1.1 Heegner points and multiplicity one

Let $A$ be a simple abelian variety of $\mathbf{GL}_{2}$ -type over a totally real field $F$ ; recall that this means that $M:=\operatorname{End}^{0}(A)$ is a field of dimension equal to the dimension of $A$ . One knows how to systematically construct points on $A$ when $A$ admits parametrisations by Shimura curves in the following sense. Let $\mathbf{B}$ be a quaternion algebra over the adèle ring $\mathbf{A}=\mathbf{A}_{F}$ of $F$ , and assume that $\mathbf{B}$ is incoherent, i.e. that its ramification set $\unicode[STIX]{x1D6F4}_{\mathbf{B}}$ has odd cardinality. We further assume that $\unicode[STIX]{x1D6F4}_{\mathbf{B}}$ contains all the archimedean places of $F$ . Under these conditions there is a tower of Shimura curves $\{X_{U}\}$ over $F$ indexed by the open compact subgroups $U\subset \mathbf{B}^{\infty \times }$ ; let $X=X(\mathbf{B}):=\mathop{\varprojlim }\nolimits_{U}X_{U}$ . For each $U$ , there is a canonical Hodge class $\unicode[STIX]{x1D709}_{U}\in \operatorname{Pic}(X_{U})_{\mathbf{Q}}$ having degree $1$ in each connected component, inducing a compatible family $\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D709}}=(\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D709},U})_{U}$ of quasi-embeddingsFootnote ¹   $\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D709},U}:X_{U}{\hookrightarrow}J_{U}:=\operatorname{Alb}X_{U}$ . We write $J:=\varprojlim J_{U}$ . The $M$ -vector space

$$\begin{eqnarray}\unicode[STIX]{x1D70B}=\unicode[STIX]{x1D70B}_{A}=\unicode[STIX]{x1D70B}_{A}(\mathbf{B}):=\mathop{\varinjlim }\nolimits_{U}\operatorname{Hom}^{0}(J_{U},A)\end{eqnarray}$$

is either zero or a smooth irreducible admissible representation of $\mathbf{B}^{\infty \times }$ . It comes with a natural stable lattice $\unicode[STIX]{x1D70B}_{\mathbf{Z}}\subset \unicode[STIX]{x1D70B}$ , and its central character

$$\begin{eqnarray}\unicode[STIX]{x1D714}_{A}:F^{\times }\backslash \mathbf{A}^{\times }\rightarrow M^{\times }\end{eqnarray}$$

corresponds, up to twist by the cyclotomic character, to the determinant of the Tate module under the class field theory isomorphism. When $\unicode[STIX]{x1D70B}_{A}$ is non-zero, $A$ is said to be parametrised by $X(\mathbf{B})$ . Under the conditions we are going to impose on $A$ , the existence of such a parametrisation, for a suitable choice of $\mathbf{B}$ (see below), is equivalent to the modularity conjecture. Recall that the latter asserts the existence of a unique $M$ -rational (Definition 1.2.1 below) automorphic representation $\unicode[STIX]{x1D70E}_{A}$ of weight  $2$ such that there is an equality of $L$ -functions $L(A,s+1/2)=L(s,\unicode[STIX]{x1D70E}_{A})$ . The conjecture is known to be true for ‘almost all’ elliptic curves $A$ (see [Reference Le HungLeH14]), and when $A_{\overline{F}}$ has complex multiplication.

Heegner points. Let $A$ be parametrised by $X(\mathbf{B})$ and let $E$ be a CM extension of $F$ admitting an $\mathbf{A}^{\infty }$ -embedding $E_{\mathbf{A}^{\infty }}{\hookrightarrow}\mathbf{B}^{\infty }$ , which we fix; we denote by $\unicode[STIX]{x1D702}$ the associated quadratic character and by $D_{E}$ its absolute discriminant. Then $E^{\times }$ acts on $X$ and by the theory of complex multiplication each closed point of the subscheme $X^{E^{\times }}$ is defined over $E^{\text{ab}}$ , the maximal abelian extension of $E$ . We fix one such CM point $P$ . Let $L(\unicode[STIX]{x1D712})$ be a field extension of $M$ and let

$$\begin{eqnarray}\unicode[STIX]{x1D712}:E^{\times }\backslash E_{\boldsymbol{ A}^{\infty }}^{\times }\rightarrow L(\unicode[STIX]{x1D712})^{\times }\end{eqnarray}$$

be a finite-order Hecke character such that

$$\begin{eqnarray}\unicode[STIX]{x1D714}_{A}\cdot \unicode[STIX]{x1D712}|_{\mathbf{A}^{\infty ,\times }}=1.\end{eqnarray}$$

We can view $\unicode[STIX]{x1D712}$ as a character of $\mathscr{G}_{E}:=\operatorname{Gal}(\overline{E}/E)$ via the reciprocity map of class field theory (normalised, in this work, by sending uniformisers to geometric Frobenii). For each $f\in \unicode[STIX]{x1D70B}_{A}$ , we then have a Heegner point

$$\begin{eqnarray}P(f,\unicode[STIX]{x1D712})=\int _{\operatorname{ Gal}(E^{\text{ab}}/E)}f(\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D709}}(P)^{\unicode[STIX]{x1D70F}})\otimes \unicode[STIX]{x1D712}(\unicode[STIX]{x1D70F})\,d\unicode[STIX]{x1D70F}\in A(\unicode[STIX]{x1D712}).\end{eqnarray}$$

Here the integration uses the Haar measure of total volume  $1$ , and

$$\begin{eqnarray}A(\unicode[STIX]{x1D712}):=(A(E^{\text{ab}})\otimes _{M}L(\unicode[STIX]{x1D712})_{\unicode[STIX]{x1D712}})^{\operatorname{Gal}(E^{\text{ab}}/E)},\end{eqnarray}$$

where $L(\unicode[STIX]{x1D712})_{\unicode[STIX]{x1D712}}$ denotes the one-dimensional Galois module $L(\unicode[STIX]{x1D712})$ with action given by  $\unicode[STIX]{x1D712}$ . The functional $f\mapsto P(f,\unicode[STIX]{x1D712})$ defines an element of

$$\begin{eqnarray}\operatorname{Hom}_{E_{\mathbf{A}^{\infty }}^{\times }}(\unicode[STIX]{x1D70B}\otimes \unicode[STIX]{x1D712},L(\unicode[STIX]{x1D712}))\otimes _{L(\unicode[STIX]{x1D712})}A(\unicode[STIX]{x1D712}).\end{eqnarray}$$

A foundational local result of Tunnell and Saito [Reference TunnellTun83, Reference SaitoSai93] asserts that, for any irreducible representation $\unicode[STIX]{x1D70B}$ of $\mathbf{B}^{\times }$ , the $L(\unicode[STIX]{x1D712})$ -dimension of

$$\begin{eqnarray}\text{H}(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D712})=\operatorname{Hom}_{E_{\mathbf{A}^{\infty }}^{\times }}(\unicode[STIX]{x1D70B}\otimes \unicode[STIX]{x1D712}L(\unicode[STIX]{x1D712}))\end{eqnarray}$$

is either zero or one. It is one exactly when, for all places $v$ of $F$ , the local condition

(1.1.1) $$\begin{eqnarray}\unicode[STIX]{x1D700}(1/2,\unicode[STIX]{x1D70B}_{E,v}\otimes \unicode[STIX]{x1D712}_{v})=\unicode[STIX]{x1D712}_{v}(-1)\unicode[STIX]{x1D702}_{v}(-1)\unicode[STIX]{x1D700}(\mathbf{B}_{v})\end{eqnarray}$$

holds, where $\unicode[STIX]{x1D70B}_{E}$ is the base-change of $\unicode[STIX]{x1D70B}$ to $E$ , $\unicode[STIX]{x1D702}=\unicode[STIX]{x1D702}_{E/F}$ is the quadratic character of $\mathbf{A}^{\times }$ associated to $E$ , and $\unicode[STIX]{x1D700}(\mathbf{B}_{v})=+1$ if $\mathbf{B}_{v}$ is split and $-1$ if $\mathbf{B}_{v}$ is ramified. In this case, denoting by $\unicode[STIX]{x1D70B}^{\vee }$ the $M$ -contragredient representation, there is an explicit generator

$$\begin{eqnarray}Q=\mathop{\prod }_{v\nmid \infty }Q_{v}\in \text{H}(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D712})\otimes _{L(\unicode[STIX]{x1D712})}\text{H}(\unicode[STIX]{x1D70B}^{\vee },\unicode[STIX]{x1D712}^{-1})\end{eqnarray}$$

defined by integration of local matrix coefficients

(1.1.2) $$\begin{eqnarray}Q_{v}(f_{1,v},f_{2,v},\unicode[STIX]{x1D712})=\frac{L(1,\unicode[STIX]{x1D702}_{v})L(1,\unicode[STIX]{x1D70B}_{v},\operatorname{ad})}{\unicode[STIX]{x1D701}_{F,v}(2)L(1/2,\unicode[STIX]{x1D70B}_{E,v}\otimes \unicode[STIX]{x1D712}_{v})}\int _{E_{v}^{\times }/F_{v}^{\times }}\unicode[STIX]{x1D712}_{v}(t_{v})(\unicode[STIX]{x1D70B}(t_{v})f_{1,v},f_{2,v})_{v}\,dt_{v}\end{eqnarray}$$

for a decomposition $(\cdot ,\cdot )=\bigotimes _{v}(\cdot ,\cdot )_{v}$ of the pairing $\unicode[STIX]{x1D70B}\otimes _{M}\unicode[STIX]{x1D70B}^{\vee }\rightarrow M$ , and Haar measures $dt_{v}$ assigning to $\mathscr{O}_{E,v}^{\times }/\mathscr{O}_{F_{v}}^{\times }$ the volume $1$ if $v$ is unramified in $E$ and $2$ if $v$ ramifies in $E$ . The normalisation is such that given $f_{1}$ , $f_{2}$ , all but finitely many terms in the product are equal to  $1$ . The pairings $Q_{v}$ in fact depend on the choice of decomposition, which in general needs an extension of scalars; the global pairing is defined over $M$ and independent of choices.

Note that the local root numbers are unchanged if one replaces $\unicode[STIX]{x1D70B}$ by its Jacquet–Langlands transfer to another quaternion algebra, and that when $\unicode[STIX]{x1D70B}=\unicode[STIX]{x1D70B}_{A}$ they equal the local root numbers $\unicode[STIX]{x1D700}(A_{E,v},\unicode[STIX]{x1D712}_{v})$ of the motive $H_{1}(A\times _{\operatorname{Spec}F}\operatorname{Spec}E)\otimes _{M}\unicode[STIX]{x1D712}$ [Reference GrossGro91]. In this way one can view the local conditions

$$\begin{eqnarray}\unicode[STIX]{x1D700}(A_{E,v},\unicode[STIX]{x1D712}_{v})=\unicode[STIX]{x1D712}_{v}(-1)\unicode[STIX]{x1D702}_{v}(-1)\unicode[STIX]{x1D700}(\mathbf{B}_{v})\end{eqnarray}$$

as determining a unique totally definite quaternion algebra $\mathbf{B}\supset E_{\mathbf{A}}$ over $\mathbf{A}$ , which is incoherent precisely when the global root number $\unicode[STIX]{x1D700}(A_{E},\unicode[STIX]{x1D712})=-1$ . In this case, $A$ is parametrised by $X(\mathbf{B})$ in the sense described above if and only if $A$ is modular in the sense that the Galois representation afforded by its Tate module is attached to a cuspidal automorphic representation of $\mathbf{GL}_{2}(\mathbf{A}_{F})$ of parallel weight $2$ . We assume this to be the case.

Gross–Zagier formulas. There is a natural identification $\unicode[STIX]{x1D70B}^{\vee }=\unicode[STIX]{x1D70B}_{A^{\vee }}$ , where $A^{\vee }$ is the dual abelian variety (explicitly, this is induced by the perfect $M=\operatorname{End}^{0}(A)$ -valued pairing $f_{1,U}\otimes f_{2,U}\mapsto \operatorname{vol}(X_{U})^{-1}f_{1,U}\circ f_{2}^{\vee }$ using the canonical autoduality of $J_{U}$ for any sufficiently small $U$ ; the normalising factor $\operatorname{vol}(X_{U})\in \mathbf{Q}^{\times }$ is the hyperbolic volume of $X_{U}(\mathbf{C}_{\unicode[STIX]{x1D70F}})$ for any $\unicode[STIX]{x1D70F}:F{\hookrightarrow}\mathbf{C}$ ; see [Reference Yuan, Zhang and ZhangYZZ12, §1.2.2]). Similarly to the above, we have a Heegner point functional $P^{\vee }(\cdot ,\unicode[STIX]{x1D712}^{-1})\in \text{H}(\unicode[STIX]{x1D70B}^{\vee },\unicode[STIX]{x1D712}^{-1})\otimes _{L}A^{\vee }(\unicode[STIX]{x1D712}^{-1})$ . Then the multiplicity-one result of Tunnell and Saito implies that for each bilinear pairing

$$\begin{eqnarray}\langle \,,\rangle :A(\unicode[STIX]{x1D712})\otimes _{L(\unicode[STIX]{x1D712})}A^{\vee }(\unicode[STIX]{x1D712}^{-1})\rightarrow V\end{eqnarray}$$

with values in an $L(\unicode[STIX]{x1D712})$ -vector space $V$ , there is an element $\mathscr{L}\in V$ such that

$$\begin{eqnarray}\langle P(f_{1},\unicode[STIX]{x1D712}),P(f_{2},\unicode[STIX]{x1D712}^{-1})\rangle =\mathscr{L}\cdot Q(f_{1},f_{2},\unicode[STIX]{x1D712})\end{eqnarray}$$

for all $f_{1}\in \unicode[STIX]{x1D70B}$ , $f_{2}\in \unicode[STIX]{x1D70B}^{\vee }$ .

