infty }^{\mathrm {ord}}\) of (A.2.1). Fix the models and pairing described above. Then \(W^{N}\) is spanned by \(x^{k-2}\) and \(W_{N}\) is spanned by the image of \(y^{k-2}\), and

$$\begin{aligned} (w_\mathrm{a}^{\mathrm {ord}} (x^{k-2}), x^{k-2})=1. \end{aligned}$$

The map \(\gamma _{H'}^{\mathrm {ord}} \) is unitary on algebraic representations

We start with a lemma completing the proof of Proposition 6.3.2.

Suppose that \(M_{p}=M_{p, 0}\otimes W_{p}\) is a decomposition of a locally algebraic coadmissible right \((G\times H)'_{p}\)-representation over L, into the product of a smooth and an irreducible algebraic representation, respectively. Let \(W^{\vee }_{\infty }\) be the dual representation to \(W_{p}\), viewed as a right representation of \((G\times H)'_{\infty }\). Assume that L is a p-adic field and that the \((G\times H)'\)-module \(M_{p}\otimes W^{\vee }_{\infty }\) is p-adic coadmissible. Then the operator \(\gamma _{H'}^{\mathrm {ord}} \) on it (whose definition of Proposition A.2.4 extends verbatim to the case where \(M_{p}\) is only locally algebraic) decomposes as

$$\begin{aligned} \gamma _{H'}^{\mathrm {ord}} = \lim _{r\rightarrow \infty } (p^{r[F:\mathbf {Q}]}\cdot \gamma _{r, p}\mathrm {U}_{p}^{-r})\otimes \gamma _{r, p}\mathrm {U}_{p}^{-r}\otimes c(W)^{-1} \gamma _{0, \infty }^{\iota }. \end{aligned}$$

according to the decomposition \(M_{p}\otimes W^{\vee }_{\infty }=M_{p, 0}\otimes W_{p}\otimes W^{\vee }_{\infty }\)

Lemma A.4.2

In relation to the situation just described, the operator

$$\begin{aligned} {}^\mathrm{alg}\gamma _{H'}^{\mathrm {ord}} := \lim _{r\rightarrow \infty } \gamma _{r, p}\mathrm {U}_{p}^{-r}\otimes c(W)^{-1}\gamma _{0, \infty }^{\iota }:W^{H'}\otimes W^{\vee , H'} \rightarrow W^{N}\otimes W_{N} \end{aligned}$$

is unitary. That is, for any invariant pairing \((\ , \ )\) on \(W\otimes W^{\vee }\) and \(\xi \in W^{H'}\), \(\xi ^{\vee }\in W^{\vee , H'}\), the images of \(\xi \otimes \xi ^{\vee }\) and \({}^\mathrm{alg}\gamma _{H'}^{\mathrm {ord}} (\xi \otimes \xi ^{\vee })\) under the pairings induced by \((\ , \ )\) coincide.

Proof

We may fix a place \(v\vert p\), and consider the factor representations \(W_{v}\otimes W_{v,\infty }^{\vee }\) of \((G\times H)'_{v} \times (G\times H)'_{v, \infty }\). After extension of scalars, we may decompose \(W_{v}=\bigotimes _{\sigma :F\rightarrow \overline{Q}_{p}}W_{v}^{\sigma }\) where each \(W_{v}^{\sigma }\) is one of the representations (A.2.2) for suitable integers wkl. Thus we are reduced to proving the unitarity of the relevant component of \({}^\mathrm{alg}\gamma _{H'}^{\mathrm {ord}} \) on the representation \(W_{v}^{\sigma }\otimes W_{v, \infty }^{\vee , \sigma }\). We omit all subscripts.

