THE p-ADIC GROSS–ZAGIER FORMULA ON SHIMURA CURVES, II: NONSPLIT PRIMES

Abstract

The formula of the title relates p-adic heights of Heegner points and derivatives of p-adic L-functions. It was originally proved by Perrin-Riou for p-ordinary elliptic curves over the rationals, under the assumption that p splits in the relevant quadratic extension. We remove this assumption, in the more general setting of Hilbert-modular abelian varieties.

1 Introduction and statement of the main result

The p-adic Gross–Zagier formula of Perrin-Riou relates p-adic heights of Heegner points and derivatives of p-adic L-functions. In its original form [26], it concerns (modular) elliptic curves over $\mathbf {Q}$ , and it is proved under two main assumptions: first, that the elliptic curve is p-ordinary; second, that p splits in the field E of complex multiplications of the Heegner points. The formula has applications to both the p-adic and the classical Birch and Swinnerton-Dyer conjecture.

The first assumption was removed by Kobayashi [19] (see also [4]). The purpose of this work is to remove the second assumption.

We work in the context of our previous work [9], which will enable us in [13] to deduce, from the formula presented here, the analogous one for higher-weight (Hilbert-) modular motives, as well as a version in the universal ordinary family with some new applications. Nevertheless, the new idea we introduce is essentially orthogonal to previous innovations, including those of [9] (and in fact it can be applied, at least in principle, to the nonordinary case as well). For this reason we start in §1.1 by informally discussing it in the simplest classical case of elliptic curves over $\mathbf {Q}$ . The general form of our results is presented in §1.2.

1.1 The main ideas in a classical context

Classically, Heegner points on the elliptic curve $A_{/\mathbf {Q}}$ are images of CM points (or divisors) on a modular curve X, under a parametrization $f\colon X\to A$ . More precisely, choosing an imaginary quadratic field E, for each ring class character $\chi \colon \mathrm {Gal}\left (\overline {E}/E\right )\to \overline {\mathbf {Q}}^{\times }$ one can construct a point $P(f, \chi )\in A_{E}(\chi )$ , the $\chi $ -isotypic part of $A\left (\overline {E}\right )_{\overline {\mathbf {Q}}}$ . The landmark formula of Gross and Zagier [17] relates the height of $P(f,\chi )$ to the derivative $L'(A_{E}\otimes \chi , 1)$ of the L-function of a twisted base change of A. The analogous formula in p-adic coefficients, ¹

(1.1.1) $$ \begin{align}\begin{aligned} \left\langle P(f,\chi), P\left(f,{\chi}^{-1}\right)\right\rangle\doteq {d\over ds}_{\rvert s=0} L_{p}\left(A_{E}, \chi\cdot\chi_{\mathrm{cyc}}^{s}\right), \end{aligned}\end{align} $$

relates cyclotomic derivatives of p-adic L-functions to p-adic height pairings $\langle \ , \ \rangle $ . We outline its proof for p-ordinary elliptic curves.

Review of Perrin-Riou’s proof

Our basic strategy is still Perrin-Riou’s variant of the one of Gross and Zagier; we briefly and informally review it, ignoring, for simplicity of exposition, the role of the character $\chi $ . Throughout the following discussion, we include pointers to corresponding statements in the main body of the paper, as guideposts meant to assist the reader’s navigation through the more technical framework used there.

Denoting by $\varphi $ the ordinary eigenform attached to A, each side of formula (1.1.1) is expressed as the image under a functional ‘p-adic Petersson product with $\varphi $ ’, denoted by $\ell _{\varphi } =formula~(3.1.6)$ , of a certain kernel function (a p-adic modular form).

For the left-hand side of formula (1.1.1), the form in question is the generating series (compare formula (3.3.4))

(1.1.2) $$ \begin{align}\begin{aligned} Z = \sum_{m\geq 1} \left\langle P^{0}, T_{m}P^{0}\right\rangle_{X} {\textbf{q}}^{m} =\sum_{v} Z_{v}, \end{aligned} \end{align} $$

where $P^{0}\in \mathrm {Div}^{0}(X)$ is a degree $0$ modification of the CM point $P\in X$ , $\langle \ , \ \rangle _{X}$ is a p-adic height pairing on X compatible with the one on A, and the decomposition into a sum running over all the finite places of $\mathbf {Q}$ (compare equation (3.6.1)) follows from a general decomposition of the global height pairing into a sum of local ones. More precisely, global height pairings are valued in the completed tensor product $H^{\times }\backslash {H}_{\mathbf {A}^{\infty }}^{\times }\hat {\otimes } L$ of the finite idèles of the Hilbert class field H of E, and of a suitable finite extension L of $\mathbf {Q}_{p}$ . The series $Z_{v}$ collects the local pairings at $w\mid v$ , each valued in $H_{w}^{\times }\hat {\otimes }L$ .

The analytic kernel $\mathscr {I}'$ giving the right-hand side of formula (1.1.1) is the derivative of a p-adic family of mixed theta-Eisenstein series (compare definition (3.2.5)). It also enjoys a decomposition

$$ \begin{align*} \mathscr{I}'= \sum_{v\neq p} \mathscr{I}^{\prime}_{v}, \end{align*} $$

where, unlike in equation (1.1.2), the sum runs over the finite places of $\mathbf {Q}$ different from p (compare equation (3.5.2)). Once we have established that $Z_{v}\doteq \mathscr {I}_{v}'$ for $v\neq p$ by computations similar to those of Gross and Zagier (compare Theorem 3.6.1), it remains to show that the p-adic modular form $Z_{p}$ is annihilated by $\ell _{\varphi }$ (compare Proposition 3.6.2).

In order to achieve this, one aims at showing that after acting on $Z_{p}$ by a Hecke operator to replace $P^{0}$ by $P^{\left [\varphi \right ]}$ (a lift of the component of its image in the $\varphi $ -part of ${\mathrm{Jac}}(X)$ ), the resulting form $Z_{p}^{\left [\varphi \right ]}$ is p-critical (compare Proposition 3.6.3). That is, its coefficients

$$ \begin{align*} a_{mp^{s}}:=\left\langle P^{\left[\varphi\right]}, T_{mp^{s}}P^{0} \right\rangle_{X,p} \end{align*} $$

decay p-adically no more slowly than a constant multiple of $p^{s}$ . The p-shift of Fourier coefficients extends the action on modular forms of the operator $\mathrm {U}_{p}$ – which, in contrast, acts by a p-adic unit on the ordinary form $\varphi $ : this implies that p-critical forms are annihilated by $\ell _{\varphi }$ .

To study the terms $a_{mp^{s}}$ , one constructs a sequence of points $P_{s}\in X_{H}$ whose fields of definitions are the layers $H_{s}$ of the anticyclotomic $p^{\infty }$ -extension of E. The relations they satisfy allow us to express

$$ \begin{align*} a_{mp^{s}} = \left\langle P^{\left[\varphi\right]}, D_{m,s}\right\rangle_{X,p}, \end{align*} $$

where $D_{m,s}$ is a degree $0$ divisor supported at Hecke-translates of $P_{s}$ , which are all essentially CM points of conductor $p^{s}$ defined over $H_{s}$ (compare Proposition 4.1.4). By a projection formula for $X_{H_{s}}\to X_{H}$ , the height $a_{mp^{s}}$ is then a sum, over primes w of H above p, of the images

$$ \begin{align*} N_{s,w}\left(h_{m,s,w}\right) \end{align*} $$

of heights $h_{m,s,w}$ computed on $X_{H_{s,w}}$ , under the norm map $N_{s,w}\colon H_{s, w}^{\times }\hat {\otimes }L \to H_{w}^{\times } \hat {\otimes } L$ . Moreover, it can be shown that the L-denominators of $h_{m,s,w}\in H_{s, w}^{\times }\hat {\otimes }L$ are uniformly bounded (compare Proposition 4.4.1), so that here we may ignore them and think of $h_{m,s,w}\in H_{s, w}^{\times }\hat {\otimes }\mathscr {O}_{L}$ .