In this framework, we may call the ‘Gross–Zagier formula’ a formula for $\mathscr{L}$ in terms of $L$ -functions. When $\langle \,,\rangle$ is the Néron–Tate height pairing valued in $\mathbf{C}\stackrel{\unicode[STIX]{x1D704}}{{\hookleftarrow}}M$ for an archimedean place $\unicode[STIX]{x1D704}$ , the generalisation by Yuan–Zhang–Zhang [Reference Yuan, Zhang and ZhangYZZ12] of the classical Gross–Zagier formula [Reference Gross and ZagierGZ86, Reference ZhangZha01a, Reference ZhangZha01b, Reference ZhangZha04] yields

(1.1.3) $$\begin{eqnarray}\mathscr{L}=\frac{c_{E}}{2}\cdot \frac{\unicode[STIX]{x1D70B}^{2[F:\mathbf{Q}]}|D_{F}|^{1/2}L^{\prime }(1/2,\unicode[STIX]{x1D70E}_{A,E}^{\unicode[STIX]{x1D704}}\otimes \unicode[STIX]{x1D712}^{\unicode[STIX]{x1D704}})}{2L(1,\unicode[STIX]{x1D702})L(1,\unicode[STIX]{x1D70E}_{A}^{\unicode[STIX]{x1D704}},\operatorname{ad})},\end{eqnarray}$$

where

(1.1.4) $$\begin{eqnarray}c_{E}:=\frac{\unicode[STIX]{x1D701}_{F}(2)}{(\unicode[STIX]{x1D70B}/2)^{[F:\mathbf{Q}]}|D_{E}|^{1/2}L(1,\unicode[STIX]{x1D702})}\in \mathbf{Q}^{\times }\end{eqnarray}$$

and, in the present Introduction, $L$ -functions are as usual Euler products over all the finite places.Footnote ² (However in the main body of the paper we will embrace the convention of [Reference Yuan, Zhang and ZhangYZZ12] of including the archimedean factors.) The most important factor is the central derivative of the $L$ -function $L(s,\unicode[STIX]{x1D70E}_{A,E}^{\unicode[STIX]{x1D704}}\otimes \unicode[STIX]{x1D712})$ .

When $\langle \,,\rangle$ is the product of the $v$ -adic logarithms on $A(F_{v})$ and $A^{\vee }(F_{v})$ , for a prime $v$ of $F$ which splits in $E$ , the $v$ -adic Waldspurger formula of Liu–Zhang–Zhang [Reference Liu, Zhang and ZhangLZZ15] (generalising [Reference Bertolini, Darmon and PrasannaBDP13]) identifies $\mathscr{L}$ with the special value of a $v$ -adic Rankin–Selberg $L$ -function obtained by interpolating the values $L(1/2,\unicode[STIX]{x1D70E}_{A,E}\otimes \unicode[STIX]{x1D712}^{\prime \prime })$ at anticyclotomic Hecke characters $\unicode[STIX]{x1D712}^{\prime \prime }$ of $E$ of higher weight at $v$ (in particular, the central value for the given character $\unicode[STIX]{x1D712}$ lies outside the range of interpolation).

The object of this paper is a formula for $\mathscr{L}$ when $\langle \,,\rangle$ is a $p$ -adic height pairing. In this case $\mathscr{L}$ is given by the central derivative of a $p$ -adic Rankin–Selberg $L$ -function obtained by interpolation of $L(1/2,\unicode[STIX]{x1D70E}_{A,E},\unicode[STIX]{x1D712}^{\prime })$ at finite-order Hecke characters of $E$ , precisely up to the factor $c_{E}/2$ of (1.1.3). We describe in more detail the objects involved.

1.2 The $p$ -adic $L$ -function

We construct the relevant $p$ -adic $L$ -function as a function on a space of $p$ -adic characters (which can be regarded as an abelian eigenvariety), characterised by an interpolation property at locally constant characters. It further depends on a choice of local models at  $p$ (in the present case, additive characters); this point is relevant for the study of fields of rationality and does not seem to have received much attention in the literature on $p$ -adic $L$ -functions.

We fix from now on a rational prime $p$ .

Fix an $M$ -rational cuspidal automorphic representation $\unicode[STIX]{x1D70E}^{\infty }$ of $\mathbf{GL}_{2}(\mathbf{A}^{\infty })$ of weight $2$ ; if there is no risk of confusion we will lighten the notation and write $\unicode[STIX]{x1D70E}$ instead of $\unicode[STIX]{x1D70E}^{\infty }$ . Let $\unicode[STIX]{x1D714}:F^{\times }\backslash \mathbf{A}^{\times }\rightarrow M^{\times }$ be the central character of $\unicode[STIX]{x1D70E}$ , which is necessarily of finite order.

Fix moreover a prime $\mathfrak{p}$ of $M$ above $p$ and assume that for all $v|p$ the local components $\unicode[STIX]{x1D70E}_{v}$ of $\unicode[STIX]{x1D70E}$ are nearly $\mathfrak{p}$ -ordinary with respective characters $\unicode[STIX]{x1D6FC}_{v}$ ; we write $\unicode[STIX]{x1D6FC}$ to denote the collection $(\unicode[STIX]{x1D6FC}_{v})_{v|p}$ . We replace $L$ by its subfield $M_{\mathfrak{p}}(\unicode[STIX]{x1D6FC})$ generated by the values of all the $\unicode[STIX]{x1D6FC}_{v}$ , and we similarly let $M(\unicode[STIX]{x1D6FC})\subset L$ be the finite extension of $M$ generated by the values of all the $\unicode[STIX]{x1D6FC}_{v}$ .

Spaces of $p$ -adic and locally constant characters. Fix throughout this work an arbitrary compact open subgroup $V^{p}\subset \widehat{\mathscr{O}}_{E}^{p,\times }:=\prod _{w\nmid p}\mathscr{O}_{E,w}^{\times }$ . Let

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}=E_{\mathbf{A}^{\infty }}^{\times }/\overline{E^{\times }V^{p}},\quad \unicode[STIX]{x1D6E4}_{F}=\mathbf{A}^{\infty ,\times }/\overline{F^{\times }\widehat{\mathscr{O}}_{F}^{p,\times }}.\end{eqnarray}$$

Then we have rigid spaces $\mathscr{Y}^{\prime }=\mathscr{Y}_{\unicode[STIX]{x1D714}}^{\prime }(V^{p})$ , $\mathscr{Y}=\mathscr{Y}_{\unicode[STIX]{x1D714}}(V^{p})$ , $\mathscr{Y}_{F}$ of respective dimensions $[F:\mathbf{Q}]+1+\unicode[STIX]{x1D6FF}$ , $[F:\mathbf{Q}]$ , $1+\unicode[STIX]{x1D6FF}$ (where $\unicode[STIX]{x1D6FF}\geqslant 0$ is the Leopoldt defect of $F$ , conjectured to be zero) representing the functors on $L$ -affinoid algebras

$$\begin{eqnarray}\displaystyle \mathscr{Y}_{\unicode[STIX]{x1D714}}^{\prime }(V^{p})(A) & = & \displaystyle \{\unicode[STIX]{x1D712}^{\prime }:\unicode[STIX]{x1D6E4}\rightarrow A^{\times }:\unicode[STIX]{x1D714}\cdot \unicode[STIX]{x1D712}^{\prime }|_{\widehat{\mathscr{O}}_{F}^{p,\times }}=1\},\nonumber\\ \displaystyle \mathscr{Y}_{\unicode[STIX]{x1D714}}(V^{p})(A) & = & \displaystyle \{\unicode[STIX]{x1D712}:\unicode[STIX]{x1D6E4}\rightarrow A^{\times }:\unicode[STIX]{x1D714}\cdot \unicode[STIX]{x1D712}|_{\mathbf{A}^{\infty ,\times }}=1\},\nonumber\\ \displaystyle \mathscr{Y}_{F}(A) & = & \displaystyle \{\unicode[STIX]{x1D712}_{F}:\unicode[STIX]{x1D6E4}_{F}\rightarrow A^{\times }\},\nonumber\end{eqnarray}$$

where the sets on the right-hand sides are intended to consist of continuous homomorphisms. The inclusion $\mathscr{Y}\subset \mathscr{Y}^{\prime }$ sits in the Cartesian diagram

(1.2.1)

where the vertical maps are given by $\unicode[STIX]{x1D712}^{\prime }\mapsto \unicode[STIX]{x1D712}_{F}=\unicode[STIX]{x1D714}\cdot \unicode[STIX]{x1D712}^{\prime }|_{\mathbf{A}^{\infty ,\times }}$ . When $\unicode[STIX]{x1D714}=\mathbf{1}$ , $\mathscr{Y}_{\mathbf{1}}$ is a group object (the ‘Cartier dual’ of $\unicode[STIX]{x1D6E4}/\unicode[STIX]{x1D6E4}_{F}$ ); in general, $\mathscr{Y}_{\unicode[STIX]{x1D714}}$ is a principal homogeneous space for $\mathscr{Y}_{\mathbf{1}}$ under the action $\unicode[STIX]{x1D712}_{0}\cdot \unicode[STIX]{x1D712}=\unicode[STIX]{x1D712}_{0}\unicode[STIX]{x1D712}$ .

Let $\boldsymbol{\unicode[STIX]{x1D707}}_{\mathbf{Q}}$ denote the ind-scheme over $\mathbf{Q}$ of all roots of unity and $\boldsymbol{\unicode[STIX]{x1D707}}_{M}$ its base-change to $M$ . Then there are ind-schemes $\mathscr{Y}^{\prime \,\text{l.c.}}$ , $\mathscr{Y}^{\text{l.c.}}$ , $\mathscr{Y}_{F}^{\text{l.c.}}$ , ind-finite over $M$ , representing the functors on $M$ -algebras

$$\begin{eqnarray}\displaystyle \mathscr{Y}^{\prime \,\text{l.c.}}(A) & = & \displaystyle \{\unicode[STIX]{x1D712}^{\prime }:\unicode[STIX]{x1D6E4}\rightarrow \boldsymbol{\unicode[STIX]{x1D707}}_{M}(A):\unicode[STIX]{x1D714}\cdot \unicode[STIX]{x1D712}^{\prime }|_{\widehat{\mathscr{O}}_{F}^{p,\times }}=1\},\nonumber\\ \displaystyle \mathscr{Y}^{\text{l.c.}}(A) & = & \displaystyle \{\unicode[STIX]{x1D712}:\unicode[STIX]{x1D6E4}\rightarrow \boldsymbol{\unicode[STIX]{x1D707}}_{M}(A):\unicode[STIX]{x1D714}\cdot \unicode[STIX]{x1D712}|_{\mathbf{A}^{\infty ,\times }}=1\},\nonumber\\ \displaystyle \mathscr{Y}_{F}^{\text{l.c.}}(A) & = & \displaystyle \{\unicode[STIX]{x1D712}_{F}:\unicode[STIX]{x1D6E4}_{F}\rightarrow \boldsymbol{\unicode[STIX]{x1D707}}_{M}(A)\},\nonumber\end{eqnarray}$$

where the sets on the right-hand sides are intended to consist of locally constant (equivalently, finite-order) characters.

In the situation of the definition, we will abusively still denote by $G$ the function $G^{\prime }$ on $\mathscr{Y}_{M^{\prime }}^{?,\text{l.c.}}$ .