Split case. Suppose first that v splits in E. Then \(W^{H'}= Lx^{(k-2-l)/2} y^{(k-2+l)/2}\), and if

$$\begin{aligned} \xi := x^{(k-2-l)/2} y^{(k-2+l)/2} \end{aligned}$$

then

$$\begin{aligned} \xi ^{\vee }:= (-1)^{(k-2+l)/2} {k-2 \atopwithdelims (){(k-2-l)/2}} x^{(k-2+l)/2}y^{ (k-2-l)/2} \end{aligned}$$

satisfies \(( \xi , \xi ^{\vee })=1\). We have

$$\begin{aligned} \xi {\gamma _{H'}^{\mathrm {ord}} }_{, p}:=\lim _{r\rightarrow \infty } \xi \gamma _{r, p}\mathrm {U}_{p}^{-r} = y^{k-2}, \end{aligned}$$

and

$$\begin{aligned} \xi ^{\vee }{\gamma _{H'}^{\mathrm {ord}} }_{,\infty }&= (-1)^{(k-2+l)/2} c(W)^{-1} {k-2 \atopwithdelims (){(k-2-l)/2}} x^{(k-2+l)/2}\\&\quad \cdot (-x+y)^{ (k-2-l)/2} \end{aligned}$$

projects into \(W^{\vee }_{N} {\mathop {\leftarrow }\limits ^{\cong }} Lx^{k-2}\) to

$$\begin{aligned} \xi ^{\vee }{\gamma _{H'}^{\mathrm {ord}} }_{,\infty } = (-1)^{k-2} c(W)^{-1} {k-2 \atopwithdelims (){(k-2-l)/2}} x^{k-2}. \end{aligned}$$

Hence

$$\begin{aligned} ( \xi {\gamma _{H'}^{\mathrm {ord}} }_{, p}, \xi ^{\vee }{\gamma _{H'}^{\mathrm {ord}} }_{,\infty })= c(W)^{-1} {k-2 \atopwithdelims (){(k-2-l)/2}} =1. \end{aligned}$$

Nonsplit case. Suppose now that v does not split in E. Let \(z:= x+\theta ^{c} y\), \(\overline{z}:=x+\theta y\). Then \(W^{H'} = L z^{(k-2-l)/2}\overline{z}^{ (k-2+l)/2}\), and if

$$\begin{aligned} \xi&:=z^{(k-2-l)/2}\overline{z}^{ (k-2+l)/2}\\&= x^{(k-2-l)/2}y^{ (k-2+l)/2} \cdot \left( \begin{array}{cc}1&{}\theta ^{c}\\ 1&{}\theta \end{array}\right) (-\mathrm{j})^{(w+k-2)/2} \in W^{H'} \end{aligned}$$

then

$$\begin{aligned} \xi ^{\vee }&:=(-1)^{(k-2+l)/2} {k-2 \atopwithdelims (){(k-2-l)/2}} x^{(k-2+l)/2}y^{ (k-2-l)/2} \\&\quad \cdot \left( \begin{array}{cc}1&{}\theta ^{c}\\ 1&{}\theta \end{array}\right) (-\mathrm{j})^{(-w-k+2)/2} \in W^{\vee , H'} \end{aligned}$$

satisfies \(( \xi , \xi ^{\vee })=1\). We have

$$\begin{aligned} \xi {\gamma _{H'}^{\mathrm {ord}} }_{, p}= \mathrm{N}^{(w-k+2)/2} \theta ^{c,(k-2-l)/2} \theta ^{(k-2+l)/2} y^{k-2} , \end{aligned}$$

and

$$\begin{aligned} \xi ^{\vee }{\gamma _{H'}^{\mathrm {ord}} }_{,\infty }&= (-1)^{(k-2-l)/2} (-\mathrm{j})^{-(w+k-2)/2} c(W)^{-1}\\&\quad \cdot {k-2 \atopwithdelims (){(k-2-l)/2}} x^{(k-2+l)/2}y^{ (k-2-l)/2} \left( \begin{array}{cc}1&{}\mathrm {N}^{-1}\theta ^{c}\\ 1&{}\mathrm {N}^{-1}\theta \end{array}\right) \end{aligned}$$

projects into \(W^{\vee }_{N} {\mathop {\leftarrow }\limits ^{\cong }} Lx^{k-2}\) to

$$\begin{aligned} \xi ^{\vee }{\gamma _{H'}^{\mathrm {ord}} }_{,\infty }&= (-1)^{(k-2-l)/2} (-\mathrm{j})^{-w-k+2} c(W)^{-1} \\&\quad \cdot {k-2 \atopwithdelims (){(k-2-l)/2}} \mathrm {N}^{(w+k-2)/2} x^{k-2}. \end{aligned}$$