A simple observation from [9] is that the valuation $w\left (h_{m,s,w}\right )$ equals

(1.1.3) $$ \begin{align}\begin{aligned} m_{X_{H_{w}}}\left( P^{\left[\varphi\right]} ,D_{m,s}\right), \end{aligned}\end{align} $$

the intersection multiplicity of the flat extensions (§4.2) of those divisors to some regular integral model $\mathscr {X}$ of $X_{H_{w}}$ . In the split case, it is almost immediate to see that this intersection multiplicity vanishes. This implies that

(1.1.4) $$ \begin{align}\begin{aligned} N_{s,w}\left(h_{m,s} \right)\in N_{s,w}\left(\mathscr{O}_{H_{s,w}}^{\times}\right)\hat{\otimes}\mathscr{O}_{L}\subset H_{w}^{\times}\hat{\otimes}\mathscr{O}_{L}.\end{aligned}\end{align} $$

Since the extension $H_{s,w}/H_{w}$ is totally ramified of degree $p^{s}$ , the subset in formula (1.1.4) is $ p^{s} \left (\mathscr {O}_{H_{w}}^{\times }\hat {\otimes }\mathscr {O}_{L}\right )\subset p^{s} \left (H_{w}^{\times }\hat {\otimes }\mathscr {O}_{L}\right ) $ , as desired.

The nonsplit case

In the nonsplit case, the p-adic intersection multiplicity has no reason to vanish. However, the foregoing argument will still go through if we more modestly show that expression (1.1.3) itself decays at least like a multiple of $p^{s}$ (compare Lemma 4.4.3). The idea to prove this is very simple: we show that if s is large, then for the purposes of computing intersection multiplicities with other divisors $\mathscr {D}$ on $\mathscr {X}$ , the Zariski closure of a CM point of conductor at least $p^{s}$ can almost be approximated by some irreducible component V of the special fiber of $\mathscr {X}$ ; hence the multiplicity will be $0$ if $\mathscr {D}$ arises as a flat extension of its generic fiber. The qualifier ‘almost’ means that the foregoing holds except if $\lvert \mathscr {D}\rvert $ contains V itself, which will be responsible for a multiplicity error term equal to a constant multiple of $ p^{s}$ .

The approximation result (Proposition 4.3.3) is precisely formulated in an (ultra)metric space of irreducible divisors on the local ring of a regular arithmetic surface, which we introduce following a recent work of García Barroso, González Pérez, and Popescu-Pampu [15]. The proof of the result is also rather simple (albeit not effective), relying on Gross’s theory of quasicanonical liftings [18]. The problem of effectively identifying the approximating divisor V is treated in [11].

Subtleties

This description ignores several difficulties of a relatively more technical nature, most of which we deal with by the representation-theoretic approach of [9] (in turn adapted from Yuan, Zhang, and Zhang [29]). Namely, we allow for arbitrary modular parametrizations f, resulting in an extra parameter $\phi $ in the kernels Z and $\mathscr {I}'$ . By representation-theoretic results, one is free to some extent to choose the parameter $\phi $ to work with without losing generality. A fine choice (or rather a pair of choices) for its p-adic component is dictated by the goal of interpolation, while imposing suitable conditions on its other components allows us to circumvent many obstacles in the proof.

1.2 Statement

We now describe our result in the general context in which we prove it – which is the same as that of [9] (and [29]), to which we refer for a less terse discussion of the background. (At some points, we find some slightly different formulations or normalizations from those of [9], to be more natural: see §2.2 for the equivalence.)

Abelian varieties parametrized by Shimura curves

Let F be a totally real field and let $A_{/F}$ be a simple abelian variety of $\mathrm {GL}_{2}$ -type. Assume that $L(A,s)$ is modular (this is known in many cases if A is an elliptic curve). Let $\mathbf {B}$ be a quaternion algebra over the adèles $\mathbf {A}=\mathbf {A}_{F}$ of F, whose ramification set $\Sigma _{\mathbf {B}}$ has odd cardinality and contains all the infinite places. To $\mathbf {B}$ is attached a tower of Shimura curves $\left (X_{U/F}\right )_{U\subset \mathbf {B}^{\infty \times }}$ , with respective Albanese varieties $J_{U}$ . It carries a canonical system of divisor classes $\xi _{U}\in {\mathrm{Cl}}(X_{U})_{\mathbf {Q}}$ of degree $1$ , providing a system $\iota _{\xi }$ of maps $\iota _{\xi , U}\in \mathrm {Hom}\,_{F}(X_{U}, J_{U})_{\mathbf {Q}}$ defined by $P\mapsto P - \deg (P) \xi _{U}$ .

The space

$$ \begin{align*} \pi=\pi_{A, \mathbf{B}}=\varinjlim_{U} \mathrm{Hom}\,^{0}(J_{U}, A) \end{align*} $$

is either zero or a smooth irreducible representation of $\mathbf {B}^{\times }$ (trivial at the infinite places), with coefficients in the number field $M:=\mathrm {End}\,^{0}(A)$ . We assume we are in the case $\pi =\pi _{A , \mathbf {B}}\neq 0$ , which under the modularity assumption and condition (1.2.2) can be arranged by suitably choosing $\mathbf {B}$ . Then for all places $v\nmid \infty $ , we have $L_{v}(A, s)=L_{v}(s-1/2 , \pi )$ in $M\otimes \mathbf {C}$ . We denote by $\omega \colon F^{\times }\backslash \mathbf {A}^{\times } \to M^{\times }$ the central character of $\pi $ . We have a canonical isomorphism $\pi _{A^{\vee }, \mathbf {B}}\cong \pi _{A,\mathbf {B}}^{\vee }$ [29, §1.2.2], and we denote by $( \ , \ )_{\pi }\colon \pi _{A, \mathbf {B}}\otimes \pi _{A^{\vee }, \mathbf {B}}\to M $ the duality pairing.

Heegner points

Let $E/F$ be a CM quadratic extension with associated quadratic character $\eta $ , and assume that E admits an $\mathbf {A}$ -embedding $E_{\mathbf {A}^{}}\hookrightarrow \mathbf {B}^{}$ , which we fix. Then $E^{\times }$ acts on the right on $X=\varprojlim _{U}X_{U}$ . The fixed-points subscheme $X^{E^{\times }}\subset X$ is F-isomorphic to $\mathrm {Spec}\, E^{\mathrm{ab}}$ , and we fix a point $P\in X^{E^{\times }}\left (E^{\mathrm{ab}}\right )$ . Let

$$ \begin{align*} \chi\colon E^{\times}\backslash E_{\mathbf{A}^{\infty}}^{\times} \cong \mathrm{Gal}\left(E^{\mathrm{ab}}/E\right) \to L(\chi)^{\times} \end{align*} $$

be a character valued in a field extension of $L(\chi )\supset M$ , satisfying

$$ \begin{align*} \omega\cdot\chi_{\rvert\mathbf{A}^{\infty\times}}=\mathbf{1}, \end{align*} $$

and define

$$ \begin{align*} A_{E}(\chi):=\left(A\left(E^{\mathrm{ab}}\right)\otimes_{M}L(\chi)_{\chi}\right)^{\mathrm{Gal}\left(E^{\mathrm{ab}}/E\right)}, \end{align*} $$

where $L(\chi )_{\chi }$ is an $L(\chi )$ -line with Galois action by $\chi $ .