Local additive character. Let $v$ be a non-archimedean place of $F$ , $p_{v}\subset \mathscr{O}_{F,v}$ the maximal ideal, and $d_{v}\subset \mathscr{O}_{F,v}$ the different. We define the space of additive characters of $F_{v}$ of level $0$ to be

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}_{v}:=\operatorname{Hom}(F_{v}/d_{v}^{-1}\mathscr{O}_{F,v},\boldsymbol{\unicode[STIX]{x1D707}}_{\mathbf{Q}})-\operatorname{Hom}(F_{v}/p_{v}^{-1}d_{v}^{-1}\mathscr{O}_{F,v},\boldsymbol{\unicode[STIX]{x1D707}}_{\mathbf{Q}}),\end{eqnarray}$$

where we regard $\operatorname{Hom}(F_{v}/p_{v}^{n}\mathscr{O}_{F,v},\boldsymbol{\unicode[STIX]{x1D707}}_{\mathbf{Q}})$ as a profinite group scheme over $\mathbf{Q}$ .Footnote ⁶ The scheme $\unicode[STIX]{x1D6F9}_{v}$ is a torsor for the action of $\mathscr{O}_{F,v}^{\times }$ (viewed as a constant profinite group scheme over $\mathbf{Q}$ ) by $a.\unicode[STIX]{x1D713}(x):=\unicode[STIX]{x1D713}(ax)$ .

If $\unicode[STIX]{x1D714}_{v}^{\prime }:\mathscr{O}_{F,v}^{\times }\rightarrow \mathscr{O}(\mathscr{X})^{\times }$ is a continuous character for a scheme or rigid space $\mathscr{X}$ , we denote by $\mathscr{O}_{\mathscr{X}\times \unicode[STIX]{x1D6F9}_{v}}(\unicode[STIX]{x1D714}_{v}^{\prime })\subset \mathscr{O}_{\mathscr{X}\times \unicode[STIX]{x1D6F9}_{v}}$ the subsheaf of functions $G$ satisfying $G(x,a.\unicode[STIX]{x1D713})=\unicode[STIX]{x1D714}_{v}^{\prime }(a)(x)G(x,\unicode[STIX]{x1D713})$ for $a\in \mathscr{O}_{F,v}^{\times }$ . By the defining property, we can identify $\mathscr{O}_{\mathscr{X}\times \unicode[STIX]{x1D6F9}_{v}}(\unicode[STIX]{x1D714}_{v}^{\prime })$ with $\text{p}_{\mathscr{X}\ast }\mathscr{O}_{\mathscr{X}\times \unicode[STIX]{x1D6F9}_{v}}(\unicode[STIX]{x1D714}_{v}^{\prime })$ (where $\text{p}_{\mathscr{X}}:\mathscr{X}\times \unicode[STIX]{x1D6F9}_{v}\rightarrow \mathscr{X}$ is the projection), a locally free rank- $1$ $\mathscr{O}_{\mathscr{X}}$ -module with action by $\mathscr{G}_{\mathbf{Q}}:=\operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ . Finally, we denote $\unicode[STIX]{x1D6F9}_{p}:=\prod _{v|p}\unicode[STIX]{x1D6F9}_{v}$ and, if $\unicode[STIX]{x1D714}_{p}^{\prime }=\prod _{v|p}\unicode[STIX]{x1D714}_{v}^{\prime }:\mathscr{O}_{F,p}^{\times }\rightarrow \mathscr{O}(\mathscr{X})^{\times }$ ,

$$\begin{eqnarray}\mathscr{O}_{\mathscr{X}\times \unicode[STIX]{x1D6F9}_{p}}(\unicode[STIX]{x1D714}_{p}^{\prime })=\bigotimes _{v|p}\mathscr{O}_{\mathscr{X}\times \unicode[STIX]{x1D6F9}_{v}}(\unicode[STIX]{x1D714}_{v}^{\prime }),\end{eqnarray}$$

where the tensor product is in the category of $\mathscr{O}_{\mathscr{X}}$ -modules. Its space of global sections over $\mathscr{X}$ will be denoted by $\mathscr{O}_{\mathscr{X}\times \unicode[STIX]{x1D6F9}_{p}}(\mathscr{X}\times \unicode[STIX]{x1D6F9}_{p},\unicode[STIX]{x1D714}_{p}^{\prime })$ or simply $\mathscr{O}_{\mathscr{X}\times \unicode[STIX]{x1D6F9}_{p}}(\mathscr{X},\unicode[STIX]{x1D714}_{p}^{\prime })$ .

These sheaves will appear in the next theorem with $\unicode[STIX]{x1D714}_{v}^{\prime }=\unicode[STIX]{x1D714}_{v}\unicode[STIX]{x1D712}_{F,\text{univ},v}^{-1}:\mathscr{O}_{F,v}^{\times }\rightarrow \mathscr{O}(\mathscr{Y}^{\prime })^{\times }$ , where $\unicode[STIX]{x1D714}_{v}$ is the central character of $\unicode[STIX]{x1D70E}_{v}$ and $\unicode[STIX]{x1D712}_{F,\text{univ},v}:\mathscr{O}_{F,v}^{\times }\rightarrow \mathscr{O}(\mathscr{Y}_{F})^{\times }\rightarrow \mathscr{O}(\mathscr{Y}^{\prime })^{\times }$ comes from the restriction of the universal $\mathscr{O}(\mathscr{Y}_{F})^{\times }$ -valued character of $\unicode[STIX]{x1D6E4}_{F}$ . As $\unicode[STIX]{x1D6F9}_{p}$ is a scheme over $\mathbf{Q}$ and $\unicode[STIX]{x1D712}_{F,\text{univ}}$ is obviously algebraic on $\mathscr{Y}_{F}^{\text{l.c.}}$ , the notion of Definition 1.2.3 extends to define $\mathscr{Y}_{M^{\prime }}^{\prime \,\text{l.c.}}$ -algebraicity of sections of $\mathscr{O}_{\mathscr{Y}^{\prime }\times \unicode[STIX]{x1D6F9}_{p}}(\unicode[STIX]{x1D714}_{v}^{\prime })$ (and we use the terminology ‘algebraic on $\mathscr{Y}_{M^{\prime }}^{\prime \,\text{l.c.}}\times \unicode[STIX]{x1D6F9}_{p}(\unicode[STIX]{x1D714}_{v}^{\prime })$ ’).

As a last preliminary, we introduce notation for bounded functions: if $\mathscr{X}$ is a rigid space, then $\mathscr{O}_{\mathscr{X}}(\mathscr{X})^{\text{b}}\subset \mathscr{O}_{\mathscr{X}}(\mathscr{X})$ is the space of global sections $G$ such that $\sup _{x\in \mathscr{X}}|G(x)|$ is finite; similarly, in the above situation, we let

$$\begin{eqnarray}\mathscr{O}_{\mathscr{X}\times \unicode[STIX]{x1D6F9}_{p}}(\mathscr{X},\unicode[STIX]{x1D714}_{p}^{\prime })^{\text{b}}:=\biggl\{G\in \mathscr{O}_{\mathscr{ X}\times \unicode[STIX]{x1D6F9}_{p}}(\mathscr{X},\unicode[STIX]{x1D714}_{p}^{\prime }):\sup _{x\in \mathscr{X}}|G(x,\unicode[STIX]{x1D713})|~\text{is finite for some }\unicode[STIX]{x1D713}\in \unicode[STIX]{x1D6F9}_{p}\biggr\}.\end{eqnarray}$$

As $\unicode[STIX]{x1D714}_{p}^{\prime }$ is continuous, $\unicode[STIX]{x1D714}_{p}^{\prime }(a)$ is bounded in $a\in \mathscr{O}_{F,p}^{\times }$ : we could then equivalently replace ‘is finite for some $\unicode[STIX]{x1D713}\in \unicode[STIX]{x1D6F9}_{p}$ ’ with ‘is uniformly bounded for all $\unicode[STIX]{x1D713}\in \unicode[STIX]{x1D6F9}_{p}$ ’.

Theorem A. There is a bounded analytic function

$$\begin{eqnarray}L_{p,\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D70E}_{E})\in \mathscr{O}_{\mathscr{Y}^{\prime }\times \unicode[STIX]{x1D6F9}_{p}}(\mathscr{Y}^{\prime },\unicode[STIX]{x1D714}_{p}^{-1}\unicode[STIX]{x1D712}_{F,\text{univ},p})^{\text{b}}\end{eqnarray}$$

uniquely determined by the following property: $L_{p,\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D70E}_{E})$ is algebraic on $\mathscr{Y}_{M(\unicode[STIX]{x1D6FC})}^{\prime \,\text{l.c.}}\times \unicode[STIX]{x1D6F9}_{p}$ and, for each $\mathbf{C}$ -valued geometric point

$$\begin{eqnarray}(\unicode[STIX]{x1D712}^{\prime },\unicode[STIX]{x1D713}_{p})\in \mathscr{Y}_{M(\unicode[STIX]{x1D6FC})}^{\prime \,\text{l.c.}}\times \unicode[STIX]{x1D6F9}_{p}(\mathbf{C}),\end{eqnarray}$$

letting $\unicode[STIX]{x1D704}:M(\unicode[STIX]{x1D6FC}){\hookrightarrow}\mathbf{C}$ be the embedding induced by the composition $\operatorname{Spec}\mathbf{C}\stackrel{\unicode[STIX]{x1D712}^{\prime }}{\rightarrow }\mathscr{Y}_{M(\unicode[STIX]{x1D6FC})}^{\prime \,\text{l.c.}}\rightarrow \operatorname{Spec}M(\unicode[STIX]{x1D6FC})$ , we have

$$\begin{eqnarray}L_{p,\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D70E}_{E})(\unicode[STIX]{x1D712}^{\prime },\unicode[STIX]{x1D713}_{p})=\mathop{\prod }_{v|p}Z_{v}^{\circ }(\unicode[STIX]{x1D712}_{v}^{\prime },\unicode[STIX]{x1D713}_{v})\frac{\unicode[STIX]{x1D70B}^{2[F:\mathbf{Q}]}|D_{F}|^{1/2}L(1/2,\unicode[STIX]{x1D70E}_{E}^{\unicode[STIX]{x1D704}}\otimes \unicode[STIX]{x1D712}^{\prime })}{2L(1,\unicode[STIX]{x1D702})L(1,\unicode[STIX]{x1D70E}^{\unicode[STIX]{x1D704}},\operatorname{ad})}\end{eqnarray}$$

in $\mathbf{C}$ . The interpolation factor is explicitly

$$\begin{eqnarray}Z_{v}^{\circ }(\unicode[STIX]{x1D712}_{v}^{\prime },\unicode[STIX]{x1D713}_{v}):=\frac{\unicode[STIX]{x1D701}_{F,v}(2)L(1,\unicode[STIX]{x1D702}_{v})^{2}}{L(1/2,\unicode[STIX]{x1D70E}_{E,v}\otimes \unicode[STIX]{x1D712}_{v}^{\prime })}\mathop{\prod }_{w|v}Z_{w}(\unicode[STIX]{x1D712}_{w}^{\prime },\unicode[STIX]{x1D713}_{v})\end{eqnarray}$$

with

$$\begin{eqnarray}Z_{w}(\unicode[STIX]{x1D712}_{w}^{\prime },\unicode[STIX]{x1D713}_{v})=\left\{\begin{array}{@{}l@{}}\displaystyle \unicode[STIX]{x1D6FC}_{v}(\unicode[STIX]{x1D71B}_{v})^{-v(D)}\unicode[STIX]{x1D712}_{w}^{\prime }(\unicode[STIX]{x1D71B}_{w})^{-v(D)}\frac{1-\unicode[STIX]{x1D6FC}_{v}(\unicode[STIX]{x1D71B}_{v})^{-f_{w}}\unicode[STIX]{x1D712}_{w}^{\prime }(\unicode[STIX]{x1D71B}_{w})^{-1}}{1-\unicode[STIX]{x1D6FC}_{v}(\unicode[STIX]{x1D71B}_{v})^{f_{w}}\unicode[STIX]{x1D712}_{w}^{\prime }(\unicode[STIX]{x1D71B}_{w})q_{F,v}^{-f_{w}}}\quad \\ \qquad \text{if }\unicode[STIX]{x1D712}_{w}^{\prime }\cdot \unicode[STIX]{x1D6FC}_{v}\circ q_{w}\text{ is unramified},\quad \\ \unicode[STIX]{x1D70F}(\unicode[STIX]{x1D712}_{w}^{\prime }\cdot \unicode[STIX]{x1D6FC}_{v}\circ q,\unicode[STIX]{x1D713}_{E_{w}})\quad \\ \qquad \text{if }\unicode[STIX]{x1D712}_{w}^{\prime }\cdot \unicode[STIX]{x1D6FC}_{v}\circ q_{w}\text{ is ramified}.\quad \end{array}\right.\end{eqnarray}$$

Here $d$ , $D\in \mathbf{A}^{\infty ,\times }$ are generators of the different of $F$ and the relative discriminant of $E/F$ , respectively, $q_{w}$ is the relative norm of $E/F$ , $f_{w}$ is the inertia degree of $w|v$ , and $q_{F,v}$ is the cardinality of the residue field at $v$ ; finally, for any character $\widetilde{\unicode[STIX]{x1D712}}_{w}^{\prime }$ of $E_{w}^{\times }$ of conductor $\mathfrak{f}$ ,

$$\begin{eqnarray}\unicode[STIX]{x1D70F}(\widetilde{\unicode[STIX]{x1D712}}_{w}^{\prime },\unicode[STIX]{x1D713}_{E_{w}}):=\int _{w(t)=-w(\mathfrak{f})}\widetilde{\unicode[STIX]{x1D712}}_{w}^{\prime }(t)\unicode[STIX]{x1D713}_{E,w}(t)\,dt\end{eqnarray}$$

with $dt$ the additive Haar measure on $E_{w}$ giving $\operatorname{vol}(\mathscr{O}_{E_{w}},dt)=1$ , and $\unicode[STIX]{x1D713}_{E,w}=\unicode[STIX]{x1D713}_{F,v}\circ \operatorname{Tr}_{E_{w}/F_{v}}$ .