Then

$$\begin{aligned} ( \xi {\gamma _{H'}^{\mathrm {ord}} }_{, p}, \xi ^{\vee }{\gamma _{H'}^{\mathrm {ord}} }_{,\infty })&= (-\mathrm{j})^{-w-k+2 } \theta ^{c,(k-2-l)/2} \theta ^{(k-2+l)/2}\\&\quad \cdot {k-2 \atopwithdelims (){(k-2-l)/2}} c(W)^{-1}=1 . \end{aligned}$$

\(\square \)

Algebraic toric period

Let \(W =W_{G} \otimes W_{H}\) be an algebraic representation of \((G\times H)'_{v,\infty }\) over L. For any \(\iota :L\hookrightarrow \mathbf {C}\), let \(\iota V_{ }\) (respectively \(\iota V_{G}\)) be the Hodge structure associated with W (respectively \(W_{G}\)), and letFootnote 32

$$\begin{aligned} \mathscr {L}(V_{(W_{G}, W_{H})}, 0):= \iota ^{-1}\left( {\pi ^{-[F_{v}:\mathbf {Q}_{p}]}L(\iota V_{}, 0) \over L(\mathrm {ad}(\iota V_{\mathrm {G}), \infty }), 1)}\right) . \end{aligned}$$

Let dt be a ‘measure’ on \(H'_{v, \infty }\), by which we simply mean a value \(\mathrm {vol}(H'_{v, \infty }, dt)\) similarly to Sect. 1.2.6, and set as in (4.3.1)

$$\begin{aligned} {\mathrm {vol}^{\circ }(H'_{v}, dt_{})} :=2^{-[F_{v}:\mathbf {Q}]} \mathrm {vol}(H'_{v, \infty }, dt). \end{aligned}$$

Let \((\ , \ )=(\ , \ )_{W_{G}}\cdot (\, \ )_{W_{H}}\) be a nondegenerate invariant pairing on \(W. \otimes W^{\vee }\).

Then for all \(f_{1}, f_{3}\in W\), \(f_{2},f_{4}\in W^{\vee }\) with \((f_{3}, f_{4})\ne 0\), we define

$$\begin{aligned} Q_{dt}^{}\left( {f_{1}\otimes f_{2} \over f_{3}\otimes f_{4}}\right) : = \mathscr {L}(V_{(W_{G}, W_{H})}, 0)^{-1}\cdot \mathrm {vol}^{}(H'_{v, \infty }, dt) \cdot {( \mathrm{p}_{H'} (f_{1}), \mathrm{p}_{H'} (f_{2})) \over (f_{3}, f_{4})}, \end{aligned}$$
(A.4.3)

where \(\mathrm{p}_{H'}\) denotes the idempotent projector onto \(H'_{v, \infty }\)-invariants.

Let \(\sigma _{W_{G}}:F_{v}^{\times }\rightarrow L^{\times }\) be the character giving the action of \(\left( \begin{array}{cc}F_{v}^{\times }&{}\\ &{}1\end{array}\right) \) on \(W_{G}^{N}\), let \(\chi :E_{v}^{\times }\rightarrow L^{\times } \) be the algebraic character attached to \(W_{H}\), and let

$$\begin{aligned} \mu ^{+}=\chi \cdot \sigma _{W_{G}}\circ N_{E_{v}/F_{v}}. \end{aligned}$$

Let \(\mathrm{j}_{v}=\) (A.1.2). Then for all \(f_{1}, f_{3}\in W^{N}:=W_{G}^{N}\otimes W_{H}\), \(f_{2}, f_{4}\in W^{\vee , N}\) with \(f_{3}, f_{4}\ne 0\), we define

$$\begin{aligned} Q_{dt}^{\mathrm {ord}}\left( {f_{1}\otimes f_{2} \over f_{3}\otimes f_{4}}\right) := \mu ^{+}(\mathrm{j}_{v})^{}\cdot {\mathrm {vol}^{\circ }(H'_{v}, dt_{})} \cdot {f_{1}\otimes f_{2} \over f_{3}\otimes f_{4}}. \end{aligned}$$
(A.4.4)