Then we have a Heegner point functional

(1.2.1)

(integration for the Haar measure of volume $1$ ) in the space of invariant linear functionals

$$ \begin{align*} \mathrm{H}\left(\pi_{A, \mathbf{B}},\chi\right)\otimes_{L(\chi)}A_{E}(\chi) ,\qquad \mathrm{H}(\pi,\chi):= \mathrm{Hom}_{E_{\mathbf{A}^{}}^{\times}}(\pi\otimes\chi, L(\chi)), \end{align*} $$

where $E_{\mathbf {A}^{}}^{\times }$ acts diagonally. There is a product decomposition $\mathrm {H}(\pi , \chi )=\bigotimes _{v}\mathrm {H}(\pi _{v}, \chi _{v})$ , where similarly $\mathrm {H}(\pi _{v}, \chi _{v}):=\mathrm {Hom}\,_{E_{v}^{\times }}(\pi _{v}\otimes \chi _{v}, L(\chi ))$ .

A local unit of measure for invariant functionals

By foundational local results of Waldspurger, Tunnell, and Saito, the dimension of $\mathrm {H}(\pi ,\chi )$ (for any representation $\pi $ of $\mathbf {B}^{\times }$ ) is either $0$ or $1$ . If A is modular and the global root number

(1.2.2) $$ \begin{align}\begin{aligned} \varepsilon(A_{E}\otimes\chi)=-1, \end{aligned} \end{align} $$

then the set of local root numbers determines a unique quaternion algebra $\mathbf {B}$ over $\mathbf {A}$ , satisfying the conditions already required and containing $E_{\mathbf {A}}$ , such that $\pi _{A, \mathbf {B}}\neq 0$ and $\dim _{L(\chi )} \mathrm {H}\left (\pi _{A, \mathbf {B}},\chi \right ) =1$ . ² We place ourselves in this case; then there is a canonical factorizable generator

$$ \begin{align*} Q_{\left(, \right), dt} =\prod_{v}Q_{\left(, \right)_{v}, dt_{v}} \in \mathrm{H}(\pi,\chi) \otimes \mathrm{H}\left(\pi^{\vee},\chi^{-1}\right) \end{align*} $$

depending on the choice of a pairing $(,)=\prod _{v}(, )_{v}\colon \pi \otimes \pi ^{\vee }\to L(\chi )$ and a measure $dt=\prod _{v} dt_{v}$ on $E_{\mathbf {A}^{}}^{\times }/\mathbf {A}^{\times }$ . It is defined locally as follows. Let us use symbols $V_{\left (A, \chi \right )}$ and $V_{\left (A, \chi \right ), v}$ , which we informally think of as denoting (up to abelian factors) the virtual motive over F with coefficients in $L(\chi )$

$$ \begin{align*} V_{\left(A, \chi\right)}={\mathrm{Res}}_{E/F}(h_{1}(A_{E})\otimes\chi)\ominus \mathrm{ad}(h_{1}(A)(1)) \end{align*} $$

and its local components (the associated local Galois representation or, if v is Archimedean, Hodge structure). Then for each place v of F and any auxiliary $\iota \colon L(\chi )\hookrightarrow \mathbf {C}$ , ³ we define

(1.2.3) $$ \begin{align} \mathscr{L}\left(\iota V_{\left(A, \chi\right), v},s\right)&:= {\zeta_{F,v}(2) L\left(1/2+s,\iota \pi_{E,v}\otimes \iota\chi_{v}\right) \over {L(1, \eta_{v})L(1,\iota \pi_{v}, \mathrm{ad})}} \cdot \begin{cases} 1 \quad &\text{if}\ v\ \text{is finite},\\ \pi^{-1} &\text{if}\ v\mid \infty\end{cases} \quad \in \iota L(\chi), \end{align} $$

(1.2.4) $$ \begin{align} Q_{\left(, \right)_{v}, dt_{v}}\left(f_{1, v}, f_{2,v}, \chi_{v}\right)&:= \iota^{-1}\mathscr{L}\left(\iota V_{\left(A, \chi\right),v},0\right)^{-1} \int_{E_{v}^{\times}/F_{v}^{\times}}\chi(t_{v}) \left(\pi_{v}(t_{v})f_{1,v}, f_{2, v}\right)_{v} dt_{v}. \end{align} $$

We make the situation more canonical by choosing $dt=\prod _{v} dt_{v} $ to satisfy

$$ \begin{align*} \mathrm{vol}\left(E^{\times}\backslash E_{\mathbf{A}}^{\times}/\mathbf{A}^{\times}, dt\right)=1 \end{align*} $$

and by defining, for any $f_{3}\in \pi , f_{4}\in \pi ^{\vee }$ such that $(f_{3}, f_{4})\neq 0$ ,

(1.2.5) $$ \begin{align}\begin{aligned} Q\left( { f_{1} \otimes f_{2}\over f_{3}\otimes f_{4}};\chi\right) := {Q_{\left( , \right), dt}(f_{1}, f_{2},\chi)\over ( f_{3}, f_{4})}. \end{aligned}\end{align} $$

p-adic heights

Let us fix a prime $\mathfrak {p}$ of M and denote by p the underlying rational prime. Suppose from now on that for each $v\mid p$ , $A_{F_{v}}$ has $\mathfrak {p}$ -ordinary (potentially good or semistable) reduction. That is, for a sufficiently large finite extension $L\supset M_{\mathfrak {p}}$ , the rational $\mathfrak {p}$ -Tate module $W_{v}:=V_{p}A\otimes _{M} L$ is a reducible $2$ -dimensional representation of $\mathrm {Gal}\left (\overline {F}_{v}/F_{v}\right )$ :

(1.2.6) $$ \begin{align}\begin{aligned} 0 \to W_{v}^{+} \to W_{v}\to W_{v}^{-} \to 0. \end{aligned} \end{align} $$

Fix such a coefficient field L, and for each $v\mid p$ let $\alpha _{v}\colon F_{v}^{\times }\cong \mathrm {Gal}\left (F_{v}^{\mathrm{ab}}/F_{v}\right )\to L^{\times }$ be the character giving the action on the twist $W_{v}^{+}(-1) $ . The field $L(\chi )$ will from now on be assumed to be an extension of L. Under those conditions, there is a canonical p-adic height pairing

$$ \begin{align*} \langle \ , \ \rangle\colon A_{E}(\chi) \otimes A_{E}^{\vee}\left(\chi^{-1}\right) \to \Gamma_{F}\hat{\otimes}L(\chi), \end{align*} $$

where $\Gamma _{F}:= \mathbf {A}^{\times }/\overline {F^{\times }\widehat {\mathscr {O}}^{p , \times }_{F}}$ (the bar denotes Zariski closure). It is normalized ‘over F’ as in [9, §4.1].

For $f_{1}, f_{3}\in \pi $ , $f_{2}, f_{4}\in \pi ^{\vee }$ , and $P^{\vee } \colon \pi ^{\vee }\otimes \chi \to A_{E}^{\vee }\left (\chi ^{-1}\right )$ the Heegner point functional of the dual, our result will measure the ratio $\left \langle P(f_{1}, \chi ), P^{\vee }\left (f_{2}, \chi ^{-1}\right )\right \rangle /( f_{3}, f_{4})_{\pi } $ against the value at the $f_{i}$ of the ‘unit’ Q. The size will be given by the derivative of the p-adic L-function that we now define.

The p-adic L-function

We continue to assume that A is $\mathfrak {p}$ -ordinary, and review the definition of the p-adic L-function from [9, Theorem A] (in an equivalent form). We start by defining the space on which it lives. Write $\Gamma _{F}=\varprojlim _{n}\Gamma _{F,n}$ as the limit of an inverse system of finite groups, and define

(1.2.7) $$ \begin{align}\begin{aligned} \mathscr{Y}_{F}^{\mathrm{l.c.}} :=\bigcup_{n}\mathrm{Spec} L\left[\Gamma_{F,n}\right] \subset \mathscr{Y}_{F} := \mathrm{Spec} \mathscr{O}_{L}[\kern-1pt[ \Gamma_{F}]\kern-1pt]\otimes_{\mathscr{O}_{L}}L. \end{aligned} \end{align} $$

Then $\mathscr {Y}_{F}$ is a space of continuous characters on $\Gamma _{F}$ , and the $0$ -dimensional ind-scheme $\mathscr {Y}_{F}^{\mathrm{l.c.}}$ is its subspace of locally constant (finite-order) characters.