In fact, we only construct $L_{p,\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D70E}_{E})$ as a bounded section of $\mathscr{O}_{\mathscr{Y}^{\prime }\times \unicode[STIX]{x1D6F9}_{p}}(\unicode[STIX]{x1D714}_{p}^{-1}\unicode[STIX]{x1D712}_{F,\text{univ},p})(D)$ , where $D$ is a divisor on $\mathscr{Y}^{\prime }$ supported away from $\mathscr{Y}$ (i.e. for any polynomial function $G$ on $\mathscr{Y}^{\prime }$ with divisor of zeroes ${\geqslant}D$ , the function $G\cdot L_{p,\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D70E}_{E})$ is a bounded global section of $\mathscr{O}_{\mathscr{Y}^{\prime }\times \unicode[STIX]{x1D6F9}_{p}}(\unicode[STIX]{x1D714}_{p}^{-1}\unicode[STIX]{x1D712}_{F,\text{univ},p})$ );Footnote ⁷ see Theorem 3.7.1 together with Proposition A.2.2 for the precise statement. This is sufficient for our purposes and to determine $L_{p,\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D70E}_{E})$ uniquely. One can then deduce that it is possible to take $D=0$ by comparing our $p$ -adic $L$ -function to some other construction where this difficulty does not arise. One such construction has been announced by David Hansen.

1.3 $p$ -adic Gross–Zagier formula

Let us go back to the situation in which $A$ is a modular abelian variety of $\mathbf{GL}_{2}$ -type, associated with an automorphic representation $\unicode[STIX]{x1D70E}_{A}$ of $\operatorname{Res}_{F/\mathbf{Q}}\mathbf{GL}_{2}$ of character $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{A}$ .

$p$ -adic heights. Several authors (notably Mazur–Tate, Schneider, Zarhin, Nekovář) have defined $p$ -adic height pairings on $A(\overline{F})\times A^{\vee }(\overline{F})$ for an abelian variety $A$ . These pairings are analogous to the classical Néron–Tate height pairings: in particular, they admit a decomposition into a sum of local symbols indexed by the (finite) places of $F$ ; for $v\nmid p$ such symbols can be calculated from intersections of zero-cycles and degree-zero divisors on the local integral models of $A$ .

In the general context of Nekovář [Reference NekovářNek93], adopted in this paper and recalled in § 4.1, height pairings can be defined for any geometric Galois representation $V$ over a $p$ -adic field; we are interested in the case $V=V_{\mathfrak{p}}A\otimes _{M_{\mathfrak{p}}}L$ , where $M=\operatorname{End}^{0}A$ and $L$ is a finite extension of a $p$ -adic completion $M_{\mathfrak{p}}$ of $M$ . Different from the Néron–Tate heights, $p$ -adic heights are associated with the auxiliary choice of splittings of the Hodge filtration on $\mathbf{D}_{\text{dR}}(V|_{\mathscr{G}_{F_{v}}})$ for the primes $v|p$ ; in our case, $\mathbf{D}_{\text{dR}}(V|_{\mathscr{G}_{F_{v}}})=H_{\text{dR}}^{1}(A^{\vee }/F_{v})\otimes _{M_{\mathfrak{p}}}L$ . When $V|_{\mathscr{G}_{F_{v}}}$ is potentially ordinary, meaning that it is reducible in the category of de Rham representations (see more precisely Definition 4.1.1),Footnote ⁸ there is a canonical such choice. If $A$ is modular corresponding to an $M$ -rational cuspidal automorphic representation $\unicode[STIX]{x1D70E}_{A}^{\infty }$ , it follows from [Reference CarayolCar86, Théorème A], together with [Reference NekovářNek06, (proof of) Proposition 12.11.5(iv)], that the restriction of $V=V_{\mathfrak{p}}A\otimes L$ to $\mathscr{G}_{F_{v}}$ is potentially ordinary if and only if $\unicode[STIX]{x1D70E}_{A,v}\otimes L$ is nearly $\mathfrak{p}$ -ordinary.

We assume this to be the case for all $v|p$ . One then has a canonical $p$ -adic height pairing

(1.3.1) $$\begin{eqnarray}\langle \,,\rangle :A(\overline{F})_{\mathbf{Q}}\otimes _{M}A^{\vee }(\overline{F})_{\boldsymbol{ Q}}\rightarrow \unicode[STIX]{x1D6E4}_{F}\,\hat{\otimes }\,L,\end{eqnarray}$$

whose precise definition will be recalled at the end of § 4.1. Its equivariance properties under the action of $\mathscr{G}_{F}=\operatorname{Gal}(\overline{F}/F)$ allow us to deduce from it pairings

(1.3.2) $$\begin{eqnarray}\langle \,,\rangle :A(\unicode[STIX]{x1D712})\otimes _{L(\unicode[STIX]{x1D712})}A^{\vee }(\unicode[STIX]{x1D712}^{-1})\rightarrow \unicode[STIX]{x1D6E4}_{F}\,\hat{\otimes }\,L(\unicode[STIX]{x1D712})\end{eqnarray}$$

for any character $\unicode[STIX]{x1D712}\in \mathscr{Y}_{L}^{\text{l.c.}}$ .

The formula. Let $\mathscr{Y}=\mathscr{Y}_{\unicode[STIX]{x1D714}}\subset \mathscr{Y}^{\prime }=\mathscr{Y}_{\unicode[STIX]{x1D714}}^{\prime }$ be the rigid spaces defined above. Denote by $\mathscr{I}_{\mathscr{Y}}\subset \mathscr{O}_{\mathscr{Y}^{\prime }}$ the ideal sheaf of $\mathscr{Y}$ and by $\mathscr{N}_{\mathscr{Y}/\mathscr{Y}^{\prime }}^{\ast }=(\mathscr{I}_{\mathscr{Y}}/\mathscr{I}_{\mathscr{Y}}^{2})|_{\mathscr{Y}}$ the conormal sheaf. By (1.2.1), it is canonically trivial:

$$\begin{eqnarray}\mathscr{N}_{\mathscr{Y}/\mathscr{Y}^{\prime }}^{\ast }\cong \mathscr{O}_{\mathscr{ Y}}\otimes T_{\mathbf{1}}^{\ast }\mathscr{Y}_{F}\cong \mathscr{O}_{\mathscr{Y}}\otimes (\unicode[STIX]{x1D6E4}_{F}\,\hat{\otimes }\,L).\end{eqnarray}$$

For a section $G$ of $\mathscr{I}_{\mathscr{Y}}$ , denote by $\text{d}_{F}G\in \mathscr{N}_{\mathscr{Y}/\mathscr{Y}^{\prime }}^{\ast }$ its image; it can be thought of as the differential in the $1+\unicode[STIX]{x1D6FF}$ cyclotomic variable(s).

Let $\unicode[STIX]{x1D712}\in \mathscr{Y}^{\text{l.c.},\text{an}}$ be a character such that $\unicode[STIX]{x1D700}(A_{E},\unicode[STIX]{x1D712})=-1$ ; denote by $L(\unicode[STIX]{x1D712})$ its residue field. By the interpolation property, the complex functional equation, and the constancy of local root numbers, the $p$ -adic $L$ -function $L_{p,\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D70E}_{A,E})$ is a section of $\mathscr{I}_{\mathscr{Y}}$ in the connected component of $\unicode[STIX]{x1D712}\in \mathscr{Y}^{\prime }$ (see Lemma 10.2.2). Let $\mathbf{B}$ be the incoherent quaternion algebra determined by (1.1.1) and let $\unicode[STIX]{x1D70B}_{A}=\unicode[STIX]{x1D70B}_{A}(\mathbf{B})$ , $\unicode[STIX]{x1D70B}_{A^{\vee }}=\unicode[STIX]{x1D70B}_{A^{\vee }}(\mathbf{B})$ .

Theorem B. Suppose that:

1. – for all $v|p$ , $A/F_{v}$ has potentially $\mathfrak{p}$ -ordinary good or semistable reduction;

2. – for all $v|p$ , $E_{v}/F_{v}$ is split;

3. – the sign $\unicode[STIX]{x1D700}(A_{E},\unicode[STIX]{x1D712})=-1$ and $\unicode[STIX]{x1D712}$ is not exceptional (Definition 1.3.3).

Then for all $f_{1}\in \unicode[STIX]{x1D70B}_{A}$ , $f_{2}\in \unicode[STIX]{x1D70B}_{A^{\vee }}$ , we have

$$\begin{eqnarray}\langle P(f_{1},\unicode[STIX]{x1D712}),P^{\vee }(f_{2},\unicode[STIX]{x1D712}^{-1})\rangle =\frac{c_{E}}{2}\cdot \mathop{\prod }_{v|p}Z_{v}^{\circ }(\unicode[STIX]{x1D712}_{v})^{-1}\cdot \text{d}_{F}L_{p,\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D70E}_{A,E})(\unicode[STIX]{x1D712})\cdot Q(f_{1},f_{2},\unicode[STIX]{x1D712})\end{eqnarray}$$

in $\mathscr{N}_{\mathscr{Y}/\mathscr{Y}^{\prime }}^{\ast }\text{}_{|\unicode[STIX]{x1D712}}\cong \unicode[STIX]{x1D6E4}_{F}\,\hat{\otimes }\,L(\unicode[STIX]{x1D712})$ . Here $c_{E}$ is as in (1.1.4).

In the right-hand side, we have considered Remark 1.2.4 and used the canonical isomorphism $\mathscr{O}_{\unicode[STIX]{x1D6F9}_{p}}(\unicode[STIX]{x1D714}_{p}^{-1})\otimes _{M}\mathscr{O}_{\unicode[STIX]{x1D6F9}_{p}}(\unicode[STIX]{x1D714}_{p})=M$ .

1.4 Anticyclotomic theory

Consider the setup of § 1.2. Recall that in the case $\unicode[STIX]{x1D700}(1/2,\unicode[STIX]{x1D70E}_{E},\unicode[STIX]{x1D712})=+1$ , the definite quaternion algebra $\mathbf{B}$ defined by (1.1.1) is coherent, i.e. it arises as $\mathbf{B}=B\otimes _{F}\mathbf{A}_{F}$ for a quaternion algebra $B$ over $F$ ; we may assume that the embedding $E_{\mathbf{A}}{\hookrightarrow}\mathbf{B}$ arises from an embedding $i:E{\hookrightarrow}B$ . Let $\unicode[STIX]{x1D70B}$ be the automorphic representation of $\mathbf{B}^{\times }$ attached to $\unicode[STIX]{x1D70E}$ by the Jacquet–Langlands correspondence; it is realised in the space of locally constant functions $B^{\times }\backslash \mathbf{B}^{\times }\rightarrow M$ , and this gives a stable lattice $\unicode[STIX]{x1D70B}_{\mathscr{O}_{M}}\subset \unicode[STIX]{x1D70B}$ . Then, given a character $\unicode[STIX]{x1D712}\in \mathscr{Y}^{\text{l.c.}}$ , the formalism of § 1.1 applies to the period functional $p\in \text{H}(\unicode[STIX]{x1D70B},\unicode[STIX]{x1D712})$ defined by

(1.4.1) $$\begin{eqnarray}p(f,\unicode[STIX]{x1D712}):=\int _{E^{\times }\backslash E_{\mathbf{A}^{\infty }}^{\times }}f(i(t))\unicode[STIX]{x1D712}(t)\,dt\end{eqnarray}$$

and to its dual $p^{\vee }(\cdot ,\unicode[STIX]{x1D712}^{-1})\in \text{H}(\unicode[STIX]{x1D70B}^{\vee },\unicode[STIX]{x1D712}^{-1})$ . Here $dt$ is the Haar measure of total volume $1$ .