Proposition A.4.3

Let W be a representation of \((G\times H)'_{v, \infty }\) over L. Let \({\gamma _{H'}^{\mathrm {ord}} }={\gamma _{H'}^{\mathrm {ord}} }_{,v, \infty }\) be as defined in (A.2.5), and let \(w_\mathrm{a}^{\mathrm {ord}}=w_{\mathrm{a}, v, \infty }^{\mathrm {ord}}\) be as defined in (A.2.1). Then for all \(f_{1}, f_{3}\in W^{N}\), \(f_{2}, f_{4}\in W^{\vee , N}\) with \(f_{3}, f_{4}\ne 0\),

$$\begin{aligned} Q_{dt}^{}\left( { \gamma _{H'}^{\mathrm {ord}} f_{1}\otimes \gamma _{H'}^{\mathrm {ord}} f_{2} \over w_\mathrm{a}^{\mathrm {ord}} f_{3}\otimes f_{4}}\right) = \dim W \cdot Q_{dt}^{\mathrm {ord}}\left( {f_{1}\otimes f_{2} \over f_{3}\otimes f_{4}}\right) . \end{aligned}$$

Proof

After possibly extending scalars we may assume that L splits E and pick an extensions of each \(\sigma :F\hookrightarrow L\) to a \(\sigma :E\hookrightarrow L\). We then have \(W=\bigotimes _{\sigma :F\hookrightarrow L}W_{\sigma } \) with \(W_{\sigma }=\) (A.2.2) for suitable integers \(w, k_{\sigma },l_{\sigma }\), and analogously \(\mu ^{+}(t)= \prod _{\sigma :F\hookrightarrow L}\mu _{\sigma }^{+}\) with

$$\begin{aligned} \mu _{\sigma }^{+}(t)=\sigma (t)^{(k_{\sigma }-2+l_{\sigma })/2} \sigma (t^{c})^{(k_{\sigma }-2-l_{\sigma })/2}, \quad \mu _{\sigma }^{+}(\mathrm{j})=(-1)^{(k_{\sigma }-2-l_{\sigma })/2}\cdot \mathrm{j}_{v}^{k_{\sigma }-2}. \end{aligned}$$
(A.4.5)

If v splits in E, this simplifies to \(\mu _{\sigma }^{+}(\mathrm{j}) = (-1)^{(k_{\sigma }-2+l_{\sigma })/2}\).

Moreover, \(\mathscr {L}(V, 0)= \prod _{\sigma }\mathscr {L}(V_{\sigma }, 0)\) with

$$\begin{aligned} \mathscr {L}(V_{\sigma }, 0)= {\pi ^{-1}\Gamma _{\mathbf {C}}({k_{\sigma }+l_{\sigma }\over 2})\Gamma _{\mathbf {C}}({k_{\sigma }-l_{\sigma }\over 2}) \over \Gamma _{\mathbf {C}}(k_{\sigma }) \Gamma _{\mathbf {R}}(2)} = {2\over k_{\sigma }-1} \cdot {k_{\sigma }-2 \atopwithdelims (){k_{\sigma }-2+l_{\sigma }\over 2 }}^{-1}. \end{aligned}$$

Fix a \(\sigma :F\hookrightarrow L\) for the rest of this proof, work with \(W_{\sigma } \) only, and we drop \(\sigma \) from the notation. We may assume that \(f:=f_{1}\), \(f^{\vee }:= f_{2}\) both equal \(x^{k-2}\) in the models (A.4.1), and that \(\mathrm {vol}(H', dt)=1\). By the definitions above and Lemma A.4.1, we then need to prove that

$$\begin{aligned} Q(\gamma _{H'}^{\mathrm {ord}} f_, \gamma _{H'}^{\mathrm {ord}} f^{\vee })&:= {k-1\over 2 }\cdot {k_{}-2 \atopwithdelims (){k-2+l_{}\over 2 }} \cdot ( \mathrm{p}_{H'}( \gamma _{H'}^{\mathrm {ord}} f_{1}),\\ \mathrm{p}_{H'}(\gamma _{H'}^{\mathrm {ord}} f_{2}) )&={k-1 \over 2} \cdot \mu ^{+}(\mathrm{j}). \end{aligned}$$

Recall in what follows that \(\gamma _{H'}^{\mathrm {ord}} \) contains the factor \(c(W) =\) (A.2.3).