For a character $\chi ' \colon \mathrm {Gal}\left (E^{\mathrm{ab}}/E\right )\to L^{\prime \times }$ together with an embedding $\iota \colon L'\to \mathbf {C}$ , we shall interpolate the ratio of complete L-functions

$$ \begin{align*} \mathscr{L}\left( \iota V_{\left(A, \chi'\right)}, s\right):= \prod_{v} \mathscr{L}\left( \iota V_{\left(A, \chi'\right),v}, s\right),\qquad \mathscr{L}\left(\iota V_{\left(A, \chi'\right),v}, s\right)=\text{formula~(1.2.3)}, \quad \Re(s)\gg0, \end{align*} $$

where the product runs over all places of F.

We now define the p-interpolation factors for the p-adic L-function. First, recall that the (inverse) Deligne–Langlands gamma factor of a Weil–Deligne representation $W'$ of $\mathrm {Gal}\left (\overline {F}_{v}/F_{v}\right )$ over a p-adic field $L'$ , with respect to a nontrivial character $\psi _{v}\colon F_{v}\to \mathbf {C}^{\times }$ and an embedding $\iota \colon L'\hookrightarrow \mathbf {C}$ , is defined as ⁴

$$ \begin{align*} \gamma(W',\psi_{v})^{-1}:= {L(W')\over \varepsilon(W', \psi_{v}) L(W^{*}(1))}. \end{align*} $$

Let $\psi =\prod _{v}\psi _{v}\colon F\backslash \mathbf {A} \to \mathbf {C}^{\times }$ be the standard additive character such that $\psi _{\infty }(\cdot ) = e^{2\pi i \mathrm {Tr}_{F_{\infty }/\mathbf {R}}(\cdot )}$ ; let $\psi _{E}=\prod _{w}\psi _{E, w}=\psi \circ \mathrm {Tr}_{\mathbf {A}_{E}/\mathbf {A}}$ . For a place $v\mid p $ of F, let $d_{v}$ be a generator of the different ideal of $F_{v}$ . For a character $\chi '\colon \mathrm {Gal}\left (\overline {E}/E\right )\to \mathbf {C}^{\times }$ , we define

(1.2.8) $$ \begin{align}\begin{aligned} e_{v}\left(V_{\left(A, \chi'\right)}\right)= \lvert d_{v}\rvert^{-1/2} {\prod_{w\mid v}\gamma\left(\iota{\mathrm{WD}}\left(W^{+}_{v\mid\mathscr{G}_{E, w}} \otimes \chi^{\prime}_{w}\right), \psi_{E,w}\right)^{-1} \over \gamma(\iota{\mathrm{WD}}(\mathrm{ad}(W_{v})(1))^{++}, \psi_{v})^{-1}} \cdot \mathscr{L}\left( V_{\left(A, \chi'\right),v}\right)^{-1}, \end{aligned} \end{align} $$

where $ \mathrm {ad}(W_{v})(1)^{++}:= \mathrm {Hom}\left (W^{-}_{v}, W^{+}_{ v}\right )(1)= \omega _{v}^{-1}\alpha _{v}^{2}\lvert \ \rvert _{v}^{2}$ and $\iota {\mathrm{WD}}$ is the functor from potentially semistable Galois representations to complex Weil–Deligne representations of [14].

Theorem A. There is a function

$$ \begin{align*} \mathscr{L}_{p}\left(V_{\left(A, \chi\right)}\right)\in \mathscr{O}(\mathscr{Y}_{F}) \end{align*} $$

characterized by the following property. For each complex geometric point $s=\chi _{F}\in \mathscr {Y}_{F}^{\mathrm{l.c.}}(\mathbf {C}) $ , with underlying embedding $\iota \colon L(\chi _{F})\hookrightarrow \mathbf {C}$ ,

$$ \begin{align*} \mathscr{L}_{p}\left(V_{\left(A, \chi\right)},s\right)= \iota {e}_{p}\left({ V}_{\left(A, \chi'\right)}\right) \cdot \mathscr{L}\left(\iota V_{\left(A, \chi'\right)}, 0\right), \quad\chi':= \chi\cdot \chi_{F\rvert\mathrm{Gal}\left(\overline{E}/E\right)}, \end{align*} $$

where $ {e}_{p}\left (V_{\left (A, \chi '\right )}\right ):= \prod _{v\mid p} e_{v}\left (V_{\left (A, \chi '\right )}\right )$ .

The factor $ {e}_{p}\left (V_{\left (A, \chi '\right )}\right )$ coincides with the one predicted by Coates and Perrin-Riou [7] for $V_{\left (A, \chi '\right )}$ (their conjecture motivates the denominator terms in equation (1.2.8), which are constants), up to the removal of a trivial zero from their interpolation factor for $\mathrm {ad}(W_{v})(1)$ .

The p-adic Gross–Zagier formula

We are almost ready to state our main result. Denote by $0\in \mathscr {Y}_{F}$ the point corresponding to $\chi _{F}=\mathbf {1}$ , and define

$$ \begin{align*} \mathscr{L}_{p}'\left(V_{\left(A, \chi\right)}, 0\right):= {\mathrm{d}} \mathscr{L}_{p}\left(V_{\left(A, \chi\right)}, 0\right)\in T_{0}\mathscr{Y}_{F}\cong \Gamma_{F}\hat{\otimes} L(\chi). \end{align*} $$

We say that $\chi _{p}$ is sufficiently ramified if it is nontrivial on a certain open subgroup of $\mathscr {O}_{E,p}^{\times }$ depending only on $\omega _{p}$ (see Assumption 3.4.1 for the precise definition and a comment).

Theorem B. Suppose that the abelian variety $A_{/F}$ is modular and that for all $v\mid p$ , the $\mathrm {Gal}\left (\overline {F}_{v}/F_{v}\right )$ -representation $V_{\mathfrak {p}}A$ is ordinary and potentially crystalline. Let $\chi \colon \mathrm {Gal}\left (E^{\mathrm{ab}}/E\right )\to L(\chi )^{\times }$ be a finite-order character satisfying

$$ \begin{align*} \varepsilon(A_{E}\otimes\chi)=-1, \end{align*} $$

and suppose that $\chi _{p}$ sufficiently ramified.

Then for any $f_{1},f_{3}\in \pi $ , $f_{2},f_{4}\in \pi ^{\vee }$ such that $(f_{3}, f_{4})_{\pi }\neq 0$ , we have

$$ \begin{align*} {\left\langle P(f_{1}, \chi), P^{\vee}\left(f_{2}, \chi^{-1}\right)\right\rangle \over ( f_{3}, f_{4})_{\pi} } = {e}_{p}\left(V_{\left(A, \chi\right)}\right)^{-1}\cdot \mathscr{L}_{p}'\left(V_{\left(\mathbf{A} , \chi\right)}, 0\right) \cdot Q\left( { f_{1} \otimes f_{2}\over f_{3}\otimes f_{4}}; \chi\right) \end{align*} $$

in $ \Gamma _{F}\hat {\otimes } L(\chi )$ .

Remark 1.2.1. The technical assumptions that $\chi _{p}$ is sufficiently ramified and that $V_{\mathfrak {p}}A$ is potentially crystalline ⁵ are removed by p-adic analytic continuation in [13, Theorem B], and replaced by the (necessary) assumption that $\chi _{p}$ is not exceptional for A – that is, ${e}_{p}\left (V_{\left (A, \chi \right )}\right )\neq 0$ (which in our case is implied by the potential crystallinity).