The formula expressing the decomposition of their product was proved by Waldspurger (see [Reference WaldspurgerWal85] or [Reference Yuan, Zhang and ZhangYZZ12]): for all finite-order characters $\unicode[STIX]{x1D712}:E^{\times }\backslash E_{\mathbf{A}}^{\times }\rightarrow M(\unicode[STIX]{x1D712})^{\times }$ valued in some extension $M(\unicode[STIX]{x1D712})\supset M$ , and for all $f_{1}\in \unicode[STIX]{x1D70B}$ , $f_{2}\in \unicode[STIX]{x1D70B}^{\vee }$ , we have

(1.4.2) $$\begin{eqnarray}p(f_{1},\unicode[STIX]{x1D712})p^{\vee }(f_{2},\unicode[STIX]{x1D712}^{-1})=\frac{c_{E}}{4}\cdot \frac{\unicode[STIX]{x1D70B}^{2[F:\mathbf{Q}]}|D_{F}|^{1/2}L(1/2,\unicode[STIX]{x1D70E}_{E}\otimes \unicode[STIX]{x1D712})}{2L(1,\unicode[STIX]{x1D702})L(1,\unicode[STIX]{x1D70E},\operatorname{ad})}\cdot Q(f_{1},f_{2},\unicode[STIX]{x1D712})\end{eqnarray}$$

in $M(\unicode[STIX]{x1D712})$ . Notice that here we could trivially modify the right-hand side to replace the complex $L$ -function with the $p$ -adic $L$ -function, thanks to the interpolation property defining the latter.

The $L$ -function terms of both the Waldspurger and the $p$ -adic Gross–Zagier formulas thus admit an interpolation as analytic functions (or sections of a sheaf) on $\mathscr{Y}_{\unicode[STIX]{x1D714}}$ . We can show that the other terms do as well.

Let $\unicode[STIX]{x1D70B}$ be the $M$ -rational representation of the (coherent or incoherent) quaternion algebra $\mathbf{B}^{\times }\supset E_{\mathbf{A}}^{\times }$ considered above, with central character $\unicode[STIX]{x1D714}$ . It will be convenient to denote $\unicode[STIX]{x1D70B}^{+}=\unicode[STIX]{x1D70B}$ , $\unicode[STIX]{x1D70B}^{-}=\unicode[STIX]{x1D70B}^{\vee }$ , $p^{+}=p$ , $p^{-}=p^{\vee }$ , $\mathscr{Y}_{\pm }=\mathscr{Y}_{\unicode[STIX]{x1D714}^{\pm 1}}$ , and, in the incoherent case, $A^{+}=A$ , $A^{-}=A^{\vee }$ , $P^{+}=P$ , $P^{-}=P^{\vee }$ , $\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D70E}_{A}$ .

We have a natural isomorphism $\mathscr{Y}_{+}\cong \mathscr{Y}_{-}$ given by inversion. If $\mathscr{F}$ is a sheaf on $\mathscr{Y}_{-}$ , we denote by $\mathscr{F}^{\unicode[STIX]{x1D704}}$ its pullback to a sheaf on $\mathscr{Y}_{+}$ ; the same notation is used to transfer sections of such sheaves.

Big Selmer groups and heights. Let $\unicode[STIX]{x1D712}_{\text{univ}}^{\pm }:\unicode[STIX]{x1D6E4}\rightarrow (\mathscr{O}(\mathscr{Y}_{\pm })^{\text{b}})^{\times }$ be the tautological character such that $\unicode[STIX]{x1D712}_{\text{univ}}^{\pm }(t)(\unicode[STIX]{x1D712})=\unicode[STIX]{x1D712}(t)^{\pm 1}$ for all $\unicode[STIX]{x1D712}\in \mathscr{Y}_{\pm }$ , and define an $\mathscr{O}(\mathscr{Y}_{\pm })^{\text{b}}$ -module

$$\begin{eqnarray}\mathbf{S}_{\mathfrak{p}}(A_{E}^{\pm },\unicode[STIX]{x1D712}_{\text{univ}}^{\pm },\mathscr{Y}_{\pm })^{\text{b}}:=H_{f}^{1}(E,V_{\mathfrak{ p}}A_{E}^{\pm }\otimes \mathscr{O}(\mathscr{Y}_{\pm })^{\text{b}}(\unicode[STIX]{x1D712}_{\text{univ}}^{\pm })),\end{eqnarray}$$

where $\mathscr{O}(\mathscr{Y}_{\pm })^{\text{b}}(\unicode[STIX]{x1D712}_{\text{univ}}^{\pm })$ denotes the module of bounded global sections $\mathscr{O}(\mathscr{Y}_{\pm })^{\text{b}}$ with $\mathscr{G}_{E}$ -action by $\unicode[STIX]{x1D712}_{\text{univ}}^{\pm }$ . Here, for a topological $\mathbf{Q}_{p}[\mathscr{G}_{E}]$ -module $V$ which is potentially ordinary at all $w|p$ in the sense of Definition 4.1.1 below, with exact sequences $0\rightarrow V_{w}^{+}\rightarrow V_{w}\rightarrow V_{w}^{-}\rightarrow 0$ , the (Greenberg) Selmer group $H_{f}^{1}(E,V)\subset H^{1}(E,V):=H^{1}(\mathscr{G}_{E},V)$ is the group of those continuous cohomology classes $c$ which are unramified away from $p$ and such that, for every $w|p$ , the restriction of $c$ to a decomposition group at a $w$ is in the kernel of

$$\begin{eqnarray}H^{1}(E_{w},V)\rightarrow H^{1}(E_{w},V^{-}).\end{eqnarray}$$

(In the case at hand, $V_{w}^{-}=V_{\mathfrak{p}}A_{E}^{\pm }|_{\mathscr{G}_{E,w}}^{-}$ is the maximal potentially unramified quotient of $V_{\mathfrak{p}}A_{E}^{\pm }|_{\mathscr{G}_{E,w}}$ ; cf. § 4.1.) For every non-exceptional $\unicode[STIX]{x1D712}^{\pm }\in \mathscr{Y}_{\pm }^{\text{l.c.}}$ , the specialisation $\mathbf{S}_{\mathfrak{p}}(A_{E}^{\pm },\unicode[STIX]{x1D712}_{\text{univ}}^{\pm },\mathscr{Y}_{\pm })^{\text{b}}\otimes L(\unicode[STIX]{x1D712})$ is isomorphic to the target of the Kummer map

(1.4.3) $$\begin{eqnarray}\unicode[STIX]{x1D705}:A_{E}^{\pm }(\unicode[STIX]{x1D712}^{\pm })\rightarrow H_{f}^{1}(E,V_{\mathfrak{ p}}A_{E}\otimes L(\unicode[STIX]{x1D712}^{\pm })_{\unicode[STIX]{x1D712}^{\pm }}).\end{eqnarray}$$

The work of Nekovář [Reference NekovářNek06] explains the exceptional specialisations and provides a height pairing on the big Selmer groups. The key underlying object is the Selmer complex

(1.4.4) $$\begin{eqnarray}\widetilde{\text{R}\unicode[STIX]{x1D6E4}}_{f}(E,V_{\mathfrak{p}}A_{E}^{\pm }\otimes \mathscr{O}(\mathscr{Y}_{\pm }^{\circ })^{\text{b}}(\unicode[STIX]{x1D712}_{\text{univ}}^{\pm })),\end{eqnarray}$$

an object in the derived category of $\mathscr{O}(\mathscr{Y}_{\pm })^{\text{b}}$ -modules defined as in [Reference NekovářNek06, §0.8] taking $T=V_{\mathfrak{p}}A_{E}^{\pm }\otimes \mathscr{O}(\mathscr{Y}_{\pm }^{\circ })^{\text{b}}(\unicode[STIX]{x1D712}_{\text{univ}}^{\pm })$ and $U_{w}^{+}=V_{\mathfrak{p}}A_{E}^{\pm }|_{\mathscr{G}_{E_{w}}}^{+}\otimes \mathscr{O}(\mathscr{Y}_{\pm }^{\circ })^{\text{b}}(\unicode[STIX]{x1D712}_{\text{univ}}^{\pm })$ in the notation of [Reference NekovářNek06]. Its first cohomology group

$$\begin{eqnarray}\widetilde{H}_{f}^{1}(E,V_{\mathfrak{ p}}A_{E}^{\pm }\otimes \mathscr{O}(\mathscr{Y}_{\pm }^{\circ })^{\text{b}}(\unicode[STIX]{x1D712}_{\text{univ}}^{\pm }))\end{eqnarray}$$

satisfies the following property. For every $L$ -algebra quotient $R$ of $\mathscr{O}(\mathscr{Y}_{+}^{\circ })^{\text{b}}$ , letting $\unicode[STIX]{x1D712}_{R}^{\pm }:\unicode[STIX]{x1D6E4}\rightarrow R^{\times }$ be the character deduced from $\unicode[STIX]{x1D712}_{\text{univ}}^{\pm }$ , there is an exact sequence [Reference NekovářNek06, (0.8.0.1)]

(1.4.5) $$\begin{eqnarray}\displaystyle 0 & \rightarrow & \displaystyle \displaystyle \underset{w|p}{\bigoplus }\,H^{0}(E_{w},V_{\mathfrak{p}}A_{E}^{\pm }|_{\mathscr{G}_{E,w}}^{-}\otimes R(\unicode[STIX]{x1D712}_{R}^{\pm }))\nonumber\\ \displaystyle & \rightarrow & \displaystyle \widetilde{H}_{f}^{1}(E,V_{\mathfrak{p}}A_{E}^{\pm }\otimes \mathscr{O}(\mathscr{Y}_{+}^{\circ })^{\text{b}}(\unicode[STIX]{x1D712}_{\text{univ}}^{\pm }))\otimes R\rightarrow H_{f}^{1}(E,V_{\mathfrak{p}}A_{E}^{\pm }\otimes R(\unicode[STIX]{x1D712}_{R}^{\pm }))\rightarrow 0.\end{eqnarray}$$

When $R=\mathscr{O}(\mathscr{Y}_{\pm })^{\text{b}}$ itself, each group $H^{0}(E_{w},V_{\mathfrak{p}}A_{E}^{\pm }|_{\mathscr{G}_{E,w}}^{-}\otimes \mathscr{O}(\mathscr{Y}_{\pm })^{\text{b}}(\unicode[STIX]{x1D712}_{\text{univ}}^{\pm }))$ vanishes as $\unicode[STIX]{x1D712}_{\text{univ},w}$ is infinitely ramified; hence,

$$\begin{eqnarray}\widetilde{H}_{f}^{1}(E,V_{\mathfrak{ p}}A_{E}^{\pm }\otimes \mathscr{O}(\mathscr{Y}_{\pm }^{\circ })^{\text{b}}(\unicode[STIX]{x1D712}_{\text{univ}}^{\pm }))\cong \mathbf{S}_{\mathfrak{ p}}(A_{E}^{\pm },\unicode[STIX]{x1D712}_{\text{univ}}^{\pm },\mathscr{Y}_{\pm })^{\text{b}}.\end{eqnarray}$$

When $R=L(\unicode[STIX]{x1D712})$ with $\unicode[STIX]{x1D712}\in \mathscr{Y}^{\text{l.c.}}$ , the group $H^{0}(E_{w},V_{\mathfrak{p}}A_{E}^{\pm }|_{\mathscr{G}_{E,w}}^{-}\otimes L(\unicode[STIX]{x1D712}^{\pm })_{\unicode[STIX]{x1D712}^{\pm }})$ vanishes unless $\unicode[STIX]{x1D712}_{w}\cdot \unicode[STIX]{x1D6FC}_{v}\circ q_{w}=\mathbf{1}$ on $E_{w}^{\times }$ , that is, unless $\unicode[STIX]{x1D712}_{w}$ is exceptional.

Finally, by [Reference NekovářNek06, ch. 11], there is a big height pairing

(1.4.6) $$\begin{eqnarray}\langle \,,\rangle :\mathbf{S}_{\mathfrak{p}}(A_{E}^{+},\unicode[STIX]{x1D712}_{\text{univ}}^{+},\mathscr{Y}_{+})^{\text{b}}\otimes _{\mathscr{ O}_{\mathcal{Y}_{+}}}\mathbf{S}_{\mathfrak{p}}(A_{E}^{-},\unicode[STIX]{x1D712}_{\text{univ}}^{-},\mathscr{Y}_{-})^{\text{b},\unicode[STIX]{x1D704}}\rightarrow \mathscr{N}_{\mathscr{ Y}_{+}/\mathscr{Y}_{+}^{\prime }}^{\ast }(\mathscr{Y}_{+})^{\text{b}}\end{eqnarray}$$

interpolating the height pairings on $H_{f}^{1}(E,V_{\mathfrak{p}}A\otimes L(\unicode[STIX]{x1D712}^{\pm })_{\unicode[STIX]{x1D712}^{\pm }})$ for non-exceptional $\unicode[STIX]{x1D712}\in \mathscr{Y}^{\text{l.c.}}$ (and more generally certain ‘extended’ pairings on $\widetilde{H}_{f}^{1}(E,V_{\mathfrak{p}}A\otimes L(\unicode[STIX]{x1D712}^{\pm })_{\unicode[STIX]{x1D712}^{\pm }})$ for all $\unicode[STIX]{x1D712}\in \mathscr{Y}^{\text{l.c.}}$ ; these will play no role here).