Split case. Suppose first that v splits in E. Then \(W^{H'} = L x^{(k-2-l)/ 2}y^{(k-2+l)/ 2 }\), and \(c(W)^{-1}\gamma _{0}^{\iota } f= c(W)^{-1} (x-y)^{k-2}\) projects to

$$\begin{aligned} \gamma _{H'}^{\mathrm {ord}} f =(-1)^{ (k-2+l )/ 2} c(W)^{-1} {k-2 \atopwithdelims ()(k-2-l)/2} x^{(k-2-l)/ 2}y^{(k-2+l )/ 2 } \in W^{H'}. \end{aligned}$$

It follows that

$$\begin{aligned} Q^{}(\gamma _{H'}^{\mathrm {ord}} f_, \gamma _{H'}^{\mathrm {ord}} f^{\vee })&= { k_{}-1\over 2} \cdot {k_{}-2 \atopwithdelims ()( {k_{}-2+l)/ 2 }}^{2}\cdot (-1)^{ {k-2+l\over 2}} \cdot c(W)^{-1}c(W^{\vee })^{-1}\\&= {k-1\over 2} \cdot \mu ^{+}(\mathrm{j}) . \end{aligned}$$

Nonsplit case. Suppose now that v is nonsplit in E. Let \(z:= x-\theta ^{c,-1} y\), \(\overline{z}:=x-\theta ^{ -1}y\), then \(W^{H'}=L z^{(k-2-l)/ 2}\overline{z}^{(k-2+l)/ 2 }\) and

$$\begin{aligned} \gamma _{0}^{\iota } f=c(W)^{-1} \mathrm {N}^{(w+k-2)/2} x^{k-2} =c(W)^{-1} \mathrm {N}^{(w+k-2)/2} \mathrm{j}^{2-k} (\theta ^{c}z-\theta \overline{z})^{k-2} \end{aligned}$$

projects to

$$\begin{aligned} \gamma _{H'}^{\mathrm {ord}} f&= c(W^{\vee })^{-1}{k-2 \atopwithdelims (){k-2-l\over 2}}^{} (-1)^{(k-2+l)/ 2}\\&\quad \cdot \mathrm {N}^{(w+k-2)/2} \mathrm{j}^{2-k} \theta ^{c, (k-l-2)/2} \theta ^{(k+l-2)/2}\cdot z^{(k-2-l)/ 2}\overline{z}^{(k-2+l)/ 2 }\\&= c(W^{\vee })^{-1}{k-2 \atopwithdelims (){k-2-l\over 2}}^{} (-1)^{(k-2+l)/ 2}\\&\quad \cdot \mathrm{j}^{(-w-k+2)/2} \theta ^{c, (k-l-2)/2} \theta ^{(k+l-2)/2}\\&\quad \cdot \left( \begin{array}{cc}1&{}1\\ -\theta ^{c,-1}&{}-\theta ^{-1}\end{array}\right) \cdot x^{(k-2-l)/ 2} {y}^{(k-2+l)/ 2 } . \end{aligned}$$

By the invariance of the pairing,

$$\begin{aligned} Q(\gamma _{H'}^{\mathrm {ord}} f, \gamma _{H'}^{\mathrm {ord}} f^{\vee })&= (-1)^{(k-2+l)/2} c(W)^{-1}c(W^{\vee })^{-1} {k-2 \atopwithdelims (){k-2-l\over 2}} (-\mathrm{j})^{2-k} \mathrm {N}^{k-2}\\&= (-1)^{k-2}{k-2 \atopwithdelims (){k-2-l\over 2}}^{-1} \mu ^{+}(\mathrm{j}) \end{aligned}$$

so that again

$$\begin{aligned} Q^{}(\gamma _{H'}^{\mathrm {ord}} f_, \gamma _{H'}^{\mathrm {ord}} f^{\vee })= \dim W\cdot \mu ^{+}(\mathrm{j}) . \end{aligned}$$

\(\square \)

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Disegni, D. The universal p-adic Gross–Zagier formula. Invent. math. (2022). https://doi.org/10.1007/s00222-022-01133-w

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