Note that for the removal of the first assumption, one only needs the anticyclotomic formula analogous to [9, Theorem C.4], and not the full generality of the multivariable formula in [13, Theorem D].

Remark 1.2.2. Concrete versions of the formula of Theorem B may be obtained by choosing explicit parametrizations $f_{i}$ and evaluating the term Q. This is a local problem, solved in [5]. In particular, by starting from Theorem B (as generalized to all characters following Remark 1.2.1) and applying the same steps as in the proofs of [10, Theorems 4.3.1, 4.3.3], we obtain the simple p-adic Gross–Zagier formula in anticyclotomic families for elliptic curves $A_{/\mathbf {Q}}$ proposed in [10, Conjecture 4.3.2], and similarly the direct analogue ⁶ of Perrin-Riou’s original result in [26].

The theorem has familiar applications extending to the nonsplit case from [9] (when the other ingredients are available); we leave their formulation to the interested reader, and highlight instead an application specific to this case pointed out in [10], as well as a new application to the nonvanishing conjecture for p-adic heights.

A new proof of a result of Greenberg and Stevens

As noted in [10, Remark 5.2.3], the anticylcotomic formula indicated in the previous remark, combined with a result of Bertolini and Darmon, gives yet another proof (quite likely the most complicated so far, but amenable to generalizations) of the following famous result of Greenberg and Stevens [16]. If $A_{/\mathbf {Q}}$ is an elliptic curve of split multiplicative reduction at p, with Néron period $\Omega _{A}$ and p-adic L-function $L_{p}(A, -)$ on $\mathscr {Y}_{\mathbf {Q}}$ , then $L_{p}(A, \mathbf {1})=0$ and

(1.2.9) $$ \begin{align}\begin{aligned} L_{p}'(A,\mathbf{1}) = \lambda_{p}(A) \cdot {L(A, 1)\over \Omega_{A}},\end{aligned}\end{align} $$

where $\lambda _{p}(A)$ is the $\mathcal {L}$ -invariant of Mazur, Tate, and Teitelbaum [21].

We recall a sketch of the argument, referring to [10] for more details. One chooses an imaginary quadratic field E such that p is inert in E and that the twist $A^{(E)}$ satisfies $L\left (A^{(E)}, 1\right )\neq 0$ . By the anticyclotomic p-adic Gross–Zagier formula, $L_{p}'(A_{E},\mathbf {1})$ is the value at $\chi =\mathbf {1}$ of the height of an anticyclotomic family $\mathscr {P}$ of Heegner points. It is shown in [2, §5.2] that the value $\mathscr {P}(\mathbf {1})$ equals, in an extended Selmer group, the Tate parameter $q_{A,p}$ of $A_{\mathbf {Q}_{p}}$ multiplied by a square root of $L(A_{E}, 1)/\Omega _{A_{E}}$ . The height of $q_{A,p}$ , in the ‘extended’ sense of [21, 25], essentially equals $\lambda _{p}(A)$ . This shows that after harmless multiplication by $L\left (A^{(E)},1 \right )/\Omega _{A^{(E)}}$ , the two sides of equation (1.2.9) are equal.

Exceptional cases and nonvanishing results

Suppose that $A_{/\mathbf {Q}}$ has multiplicative reduction at a prime p inert in E, and that $ L(A_{E}, 1)\neq 0$ . Then for all but finitely many anticyclotomic characters $\chi $ of p-power conductor, a Heegner point in $A_{E}(\chi )$ is nonzero and the p-adic height pairing on $A_{E}(\chi )$ is nondegenerate. This follows from noting, similarly to before, that in the p-adic Gross–Zagier formula in anticyclotomic families for $A_{E}$ , both sides are nonzero since the height side specializes, at the character $\chi =\mathbf {1}$ , to a nonzero multiple of $\lambda _{p}(A)$ , which is in turn nonzero by [1].

A similar argument, applied to the formula in Hida families of [13], will yield the following result: if $A_{/\mathbf {Q}} $ is an elliptic curve with multiplicative reduction and $L(A,1)\neq 0$ , then the Selmer group of the self-dual Hida family ${\mathbf {f}}$ through A has generic rank $1$ , and neither the height regulator nor the cyclotomic derivative of the p-adic L-function of ${\mathbf {f}}$ vanishes. The details will appear in [13].

1.3 Organization of the paper

In §2, we restate our theorems in an equivalent form, a direct generalization of the statements from [9] (up to a correction involving a factor of $2$ , discussed in Appendix B). In §3 we recall the proof strategy from [9], with suitable modifications and corrections. The new argument to treat p-adic local heights in the nonsplit case is developed in §4.

We conclude with two appendices, one dedicated to some local results and the other containing a list of errata to [9].

2 Comparison with [9]

We compare Theorems A and B with the corresponding results from [9]. We continue with the setup and notation of §1.2.

2.1 The p-adic L-function

We deduce our Theorem A from [9, Theorem A].

Let $\sigma ^{\infty }$ be the nearly $\mathfrak {p}$ -ordinary, M-rational [9, Definition 1.2.1] representation of $\mathrm {GL}_{2}(\mathbf {A})$ attached to A as in [9]. In [9, Theorem A], we constructed a p-adic L-function

$$ \begin{align*} L_{p, \alpha}(\sigma_{E}), \end{align*} $$

which is a bounded function on a rigid space $\mathscr {Y}^{\prime \rm rig}_{/L}$ (denoted by $\mathscr {Y}'$ in [9]). In the construction of that theorem (and in all this paper), we use the same additive character $\psi _{p}=\prod _{v\mid p}\psi _{v}$ as in Theorem A (and see the correction in Appendix B for the exact ring of definition of $L_{p, \alpha }(\sigma _{E})$ ).

The space $\mathscr {Y}^{\prime \rm rig}=\mathscr {Y}^{\prime \rm rig}_{\omega , V^{p}}$ parametrizes certain continuous p-adic characters of $E^{\times }\backslash E_{\mathbf {A}}^{\times }$ invariant under an arbitrarily fixed compact open subgroup $V^{p}\subset E_{\mathbf {A}^{p\infty }}^{\times }$ . Boundedness means precisely that we may (and do) identify $L_{p, \alpha }(\sigma _{E})$ with a function on a corresponding scheme

(2.1.1) $$ \begin{align}\begin{aligned} \mathscr{Y}'\subset \mathrm{Spec} \mathscr{O}_{L}\left[\kern-1pt[ E^{\times}\backslash E_{\mathbf{A}^{\infty}}^{\times}/V^{p} \right]\kern-1pt]\otimes {L} \end{aligned}\end{align} $$

that, when also viewed as a space of characters $\chi '$ , is the subscheme cut out by the closed condition $\omega \cdot \chi ^{\prime }_{\rvert \widehat {\mathscr {O}}_{F}^{p,\times }}=\mathbf {1}$ . Similarly to $\mathscr {Y}_{F}$ , the scheme $\mathscr {Y}'$ contains a $0$ -dimensional subscheme $\mathscr {Y}^{\prime \rm l.c.}$ parametrizing the locally constant characters in $\mathscr {Y}'$ . The function $L_{p, \alpha }(\sigma _{E})$ is characterized by the following property. Denote by $D_{K}$ the discriminant of a number field K. Then at all $\chi '\in \mathscr {Y}'$ with underlying embedding $\iota \colon L\hookrightarrow \mathbf {C}$ , we have

(2.1.2) $$ \begin{align}\begin{aligned} L_{p,\alpha}(\sigma_{E})(\chi') =\prod_{v\mid p}Z_{v}^{\circ}\left(\chi^{\prime}_{v}, \psi_{v}\right)\cdot { \pi^{2\left[F:\mathbf{Q}\right]} \lvert D_{F}\rvert^{1/2} \over 2\zeta_{F}(2)} \cdot \mathscr{L}\left(\iota V_{\left(A, \chi'\right)}\right) \end{aligned}\end{align} $$

for certain local factors $Z_{v}^{\circ }$ .