Heegner–theta elements and anticyclotomic formulas. Keep the assumptions that for all $v|p$ , $E_{v}/F_{v}$ is split and $\unicode[STIX]{x1D70B}_{v}\cong \unicode[STIX]{x1D70E}_{v}$ is $\mathfrak{p}$ -nearly ordinary with unit character $\unicode[STIX]{x1D6FC}_{v}$ . Then, after tensoring with $\mathscr{O}_{\unicode[STIX]{x1D6F9}_{p}}(\unicode[STIX]{x1D6F9}_{p})$ (in order to use Kirillov models at $p$ ), we will have a decomposition $\unicode[STIX]{x1D70B}^{\pm }\cong \unicode[STIX]{x1D70B}^{\pm ,p}\otimes \unicode[STIX]{x1D70B}_{p}^{\pm }$ , which is an isometry with respect to pairings $(\,,\,)^{p}$ , $(\,,\,)_{p}$ on each of the factors. By (1.1.2), for each $\unicode[STIX]{x1D712}=\unicode[STIX]{x1D712}^{p}\unicode[STIX]{x1D712}_{p}\in \mathscr{Y}_{M}^{\text{l.c.}}$ we can then define a toric period

(1.4.7) $$\begin{eqnarray}Q^{p}(f^{+,p},f^{-,p},\unicode[STIX]{x1D712})\in M(\unicode[STIX]{x1D712})\otimes \mathscr{O}_{\unicode[STIX]{x1D6F9}_{p}}(\unicode[STIX]{x1D714}_{p}^{-1}).\end{eqnarray}$$

Given $f^{\pm ,p}\in \unicode[STIX]{x1D70B}^{\pm ,p}$ , we will construct an explicit pair of elements

(1.4.8) $$\begin{eqnarray}f_{\unicode[STIX]{x1D6FC}}^{\pm }=(f_{\unicode[STIX]{x1D6FC},V_{p}}^{\pm })=(f^{\pm ,p}\otimes f_{\unicode[STIX]{x1D6FC},p,V_{p}}^{\pm })_{V_{p}}\in \unicode[STIX]{x1D70B}_{M(\unicode[STIX]{x1D6FC})}^{\pm ,p}\otimes \underset{V_{p}}{\varprojlim }\,\unicode[STIX]{x1D70B}_{p}^{\pm ,V_{p}},\end{eqnarray}$$

where the inverse system is indexed by compact open subgroups $V_{p}\subset E_{p}^{\times }\subset \mathbf{B}_{p}^{\times }$ containing $\operatorname{Ker}(\unicode[STIX]{x1D714}_{p})$ , with transition maps being given by averages under their $\unicode[STIX]{x1D70B}_{p}^{\pm }$ -action, and $f_{\unicode[STIX]{x1D6FC},p,V_{p}}^{\pm }$ are suitable elements of $\unicode[STIX]{x1D70B}_{p}^{\pm ,V_{p}}$ . We compute in Lemma 10.1.2 that we have

$$\begin{eqnarray}Q_{p}(f_{\unicode[STIX]{x1D6FC},p}^{+},f_{\unicode[STIX]{x1D6FC},p}^{-})=\unicode[STIX]{x1D701}_{F,p}(2)^{-1}\mathop{\prod }_{v|p}Z_{v}^{\circ }\end{eqnarray}$$

as sections of $\bigotimes _{v|p}\mathscr{O}_{\mathscr{ Y}_{v}^{\text{l.c.}}\times \unicode[STIX]{x1D6F9}_{v}}(\unicode[STIX]{x1D714}_{v})$ , where the left-hand side in the above expression is computed, for each $\unicode[STIX]{x1D712}_{p}\in \prod _{v|p}\mathscr{Y}_{v}^{\text{l.c.}}$ , as the limit of $Q_{p}(f_{\unicode[STIX]{x1D6FC},p,V_{p}}^{+},f_{\unicode[STIX]{x1D6FC},p,V_{p}}^{-})$ as $V_{p}\rightarrow \operatorname{Ker}(\unicode[STIX]{x1D714}_{p})$ .

For the following theorem, note that all the local signs in (1.1.1) extend to locally constant functions of $\mathscr{Y}_{+}$ (this is a simple special case of [Reference Pottharst and XiaoPX14, Proposition 3.3.4]); the quaternion algebra over $\mathbf{A}$ determined by (1.1.1) is then also constant along the connected components of $\mathscr{Y}_{+}$ . We will say that a connected component $\mathscr{Y}_{+}^{\circ }\subset \mathscr{Y}_{+}$ is of type $\unicode[STIX]{x1D700}\in \{\pm 1\}$ if $\unicode[STIX]{x1D700}(1/2,\unicode[STIX]{x1D70E}_{E},\unicode[STIX]{x1D712})=\unicode[STIX]{x1D700}$ along $\mathscr{Y}^{\circ }$ .

In parts (3) and (4), we have used the canonical isomorphism $\mathscr{O}_{\unicode[STIX]{x1D6F9}_{p}}(\unicode[STIX]{x1D714}_{p})\otimes \mathscr{O}_{\unicode[STIX]{x1D6F9}_{p}}(\unicode[STIX]{x1D714}_{p}^{-1})\cong M$ . The height pairing of part (4) is (1.4.6).

1.5 Applications

Theorem B has by now standard applications to the $p$ -adic and the classical Birch and Swinnerton-Dyer conjectures; the interested reader will have no difficulty in obtaining them as in [Reference Perrin-RiouPer87, Reference DisegniDis15]. We obtain in particular one $p$ -divisibility in the classical Birch and Swinnerton-Dyer conjecture for a $p$ -ordinary CM elliptic curve $A$ over a totally real field as in [Reference DisegniDis15, Theorem D] without the spurious assumptions of [Reference DisegniDis15] on the behaviour of $p$ in $F$ . In the rest of this subsection, we describe two other applications.

On the $p$ -adic Birch and Swinnerton-Dyer conjecture in anticyclotomic families. The next theorem, which can be thought of as a case of the $p$ -adic Birch and Swinnerton-Dyer conjecture in anticyclotomic families, combines Theorem C (4) with work of Fouquet [Reference FouquetFou13] to generalise a result of Howard [Reference HowardHow05] towards a conjecture of Perrin-Riou [Reference Perrin-RiouPer87]. We first introduce some notation: let $\unicode[STIX]{x1D6EC}:=\mathscr{O}(\mathscr{Y}_{+}^{\circ })^{\text{b}}$ , and let the anticyclotomic height regulator

(1.5.1) $$\begin{eqnarray}\mathscr{R}\subset \unicode[STIX]{x1D6EC}\,\hat{\otimes }\,\operatorname{Sym}^{r}\unicode[STIX]{x1D6E4}_{F}\end{eqnarray}$$

be the discriminant of (1.4.6) on the $\unicode[STIX]{x1D6EC}$ -module

$$\begin{eqnarray}\mathbf{S}_{\mathfrak{p}}(A_{E}^{+},\unicode[STIX]{x1D712}_{\text{univ}}^{+},\mathscr{Y}_{+}^{\circ })^{\text{b}}\otimes _{\unicode[STIX]{x1D6EC}}\mathbf{S}_{\mathfrak{ p}}(A_{E}^{-},\unicode[STIX]{x1D712}_{\text{univ}}^{-},\mathscr{Y}_{-}^{\circ })^{\text{b},\unicode[STIX]{x1D704}},\end{eqnarray}$$

where the integer $r$ in (1.5.1) is the generic rank of the finite-type $\unicode[STIX]{x1D6EC}$ -module $\mathbf{S}_{\mathfrak{p}}(A_{E}^{+},\unicode[STIX]{x1D712}_{\text{univ}}^{+},\mathscr{Y}_{+}^{\circ })^{\text{b}}$ . Recall that this module is the first cohomology of the Selmer complex $\widetilde{\text{R}\unicode[STIX]{x1D6E4}}_{f}(E,V_{\mathfrak{p}}A^{+}\otimes \mathscr{O}(\mathscr{Y}_{+}^{\circ })^{\text{b}}(\unicode[STIX]{x1D712}_{\text{univ}}^{+}))$ of (1.4.4). Let

$$\begin{eqnarray}\widetilde{H}_{f}^{2}(E,V_{\mathfrak{ p}}A^{+}\otimes \mathscr{O}(\mathscr{Y})^{\text{b}}(\unicode[STIX]{x1D712}_{\text{univ}}))_{\text{tors}}\end{eqnarray}$$

be the torsion part of the second cohomology group. Its characteristic ideal in $\unicode[STIX]{x1D6EC}$ can roughly be thought of as interpolating the $p$ -parts of the rational terms (order of the Tate–Shafarevich group, Tamagawa numbers) appearing on the algebraic side of the Birch and Swinnerton-Dyer conjecture for $A(\unicode[STIX]{x1D712})$ .

Theorem D. In the situation of Theorem C (4), assume furthermore that:

1. – $p\geqslant 5$ ;

2. – $V_{\mathfrak{p}}A$ is potentially crystalline as a $\mathscr{G}_{F_{v}}$ -representation for all $v|p$ ;

3. – the character $\unicode[STIX]{x1D714}$ is trivial and $\mathscr{Y}^{\circ }$ is the connected component of $\mathbf{1}\in \mathscr{Y}$ ;

4. – the residual representation $\overline{\unicode[STIX]{x1D70C}}:\mathscr{G}_{F}\rightarrow \operatorname{Aut}_{\mathbf{F}_{\mathfrak{p}}}(T_{\mathfrak{p}}A\otimes \mathbf{F}_{\mathfrak{p}})$ is irreducible (where $\mathbf{F}_{\mathfrak{p}}$ is the residue field of $M_{\mathfrak{p}}$ ), and it remains irreducible when restricted to the Galois group of the Hilbert class field of $E$ ;

5. – for all $v|p$ , the image of $\unicode[STIX]{x1D70C}|_{\mathscr{G}_{F,v}}$ is not scalar.

Then

$$\begin{eqnarray}\mathbf{S}_{\mathfrak{p}}(A_{E},\unicode[STIX]{x1D712}_{\text{univ}},\mathscr{Y}^{\circ })^{\text{b}},\quad \mathbf{S}_{\mathfrak{p}}(A_{E},\unicode[STIX]{x1D712}_{\text{univ}}^{-1},\mathscr{Y}^{\circ })^{\text{b}\unicode[STIX]{x1D704}}\end{eqnarray}$$

both have generic rank  $1$ over $\unicode[STIX]{x1D6EC}$ , a non-torsion element of their tensor product over $\unicode[STIX]{x1D6EC}$ is given by any $\mathscr{P}_{\unicode[STIX]{x1D6FC}}^{+}(f^{+,p})\otimes \mathscr{P}_{\unicode[STIX]{x1D6FC}}^{-}(f^{-,p})^{\unicode[STIX]{x1D704}}$ such that $\mathscr{Q}(f^{+,p},f^{-,p})\neq 0$ , and

(1.5.2) $$\begin{eqnarray}(\text{d}_{F}L_{p,\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D70E}_{E})|_{\mathscr{Y}^{\circ }})\subset \mathscr{R}\cdot \operatorname{char}_{\unicode[STIX]{x1D6EC}}\widetilde{H}_{f}^{2}(E,V_{\mathfrak{ p}}A\otimes \unicode[STIX]{x1D6EC}(\unicode[STIX]{x1D712}_{\text{univ}}))_{\text{tors}}\end{eqnarray}$$

as $\unicode[STIX]{x1D6EC}$ -submodules of $\unicode[STIX]{x1D6EC}\,\hat{\otimes }\,\unicode[STIX]{x1D6E4}_{F}$ .

The ‘potentially crystalline’ assumption for $V_{\mathfrak{p}}A$ , which is satisfied if $A$ has potentially good reduction at all $v|p$ , is imposed in order for $V_{\mathfrak{p}}A\otimes \mathscr{O}(\mathscr{Y}^{\circ })^{\text{b}}$ to be ‘non-exceptional’ in the sense of [Reference FouquetFou13] (which is more restrictive than ours); the assumption on $\unicode[STIX]{x1D714}$ allows us to invoke the results of [Reference Cornut and VatsalCV05, Reference Aflalo and NekovářAN10] on the non-vanishing of anticyclotomic Heegner points, and to write $A=A^{+}=A^{-}$ , $\mathscr{Y}=\mathscr{Y}_{+}=\mathscr{Y}_{-}=\mathscr{Y}_{\mathbf{1}}$ . See [Reference FouquetFou13, Theorem B(ii)] for the exact assumptions needed, which are slightly weaker.