Fix a finite-order character

$$ \begin{align*} \chi\colon E^{\times}\backslash E_{\mathbf{A}^{\infty}}^{\times} \cong \mathrm{Gal}\left(E^{\mathrm{ab}}/E\right) \to L(\chi)^{\times} \end{align*} $$

satisfying $\omega \cdot \chi _{\rvert \mathbf {A}^{\infty \times }}=\mathbf {1}, $ and consider the map

$$ \begin{align*}\begin{aligned} j_{\chi} \colon \mathscr{Y}_{F} &\to \mathscr{Y}',\\ \chi_{F} &\mapsto \chi\cdot \chi_{F} \circ N_{E_{\mathbf{A}^{\infty\times}/\mathbf{A}^{\infty\times}}}. \end{aligned}\end{align*} $$

Proof of Theorem A

Define

(2.1.3) $$ \begin{align}\begin{aligned} C\left(\chi_{p}'\right):={ e_{p} \left(V_{\left(A, \chi'\right)}\right) \over \prod_{v\mid p }Z_{v}^{\circ}\left(\chi^{\prime}_{v}, \psi_{v}\right)}. \end{aligned}\end{align} $$

We show in Proposition A.1.2 that this is a constant in $C\in L$ , independent of $\chi ^{\prime }_{p}$ .

Define

(2.1.4) $$ \begin{align}\begin{aligned} \mathscr{L}_{p}\left(V_{\left(A, \chi\right)}\right):= {2 \zeta_{F}(2)\over \pi^{2\left[F:\mathbf{Q}\right] } \lvert D_{F}\rvert^{1/2}} \cdot C\cdot L(1, \sigma_{v}, \mathrm{ad}) \cdot j_{\chi}^{*}L_{p, \alpha}(\sigma_{E}), \end{aligned}\end{align} $$

a function in $\mathscr {O}(\mathscr {Y}_{F})$ . It is clear from the definition and equation (2.1.2) that it satisfies the required interpolation property.

2.2 Equivalence of statements

We now restate Theorem B in a form that directly generalizes [9, Theorem B]. It is the form in which we will prove it, for convenience of reference.

We retain the setup of §1.2. Let ${\mathrm{d}}_{F}$ be the $\Gamma _{F}$ -differential defined before [9, Theorem B]. For all $v\nmid \infty $ , let $dt_{v}$ be the measure on $E_{v}^{\times }/F_{v}^{\times }$ specified in [9, paragraph after (1.1.2)] if $v\nmid \infty $ and the measure giving $\mathbf {C}^{\times }/\mathbf {R}^{\times }$ volume $2$ if $v\mid \infty $ .

Theorem 2.2.1. Retain the assumptions of Theorem B, and fix a decomposition $(, )_{\pi }=\prod _{v}( ,)_{v}$ , with $(1, 1)_{v}=1$ if $v\mid \infty $ . Then for all $f_{1}\in \pi , f_{2}\in \pi ^{\vee }$ ,

(2.2.1) $$ \begin{align}\begin{aligned} \left\langle P(f_{1}, \chi), P^{\vee}\left(f_{2}, \chi^{-1}\right)\right\rangle = {c_{E}}\cdot \prod_{v\mid p}Z_{v}^{\circ}( \chi_{v})^{-1}\cdot {\mathrm{d}}_{F} L_{p, \alpha}(\sigma_{A,E})(\chi)\cdot \prod_{v\nmid \infty} Q_{\left(, \right)_{v}, dt_{v}}(f_{1}, f_{2}, \chi)\end{aligned}\end{align} $$

in $ \Gamma _{F} \hat {\otimes } L(\chi )$ , where

$$ \begin{align*} c_{E}:={ \zeta_{F}(2)\over (\pi/2)^{\left[F:\mathbf{Q}\right]}\lvert D_{E}\rvert^{1/2} L(1, \eta) } \in\mathbf{Q}^{\times}. \end{align*} $$

Lemma 2.2.2. Theorem 2.2.1 is equivalent to Theorem B. When every prime $v\mid p$ splits in E, it specializes to [9, Theorem B] as corrected in Appendix B.

Proof. The second assertion is immediate; we prove the first one. First, we note that equation (2.2.1) is equivalent to

(2.2.2) $$ \begin{align} &\left\langle P(f_{1}, \chi), P^{\vee}\left(f_{2}, \chi^{-1}\right)\right\rangle\over (f_{3},f_{4})_{\pi} \nonumber \\ &\quad = {c_{E}}\cdot \prod_{v\mid p}Z_{v}^{\circ}( \chi_{v})^{-1}\cdot {\mathrm{d}}_{F} L_{p, \alpha}(\sigma_{A,E})(\chi) \cdot 2^{-\left[F:\mathbf{Q}\right]}\prod_{v} {Q_{\left(, \right)_{v}, dt_{v}}(f_{1}, f_{2}, \chi) \over \left(f_{3,v}, f_{4,v}\right)_{v}} \end{align} $$

for any $f_{3}\in \pi , f_{4}\in \pi ^{\vee }$ with $f_{3, \infty }=f_{4, \infty }=1$ and $(f_{3}, f_{4})_{\pi }\neq 0$ (the extra power of $2$ comes from the Archimedean places). The left-hand side of equation (2.2.2) is the same as that of the formula of Theorem B, and the product of the terms after the L-derivative in the right-hand side equals

$$ \begin{align*} & 2^{-\left[F:\mathbf{Q}\right]} {\prod_{v} dt_{v}\over dt} \cdot Q\left( { f_{1} \otimes f_{2}\over f_{3}\otimes f_{4}}; \chi\right)\\ &\quad = 2^{1-\left[F:\mathbf{Q}\right]} \left\lvert D_{E/F}\right\rvert^{1/2}\lvert D_{F}\rvert^{1/2}\pi^{-\left[F:\mathbf{Q}\right]}L(1, \eta) \cdot Q\left( { f_{1} \otimes f_{2}\over f_{3}\otimes f_{4}}; \chi\right), \end{align*} $$

because the measure $\prod _{v} dt_{v}$ (resp., $dt$ ) gives $E^{\times }\backslash \mathbf {A}_{E}^{\times }/\mathbf {A}^{\times }$ volume $2 \left \lvert D_{E/F}\right \rvert ^{1/2} \lvert D_{F}\rvert ^{1/2} \pi ^{-\left [F:\mathbf {Q}\right ]}L(1, \eta )$ (resp., $1$ ).