The proof of Theorem D will be given in § 10.3.

Generic non-vanishing of $p$ -adic heights on CM abelian varieties. The non-vanishing of (cyclotomic) $p$ -adic heights is in general, as we have mentioned, a deep conjecture (or a ‘strong suspicion’) of Schneider [Reference SchneiderSch85]. The following result provides some new evidence towards it. It is a corollary of Theorem C (4) together with the non-vanishing results for Katz $p$ -adic $L$ -functions of Hida [Reference HidaHid10], Hsieh [Reference HsiehHsi14], and Burungale [Reference BurungaleBur15] (via a factorisation of the $p$ -adic $L$ -function). The result is a special case of a finer one to appear in forthcoming joint work with Burungale. For CM elliptic curves over $\mathbf{Q}$ , it was known as a consequence of different non-vanishing results of Bertrand [Reference BertrandBer83] and Rohrlich [Reference RohrlichRoh84] (see [Reference Agboola and HowardAH06, Appendix A, by K. Rubin]).

Theorem E. In the situation of Theorem C (4), suppose that $A_{E}$ has complex multiplicationFootnote ¹⁰ and that $p\nmid 2D_{F}h_{E}^{-}$ , where $h_{E}^{-}=h_{E}/h_{F}$ is the relative class number. Let $\langle \,,\rangle _{\text{cyc}}$ be the pairing deduced from (1.4.6) by the map $\mathscr{N}_{\mathscr{Y}/\mathscr{Y}^{\prime }}(\mathscr{Y}^{\circ })^{\text{b}}\cong \mathscr{O}(\mathscr{Y}^{\circ })^{\text{b}}\,\hat{\otimes }\,\unicode[STIX]{x1D6E4}_{F}\rightarrow \mathscr{O}(\mathscr{Y}^{\circ })^{\text{b}}\,\hat{\otimes }\,\unicode[STIX]{x1D6E4}_{\text{cyc}}$ , where $\unicode[STIX]{x1D6E4}_{\text{cyc}}=\unicode[STIX]{x1D6E4}_{\mathbf{Q}}$ viewed as a quotient of $\unicode[STIX]{x1D6E4}_{F}$ via the adèlic norm map.

Then, for any $f^{\pm ,p}$ such that $\mathscr{Q}(f^{+,p},f^{-,p})\neq 0$ in $\mathscr{O}(\mathscr{Y}^{\circ })^{\text{b}}$ , we have

$$\begin{eqnarray}\langle \mathscr{P}_{\unicode[STIX]{x1D6FC}}^{+}(f^{+,p}),\mathscr{P}_{\unicode[STIX]{x1D6FC}}^{-}(f^{-,p})^{\unicode[STIX]{x1D704}}\rangle _{\text{cyc}}\neq 0\quad \text{in }\mathscr{O}(\mathscr{Y}^{\circ })^{\text{b}}\,\hat{\otimes }\,\unicode[STIX]{x1D6E4}_{\text{cyc}}.\end{eqnarray}$$

1.6 History and related work

We briefly discuss previous work towards our main theorems, and some related works. We will loosely term the ‘classical context’ the following specialisation of the setting of our main results: $A$ is an elliptic curve over $\mathbf{Q}$ with conductor $N$ and good ordinary reduction at $p$ ; $p$ is odd; the quadratic imaginary field $E$ has discriminant coprime to $N$ and it satisfies the Heegner condition: all primes dividing $N$ split in $E$ (this implies that $\mathbf{B}^{\infty }$ is split); the parametrisation $f:J\rightarrow A$ factors through the Jacobian of the modular curve $X_{0}(N)$ ; the character $\unicode[STIX]{x1D712}$ is unramified everywhere, or unramified away from $p$ .

Ancestors. In the classical context, Theorems A and B were proved by Perrin-Riou [Reference Perrin-RiouPer87]; intermediate steps towards the present generality were taken in [Reference DisegniDis15, Reference MaMa16]. When $E/F$ is split above $p$ , Theorem A can essentially be deduced from a general theorem of Hida [Reference HidaHid91] (cf. [Reference WanWan15, §7.3]), except for the location of the possible poles. Theorems C (4) and D in the classical context are due to Howard [Reference HowardHow05] (in fact, Theorems B and C (4) were first envisioned by Mazur [Reference MazurMaz83] in that context, whereas Perrin-Riou [Reference Perrin-RiouPer87] had conjectured the equality in (1.5.2)). Theorem C (3) is hardly new and has many antecedents in the literature: see e.g. [Reference Van OrderVan12] and references therein.

Relatives. Some analogues of Theorem B were proven in situations which differ from the classical context in directions which are orthogonal to those of the present work: Nekovář [Reference NekovářNek95] and Shnidman [Reference ShnidmanShn16] dealt with the case of higher weights; Kobayashi [Reference KobayashiKob13] dealt with the case of elliptic curves with supersingular reduction.

Friends. We have already mentioned two other fully general Gross–Zagier formulas in the sense of § 1.1, namely the original archimedean one of [Reference Yuan, Zhang and ZhangYZZ12] generalising [Reference Gross and ZagierGZ86], and a different $p$ -adic formula proved in [Reference Liu, Zhang and ZhangLZZ15] generalising [Reference Bertolini, Darmon and PrasannaBDP13]. The panorama of existing formulas of this type is complemented by a handful of results, mostly in the classical context, valid in the presence of an exceptional zero (the case excluded in Theorem B). We refer the reader to [Reference DisegniDis16], where we prove a new such formula for $p$ -adic heights and review other ones due to Bertolini–Darmon. It is to be expected that all of those results should be generalisable to the framework of § 1.1.

Children. Finally, explicit versions of any Gross–Zagier formula in the framework of § 1.1 can be obtained by the explicit computation of the local integrals $Q_{v}$ . This is carried out in [Reference Cai, Shu and TianCST14], where it is applied to the cases of the archimedean Gross–Zagier formula and of the Waldspurger formula; the application to an explicit version of Theorem B can be obtained in exactly the same manner. An explicit version of the anticyclotomic formulas of Theorem C can also be obtained as a consequence: see [Reference DisegniDis16] for a special case.

1.7 Outline of proofs and organisation of the paper

Let us briefly explain the main arguments and at the same time the organisation of the paper.

For the sake of simplicity, the notation used in this introductory discussion slightly differs from that of the body text, and we ignore powers of $\unicode[STIX]{x1D70B}$ , square roots of discriminants, and choices of additive character.

Construction of the $p$ -adic $L$ -function (§ 3). It is crucial for us to have a flexible construction which does not depend on choices of newforms. The starting point is Waldspurger’s [Reference WaldspurgerWal85] Rankin–Selberg integral

(1.7.1) $$\begin{eqnarray}\frac{(\unicode[STIX]{x1D711},I(\unicode[STIX]{x1D719},\unicode[STIX]{x1D712}^{\prime }))_{\text{Pet}}}{2L(1,\unicode[STIX]{x1D70E},\text{ad})/\unicode[STIX]{x1D701}_{F}(2)}=\frac{L(1/2,\unicode[STIX]{x1D70E}_{E}\otimes \unicode[STIX]{x1D712}^{\prime })}{L(1,\unicode[STIX]{x1D702})}\mathop{\prod }_{v}R_{v}^{\natural }(\unicode[STIX]{x1D711}_{v},\unicode[STIX]{x1D719}_{v},\unicode[STIX]{x1D712}_{v}^{\prime }),\end{eqnarray}$$

where $\unicode[STIX]{x1D711}\in \unicode[STIX]{x1D70E}$ , $I(\unicode[STIX]{x1D719},\unicode[STIX]{x1D712}^{\prime })$ is a mixed theta-Eisenstein series depending on a choice of an adèlic Schwartz function $\unicode[STIX]{x1D719}$ , and $R_{v}^{\natural }(\unicode[STIX]{x1D711}_{v},\unicode[STIX]{x1D719}_{v},\unicode[STIX]{x1D712}^{\prime })$ are normalised local integrals (almost all of which are equal to  $1$ ). Then, after dividing both sides by the period $2L(1,\unicode[STIX]{x1D70E},\text{ad})$ , we can:

1. – interpolate the kernel $\unicode[STIX]{x1D712}^{\prime }\mapsto I(0,\unicode[STIX]{x1D719},\unicode[STIX]{x1D712}^{\prime })$ to a $\mathscr{Y}^{\prime }$ -family $\mathscr{I}(\unicode[STIX]{x1D719}^{p\infty };\unicode[STIX]{x1D712}^{\prime })$ of $p$ -adic modular forms for any choice of the components $\unicode[STIX]{x1D719}^{p\infty }$ , and a well-chosen $\unicode[STIX]{x1D719}_{p\infty }$ (we will set $\unicode[STIX]{x1D719}_{v}(x,u)$ to be ‘standard’ at $v|\infty$ , and close to a delta function in $x$ at $v|p$ );

2. – interpolate the functional ‘Petersson product with $\unicode[STIX]{x1D711}$ ’ to a functional $\ell _{\unicode[STIX]{x1D711}^{p},\unicode[STIX]{x1D6FC}}$ on $p$ -adic modular forms, for any $\unicode[STIX]{x1D711}\in \unicode[STIX]{x1D70E}$ which is a ‘ $\text{U}_{v}$ -eigenvector of eigenvalue $\unicode[STIX]{x1D6FC}_{v}$ ’ at the places $v|p$ , and is antiholomorphic at infinity;

3. – interpolate the normalised local integrals $\unicode[STIX]{x1D712}^{\prime }\mapsto R_{v}^{\natural }(\unicode[STIX]{x1D711}_{v},\unicode[STIX]{x1D719}_{v},\unicode[STIX]{x1D712}_{v}^{\prime })$ to functions $\mathscr{R}_{v}^{\natural }(\unicode[STIX]{x1D711}_{v},\unicode[STIX]{x1D719}_{v})$ for all $v\nmid p\infty$ and any $\unicode[STIX]{x1D711}_{v}$ , $\unicode[STIX]{x1D719}_{v}$ .

To conclude, we cover $\mathscr{Y}^{\prime }$ by finitely many open subsets $\mathscr{U}_{i}$ ; for each $i$ , we choose appropriate $\unicode[STIX]{x1D711}^{p}$ , $\unicode[STIX]{x1D719}^{p}$ and we defineFootnote ¹¹

$$\begin{eqnarray}L_{p,\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D70E}_{E})|_{\mathscr{U}_{i}}=\frac{\ell _{\unicode[STIX]{x1D711}^{p},\unicode[STIX]{x1D6FC}}(\mathscr{I}(\unicode[STIX]{x1D711}^{p\infty }))}{\mathop{\prod }_{v\nmid p\infty }\mathscr{R}_{v}^{\natural }(\unicode[STIX]{x1D711}_{v},\unicode[STIX]{x1D719}_{v})}.\end{eqnarray}$$

The explicit computation of the local integrals at $p$ (in the Appendix) and at infinity yields the interpolation factor.

Proof of the Gross–Zagier formula and its anticyclotomic version. We outline the main arguments of our proof, with an emphasis on the reduction steps.

Multiplicity one. We borrow or adapt many ideas (and calculations) from [Reference Yuan, Zhang and ZhangYZZ12], in particular the systematic use of the multiplicity-one principle of § 1.1. As both sides of the formula are functionals in the same one-dimensional vector space, it is enough to prove the result for one pair $f_{1}$ , $f_{2}$ with $Q(f_{1},f_{2},\unicode[STIX]{x1D712})\neq 0$ ; finding such $f_{1}\otimes f_{2}$ is a local problem. It is equivalent to choosing functions $\unicode[STIX]{x1D711}_{v}\otimes \unicode[STIX]{x1D719}_{v}^{\prime }=\unicode[STIX]{x1D703}^{-1}(f_{1}\otimes f_{2})$ as just above, by the Shimizu lift $\unicode[STIX]{x1D703}$ realising the Jacquet–Langlands correspondence (§ 5.1). For $v|p$ , we thus have an explicit choice of such, corresponding to the one made above.Footnote ¹² For $v\nmid p$ , we can introduce several restrictions on $(\unicode[STIX]{x1D711}_{v},\unicode[STIX]{x1D719}_{v})$ as in [Reference Yuan, Zhang and ZhangYZZ12], with the effect of simplifying many calculations of local heights (§ 6).

Arithmetic theta lifting and kernel identity (§ 5). In [Reference Yuan, Zhang and ZhangYZZ12], the authors introduce an arithmetic–geometric analogue of the Shimizu lift, by means of which they are able to write also the Heegner-points side of their formula as a Petersson product with $\unicode[STIX]{x1D711}$ of a certain geometric kernel. We can adapt without difficulty their results to reduce our formula to the assertionFootnote ¹³ that

$$\begin{eqnarray}\text{d}_{F}\mathscr{I}(\unicode[STIX]{x1D719}^{p\infty };\unicode[STIX]{x1D712})-2L_{(p)}(1,\unicode[STIX]{x1D702})\widetilde{Z}(\unicode[STIX]{x1D719}^{\infty },\unicode[STIX]{x1D712})\end{eqnarray}$$

is killed by the $p$ -adic Petersson product $\ell _{\unicode[STIX]{x1D711}^{p},\unicode[STIX]{x1D6FC}}$ . Here $\widetilde{Z}(\unicode[STIX]{x1D719}^{\infty },\unicode[STIX]{x1D712})$ is a modular form depending on $\unicode[STIX]{x1D719}$ encoding the height pairings of CM points on Shimura curves and their Hecke translates; it generalises the classical generating series $\sum _{m}\langle \unicode[STIX]{x1D704}_{\unicode[STIX]{x1D709}}(P)[\unicode[STIX]{x1D712}],T(m)\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D709}}(P)\rangle \mathbf{q}^{m}$ .

Decomposition and comparison (§§ 7–8). Both terms in the kernel identity are sums of local terms indexed by the finite places of $F$ . For $v\nmid p$ , we compute both sides and show that the difference essentially coincides with the one computed in [Reference Yuan, Zhang and ZhangYZZ12]: it is either zero or, at bad places, a modular form orthogonal to all forms in $\unicode[STIX]{x1D70E}$ . In fact, we can show this only for a certain restricted set of $q$ -expansion coefficients; as the global kernels are $p$ -adic modular forms, this will suffice by a simple approximation argument (Lemma 2.1.2).

$p$ -adic Arakelov theory or analytic continuation. The argument just sketched relies on calculations of arithmetic intersections of CM points; this in general does not suffice, as we need to consider the contribution of the Hodge classes in the generating series too. It will turn out that such contribution vanishes; two approaches can be followed to show this. The first one, in analogy with [Reference ZhangZha01a, Reference Yuan, Zhang and ZhangYZZ12] and already used in a simpler context in [Reference DisegniDis15], is to make use of Besser’s $p$ -adic Arakelov theoryFootnote ¹⁴ [Reference BesserBes05] in order to separate such contribution.

We will follow an alternative approach (see Proposition 10.2.3), which exploits the generality of our context and the existence of extra variables in the $p$ -adic world. Once having constructed the Heegner–theta element $\mathscr{P}^{\pm }$ , the anticyclotomic formula of Theorem C (4) is essentially a corollary of Theorem B for all finite-order characters $\unicode[STIX]{x1D712}$ : we only need to check the compatibility $Q_{v}(f_{\unicode[STIX]{x1D6FC},v}^{+},f_{\unicode[STIX]{x1D6FC},v}^{-},\unicode[STIX]{x1D712})=\unicode[STIX]{x1D701}_{F,v}(2)^{-1}\cdot Z_{v}(\unicode[STIX]{x1D712}_{v})$ for all $v|p$ by explicit computation. Conversely, thanks to the multiplicity-one result, it is also true that Theorem B for any $\unicode[STIX]{x1D712}$ is obtained as a corollary of Theorem C (4) by specialisation. We make use of both of these observations: we first prove Theorem B for all but finitely many finite-order characters $\unicode[STIX]{x1D712}$ ; this suffices to deduce Theorem C (4) by an analytic continuation argument, which finally yields Theorem B for the remaining characters $\unicode[STIX]{x1D712}$ as well. The initially excluded characters are those (such as the trivial character when it is contemplated) for which the contribution of the Hodge classes is not already annihilated by $\unicode[STIX]{x1D712}$ -averaging; for all other characters the Arakelov-theoretic arguments just mentioned are then unnecessary.

Annihilation of $p$ -adic heights (§ 9). We are left to deal with the contribution of the places $v|p$ . We can show quite easily that this is zero for the analytic kernel. As in the original work of Perrin-Riou [Reference Perrin-RiouPer87], the vanishing of the contribution of the geometric kernel is the heart of the argument. We establish it via an elaboration of a method of Nekovář [Reference NekovářNek95] and Shnidman [Reference ShnidmanShn16]. The key new ingredient in adapting it to our semistable case is a simple integrality criterion for local heights in terms of intersections, introduced in § 4.3, after a review of the theory of heights.

Local toric period. Finally, in the Appendix we compute the local toric period $Q(\unicode[STIX]{x1D703}(\unicode[STIX]{x1D711}_{v}\otimes \unicode[STIX]{x1D719}_{v}^{\prime }),\unicode[STIX]{x1D712}_{v})$ for $v|p$ and compare it to the interpolation factor of the $p$ -adic $L$ -function. Both are highly ramified local integrals, and they turn out to differ by the multiplicative constant $L(1,\unicode[STIX]{x1D702}_{v})$ ; this completes the comparison between the kernel identity and Theorem B.

1.8 Notation

We largely follow the notation and conventions of [Reference Yuan, Zhang and ZhangYZZ12, §1.6].

$L$ -functions. In the rest of the paper (and unlike in the Introduction, where we adhere to the more standard convention), all complex $L$ - and zeta functions are complete including the $\unicode[STIX]{x1D6E4}$ -factors at the infinite places. (This is to facilitate referring to the results and calculations of [Reference Yuan, Zhang and ZhangYZZ12], where this convention is adopted.)

Fields and adèles. The fields $E$ and $F$ will be as fixed in the Introduction unless otherwise noted. The adèle ring of $F$ will be denoted $\mathbf{A}_{F}$ or simply $\mathbf{A}$ ; it contains the ring $\mathbf{A}^{\infty }$ of finite adèles. We let $D_{F}$ and $D_{E}$ be the absolute discriminants of $F$ and $E$ , respectively. We also choose an idèle $d\in \mathbf{A}^{\infty ,\times }$ generating the different of $F/\mathbf{Q}$ , and an idèle $D\in \mathbf{A}^{\infty ,\times }$ generating the relative discriminant of $E/F$ .

We use standard notation to restrict adèlic objects (groups, $L$ -functions, and so on) away from a finite set of places $S$ , e.g. $\mathbf{A}^{S}:=\prod _{v\notin S}^{\prime }F_{v}$ , whereas $F_{S}:=\prod _{v\in S}F_{v}$ . When $S$ is the set of places above $p$ (respectively $\infty$ ), we use this notation with ‘ $S$ ’ replaced by ‘ $p$ ’ (respectively ‘ $\infty$ ’).

We denote by $F_{\infty }^{+}\subset F_{\infty }$ the group of $(x_{\unicode[STIX]{x1D70F}})_{\unicode[STIX]{x1D70F}|\infty }$ with $x_{\unicode[STIX]{x1D70F}}>0$ for all $\unicode[STIX]{x1D70F}$ , and we let $\mathbf{A}_{+}^{\times }:=\mathbf{A}^{\infty ,\times }F_{\infty }^{+}$ , $F_{+}^{\times }:=F^{\times }\cap F_{\infty }^{+}$ .

For a non-archimedean prime $v$ of a number field $F$ , we denote by $q_{F,v}$ the cardinality of the residue field and by $\unicode[STIX]{x1D71B}_{v}$ a uniformiser.

Subgroups of $\mathbf{GL}_{2}$ . We consider $\mathbf{GL}_{2}$ as an algebraic group over $F$ . We denote by $P$ , respectively $P^{1}$ , the subgroup of $\mathbf{GL}_{2}$ , respectively $\mathbf{SL}_{2}$ , consisting of upper-triangular matrices; by $A\subset P\subset \mathbf{GL}_{2}$ the diagonal torus; and by $N\subset P\subset \mathbf{GL}_{2}$ the unipotent radical of $P$ . We let $n(x):=(\!\begin{smallmatrix}1 & x\\ & 1\end{smallmatrix}\!)$ and

$$\begin{eqnarray}w:=\left(\begin{array}{@{}cc@{}} & 1\\ -1 & \end{array}\right).\end{eqnarray}$$

Quadratic torus. We let $T:=\operatorname{Res}_{E/F}\mathbf{G}_{m}$ ; the embedding $T(\mathbf{A}^{\infty })\subset \mathbf{B}^{\times }$ is fixed. We let $Z:=\mathbf{G}_{m,F}$ , and view it both as a subgroup of $T$ and as the centre of $\mathbf{GL}_{2}$ .

Automorphic quotients. If $G$ is a reductive group over the totally real field $F$ , we denote

$$\begin{eqnarray}[G]:=G(F)\backslash G(\mathbf{A})/Z(\mathbf{A}).\end{eqnarray}$$

Measures. We choose local and global Haar measures as in [Reference Yuan, Zhang and ZhangYZZ12]. In particular, we have

$$\begin{eqnarray}\operatorname{vol}(\mathbf{GL}_{2}(\mathscr{O}_{F,v}))=|d|_{v}^{2}\unicode[STIX]{x1D701}_{F,v}(2)^{-1}\end{eqnarray}$$

for all non-archimedean $v$ .

We denote by $dt$ the local and global measures on $T/Z$ of [Reference Yuan, Zhang and ZhangYZZ12], which give $\operatorname{vol}([T],dt)=2L(1,\unicode[STIX]{x1D702})$ . The global measure

$$\begin{eqnarray}d^{\circ }t:=|D_{F}|^{1/2}|D_{E/F}|^{1/2}\,dt\end{eqnarray}$$

gives $\operatorname{vol}([T],d^{\circ }t)\in \mathbf{Q}^{\times }$ .

Regularised averages and integration. We borrow some notation from [Reference Yuan, Zhang and ZhangYZZ12, §1.6.7]. If $G$ is a topological group with a left Haar measure $dg$ with finite volume, we define

$$\begin{eqnarray}\unicode[STIX]{x2A0D}_{G}f\,dg:=\frac{1}{\operatorname{vol}(G)}\int _{G}f(g)\,dg.\end{eqnarray}$$

(This reduces to the usual average when $G$ is a finite group.)

If $F$ is a totally real field and $f$ is a function on $F^{\times }\backslash \mathbf{A}^{\times }$ invariant under $F_{\infty }^{\times }$ , we denote

$$\begin{eqnarray}\unicode[STIX]{x2A0D}_{\mathbf{A}^{\times }}f(z)\,dz:=\unicode[STIX]{x2A0D}_{F^{\times }\backslash \mathbf{A}^{\times }/F_{\unicode[STIX]{x1D70F}}^{\times }}f(z)\,dz,\end{eqnarray}$$

where $\unicode[STIX]{x1D70F}$ is any archimedean place of $F$ . If $f$ is further invariant under a compact open subgroup $U$ , this reduces to the average over $F^{\times }\backslash \mathbf{A}^{\times }/F_{\infty }^{\times }U$ .

Finally, let $G$ be a reductive group over $F$ with an embedding of $\mathbf{G}_{m/F}$ into the centre $G$ , and assume that $dg$ is a left Haar measure giving finite volume to $[G]=G(F)\backslash G(\mathbf{A})/Z(\mathbf{A})$ . Let $f$ be a function on $G(F)\backslash G(\mathbf{A})/Z(F_{\infty })$ ; then we define

$$\begin{eqnarray}\int _{[G]}^{\ast }f(g)\,dg:=\int _{[G]}\unicode[STIX]{x2A0D}_{Z(\mathbf{A})}f(zg)\,dz\,dg\end{eqnarray}$$

and

$$\begin{eqnarray}\unicode[STIX]{x2A0D}_{[G]}f(g)\,dg:=\frac{1}{\operatorname{vol}([G])}\int _{[G]}\unicode[STIX]{x2A0D}_{Z(\mathbf{A})}f(zg)\,dz\,dg.\end{eqnarray}$$

Note in particular that for functions which factor through a compact quotient of $G(F)\backslash G(A)$ and are locally constant there, the regularised integration reduces to a finite sum and, when using $\mathbf{Q}$ -valued measures such as the measure $d^{\circ }t$ on $T$ , it makes sense for functions taking $p$ -adic values as well.

Multi-indices. If $S$ is a set and $r\in \mathbf{Z}^{S}$ , $p\in G^{S}$ for some group $G$ , we often write $p^{r}:=\prod _{v\in S}p_{v}^{r_{v}}$ . This will typically be applied in the following situation: $S=S_{p}$ is the set of places of $F$ above $p$ , $G$ is the (semi)group of ideals of $\mathscr{O}_{F}$ , and $p_{v}$ is the ideal corresponding to $v$ .

Functions of $p$ -adic characters. When $\mathscr{Y}^{?}$ is one of the rigid spaces introduced above and $G(A)\in \mathscr{O}(\mathscr{Y}^{?})$ is a function on $\mathscr{Y}^{?}$ depending on other ‘parameters’ $A$ (e.g. a $p$ -adic $L$ -function), we write $G(A;\unicode[STIX]{x1D712})$ for the evaluation $G(A)(\unicode[STIX]{x1D712})$ .