Next, we have ${\mathrm{d}}_{F}L_{p, \alpha }(\sigma _{E})(\chi )={1\over 2} {\mathrm{d}}\left ( j_{\chi }^{*} L_{p, \alpha }(\sigma _{E}) \right )(\mathbf {1})$ , and it is clear from comparing the interpolation properties that

$$ \begin{align*} \prod_{v\mid p}Z_{v}^{\circ}( \chi_{v})^{-1} \cdot {1\over 2} {\mathrm{d}} \left( j_{\chi}^{*} L_{p, \alpha}(\sigma_{E}) \right)(\mathbf{1})= {\pi^{2\left[F:\mathbf{Q}\right] } \lvert D_{F}\rvert^{1/2} \over 2\zeta_{F}(2)} \cdot {e}_{p}\left(V_{\left(A, \chi\right)}\right)^{-1}\cdot \mathscr{L}_{p}'\left(V_{\left(A, \chi\right)}, 0\right). \end{align*} $$

It follows that the right-hand side of equation (2.2.1) equals $c \cdot {e}_{p}\left (V_{\left (A, \chi \right )}\right )^{-1}\cdot \mathscr {L}_{p}'\left (V_{\left (A, \chi \right )}, 0\right ) \cdot Q\left ( { f_{1} \otimes f_{2}\over f_{3}\otimes f_{4}}; \chi \right ),$ where

$$ \begin{align*} c= {c_{E}} \cdot {\pi^{2\left[F:\mathbf{Q}\right] } \lvert D_{F}\rvert^{1/2} \over 2\zeta_{F}(2)} \cdot 2^{1-\left[F:\mathbf{Q}\right]} \left\lvert D_{E/F}\right\rvert^{1/2}\lvert D_{F}\rvert^{1/2} \pi^{-[F:\mathbf{Q}]}L(1, \eta)=1.\\[-47pt] \end{align*} $$

3 Structure of the proof

We review the formal structure of the proof in [9], dwelling only on those points where the arguments need to be modified or corrected. For an introductory description with some more details than are given in §1.1, see [9, §1.7]. Readers interested in a detailed understanding of the present section are advised to keep a copy of [9] handy.

3.1 Notation and setup

We very briefly review some notation and definitions from [9], which will be used throughout the paper.

Galois groups

If K is a perfect field, we denote by $\mathscr {G}_{K}:=\mathrm {Gal}\left (\overline {K}/K\right )$ its absolute Galois group.

Local fields

For v finite a place of F, we denote by $\varpi _{v}$ a fixed uniformizer and by $q_{F, v}$ the cardinality of the residue field of $F_{v}$ . We denote by $d_{v}$ a generator of the absolute different of $F_{v}$ , by $D_{v}$ a generator of the relative discriminant of $E_{v}/F_{v}$ (equal to $1$ unless v ramifies in E), and by $e_{v}$ the ramification degree of $E_{v}/F_{v}$ . If $w\mid v$ is a place of E, we denote by $q_{v}\colon E_{v}\to F_{v}$ and $q_{w}\colon E_{w}\to F_{v}$ the relative norm maps.

We denote by $\psi =\prod _{v}\psi _{v}\colon F\backslash \mathbf {A} \to \mathbf {C}^{\times }$ the additive character fixed before Theorem A.

Base change of rings and schemes

If R is a ring, $R'$ is an R-algebra, M is an R-module, and S is an R-scheme, we denote $M_{R'}=M\otimes _{R}R', S_{R'}= S\times _{\mathrm {Spec} R}\mathrm {Spec} R'$ .

Groups, measures, integration

We adopt the same notation and choices of measures as in [9, §1.9], including a regularized integration $\int ^{*}$ . In particular, $T:= {\mathrm{Res}}_{E/F}{\mathbf {G}}_{m, E}, Z={\mathbf {G}}_{m, F}$ , and on the adelic points of $T/Z$ we use two measures $dt$ (the same as introduced before Theorem 2.2.1) and $d^{\circ }t$ . The measure denoted by $dt$ in the introduction will not be used.

Operators at p

Let $v\mid p$ be a place of F. We denote by $\varpi _{v}$ a fixed uniformizer at v. For $r\geq 1$ we let $K_{1}^{1}\left (\varpi _{v}^{r}\right ) \subset \mathrm {GL}_{2}\left (\mathscr {O}_{F, v}\right )$ be the subgroup of matrices which become upper unipotent upon reduction modulo $\varpi ^{r}$ . We denote by

$$ \begin{align*} \mathrm{U}_{v,*}=K^{1}_{1}\left(\varpi_{v}^{r}\right) \left(\begin{array}{cc}1&\\&\varpi_{v}^{-1}\end{array}\right)K^{1}_{1}\left(\varpi_{v}^{r}\right), \qquad \mathrm{U}_{v}^{*}=K^{1}_{1}\left(\varpi_{v}^{r}\right) \left(\begin{array}{cc}\varpi^{r}&\\&1\end{array}\right)K^{1}_{1}\left(\varpi_{v}^{r}\right) \end{align*} $$

the usual double coset operators, and define

$$ \begin{align*} w_{r,v}:= \left(\begin{array}{cc}&1\\-\varpi_{v}^{r}&\end{array}\right)\in \mathrm{GL}_{2}(F_{v}). \end{align*} $$

We also define $w_{r}:=\prod _{v\mid p}w_{r,v}\in \mathrm {GL}_{2}(F_{p})$ , and if $(\beta _{v})_{v\mid p}$ are characters of $F_{v}^{\times }$ , we denote $\beta _{p}(\varpi ):=\prod _{v\mid p}\beta _{v}(\varpi _{v})$ .

Spaces of characters

We denote by $\mathscr {Y}_{F}, \mathscr {Y}', \mathscr {Y}$ , respectively, the schemes over L defined in formulas (1.2.7) and (2.1.1) and the subscheme of $\mathscr {Y}$ cut out by the condition $\chi _{\rvert \mathbf {A}^{\infty \times }}=\omega ^{-1}$ . We add to this notation a superscript ‘l.c.’ to denote the ind-subschemes of locally constant characters (which has a model over a finite extension of M in L).

Let $\mathscr {I}_{\mathscr {Y}/\mathscr {Y}'}$ be the ideal sheaf of $\mathscr {Y}\subset \mathscr {Y}'$ . If $\mathscr {M}$ is a coherent $\mathscr {O}_{\mathscr {Y}'}$ -module, we denote

$$ \begin{align*} {\mathrm{d}}_{F}\colon \mathscr{M}\otimes_{\mathscr{O}_{\mathscr{Y}'}} \mathscr{I}_{\mathscr{Y}/\mathscr{Y}'} \to \mathscr{M} \otimes_{\mathscr{O}_{\mathscr{Y}'}} \mathscr{I}_{\mathscr{Y}/\mathscr{Y}'}/\mathscr{I}_{\mathscr{Y}/\mathscr{Y}'}^{2} =\mathscr{M}_{\rvert\mathscr{Y}}\hat{\otimes}\Gamma_{F} \end{align*} $$

the normal derivative (compare the definition before [9, Theorem B]).

Kirillov models

Let $\sigma ^{\infty }=\bigotimes _{v\nmid \infty }\sigma _{v} $ be the M-rational automorphic representation of $\mathrm {GL}_{2}(\mathbf {A})$ attached to A, and denote abusively still by $\sigma ^{\infty }$ its base change to L. For every place v the representations $\sigma _{v}$ of $\mathbf {B}_{v}^{\times }$ and $\pi _{v}$ of $\mathrm {GL}_{2}(F_{v})$ are Jacquet–Langlands correspondents.

For $v\mid p$ , we denote by

$$ \begin{align*} \mathscr{K}_{\psi_{v}}\colon \sigma_{v} \to C^{\infty}\left(F_{v}^{\times}, L\right) \end{align*} $$

a fixed rational Kirillov model.

Orthogonal spaces

We define ${\mathbf {V}}:=\mathbf {B}$ equipped with the reduced norm q, a quadratic form valued in $\mathbf {A}$ . The image of $E_{\mathbf {A}}$ is a subspace ${\mathbf {V}}_{1}$ of the orthogonal space ${\mathbf {V}}$ , and we let ${\mathbf {V}}_{2}$ be its orthogonal complement. The restriction $q_{\rvert {\mathbf {V}}_{1}}$ is the adelization of the norm of $E/F$ .

Schwartz spaces and Weil representation

If $\bf V'$ is any one of the spaces already discussed, we denote by ${\overline {\mathscr {S}}}({\mathbf {V}}'\times \mathbf {A}^{\times })=\bigotimes ^{\prime }_{v} {\overline {\mathscr {S}}}\left ({\mathbf {V}}^{\prime }_{v}\times F_{v}^{\times }\right )$ the Fock space of Schwartz functions considered in [9]. (This differs from the usual Schwartz space only at infinity.) There is a Weil representation

$$ \begin{align*} r=r_{\psi}\colon \mathrm{GL}_{2}(\mathbf{A})\times {\mathrm{O}}({\textbf{V}}, q)\to \mathrm{End}\, {\overline{\mathscr{S}}}({\textbf{V}}\times \mathbf{A}^{\times}), \end{align*} $$

defined as in [9, §3.1]. The orthogonal group of $\bf V$ naturally contains the product $T(\mathbf {A})\times T(\mathbf {A})$ acting by left and right multiplication on ${\mathbf {V}}$ . The Weil representation also depends on a choice of additive characters $\psi $ . The restriction $r_{\rvert T(\mathbf {A})\times T(\mathbf {A})}$ preserves the decomposition ${\mathbf {V}}_{1}\oplus { \bf V}_{2}$ , hence it accordingly decomposes as $r_{1}\oplus r_{2}$ .

Special data at p

We list the functions at the places $v\mid p$ that we use.

Define

(3.1.1) $$ \begin{align}\begin{aligned} W_{v}(y):= \mathbf{1}_{\mathscr{O}_{F, v}-{0}}(y)\lvert y\rvert_{v} \alpha_{v}(y),\end{aligned}\end{align} $$

the ordinary vector in the fixed Kirillov model $\mathscr {K}_{\psi _{v}}$ of $\sigma _{A,v}$ . We consider

(3.1.2) $$ \begin{align}\begin{aligned} \varphi_{v}=\varphi_{v,r}:= \mathscr{K}_{\psi_{v}}^{-1} \left(\alpha_{v}(\varpi_{v})^{-r}w_{r} W_{v}\right) \in \sigma_{v}. \end{aligned}\end{align} $$

Now we consider Schwartz functions. We let

$$ \begin{align*} {\textbf{B}}_{v}\cong M_{2}(F_{v}) \end{align*} $$

be the indefinite quaternion algebra over $F_{v}$ ; this choice is justified a posteriori by Corollary A.2.3. The following choices of functions correct and modify the ones fixed in [9] (compare the errata in Appendix B); note in particular that we will use two different functions on ${\mathbf {V}}_{2, v}$ .

Decompose orthogonally ${\mathbf {V}}_{v}={\mathbf {V}}_{1,v}\oplus {\mathbf {V}}_{2, v}$ , where ${\mathbf {V}}_{1,v}=E_{v}$ under the fixed embedding $E_{\mathbf {A}^{\infty }}\hookrightarrow \mathbf {B}^{\infty }$ . We define the following Schwartz functions on, respectively, $F_{v}^{\times }$ and its product with ${\mathbf {V}}_{1,v}$ , ${\mathbf {V}}_{2,v}$ , ${\mathbf {V}}_{v}$ :

(3.1.3) $$ \begin{align}\begin{aligned} \phi_{F,r}(u)&:=\delta_{1, U_{F, r}}(u),& \text{where } \delta_{1, U_{F, r}}(u)&:= {\mathrm{vol}\left(\mathscr{O}_{F,v}^{\times}\right)\over \mathrm{vol}\left(1+\varpi^{r}\mathscr{O}_{F,v}\right)}\mathbf{1}_{1+\varpi^{r}\mathscr{O}_{F,v}}(u),\\ \phi_{1, r}(x_{1}, u)& := \delta_{1, U_{T,r}}(x_{1}) \delta_{1, U_{F,r}}(u),& \text{where } \delta_{1, U_{T,r}}(x_{1}) &= {\mathrm{vol}\left(\mathscr{O}_{E,v}\right)\over \mathrm{vol}\left(1+\varpi^{r}\mathscr{O}_{E,v}\right)} \mathbf{1}_{1+\varpi^{r}\mathscr{O}_{E,v}}(x_{1}), \end{aligned}\end{align} $$

and

(3.1.4) $$ \begin{align}\begin{aligned} \phi_{2}^{\circ}(x_{2}, u) & := \mathbf{1}_{\mathscr{O}_{{\textbf{V}}_{2,v}}}(x_{2}) \mathbf{1}_{\mathscr{O}_{F,v}^{\times}}(u),\\ \phi_{2, r}(x_{2}, u)& := e_{v}^{-1}\lvert d\rvert_{v}\cdot \mathbf{1}_{\mathscr{O}_{{\textbf{V}}_{2,v}}\cap q^{-1}\left(-1+\varpi^{r}\mathscr{O}_{F, v}\right)}(x_{2}) \mathbf{1}_{1+\varpi^{r}\mathscr{O}_{F,v}}(u), \\ \phi_{r}(x, u)&:= \phi_{1, r}(x_{1},u) \phi_{2, r}(x_{2}, u).&& \end{aligned}\end{align} $$

p-adic modular forms and q-expansions

In [9, §2], we defined the notion of Hilbert automorphic forms and twisted Hilbert automorphic forms (the latter depend on an extra variable $u \in \mathbf {A}^{\times }$ ). We also defined the associated space of q-expansions, and a less redundant space of reduced q-expansions. When the coefficient field is a finite extension L of $\mathbf {Q}_{p}$ , these spaces are endowed with a topology. We have an (injective) reduced-q-expansion map on modular forms, denoted by

$$ \begin{align*} \varphi'\mapsto {}^{\textbf{q}}\varphi'. \end{align*} $$

The image of modular forms (resp., cusp forms) of level $K^{p}K^1_{1}(p^{\infty })\subset \mathrm {GL}_{2}(\mathbf {A}^{p\infty })$ , parallel weight $2$ , and central character $\omega ^{-1}$ is denoted by ${\mathbf {M}}={\mathbf {M}}\left (K^{p}, \omega ^{-1}\right )$ (resp., ${\mathbf {S}}$ ). The closure of ${\mathbf {M}}$ (resp., ${\mathbf {S}}$ ) in the space of q-expansions with coefficients in L is denoted ${\mathbf {M}}'$ (resp., ${\mathbf {S}}'$ ) and its elements are called p-adic modular forms (resp., cusp forms).

If $\mathscr {Y}^{?}=\mathscr {Y}_{F}, \mathscr {Y}'$ , we define the notion of a $\mathscr {Y}^{?}$ -family of modular forms by copying word for word [9, Definition 2.1.3]; the resulting notion coincides with that of bounded families on the analogous rigid spaces considered there.

For a finite set of places S disjoint from those above p, we have also defined a certain quotient space $\overline {\mathbf {S}}^{\prime }_{S}$ of cuspidal reduced q-expansions modulo those whose coefficients of index $a\in F^{\times }\mathbf {A}^{S\infty \times }$ all vanish. According to [9, Lemma 2.1.2], for any S the reduced-q-expansion map induces an injection

(3.1.5) $$ \begin{align}\begin{aligned} {\textbf{S}}\hookrightarrow \overline{\textbf{S}}_{S}'. \end{aligned}\end{align} $$

p-adic Petersson product and p-critical forms

For $\varphi ^{p}\in \sigma $ , we defined in [9, Proposition 2.4.4] a functional

(3.1.6) $$ \begin{align}\begin{aligned} \ell_{\varphi^{p},\alpha}\colon {\textbf{M}}\left(K^{p}, \omega^{-1}, L\right)\to L, \end{aligned}\end{align} $$

whose restriction to classical modular forms equals, up to an adjoint L-value, the limit as $r\to \infty $ of Petersson products with antiholomorphic forms $\varphi ^{p}\varphi _{p,r}\in \sigma $ with component $\varphi _{p, r}=\prod _{v\mid p}\varphi _{v, r}$ as in equation (3.1.2).

Set $v\mid p$ . We say that a form or q-expansion over a finite-dimensional $\mathbf {Q}_{p}$ -vector space L is v-critical if its coefficients $a_{*}$ (where $*\in \mathbf {A}^{\infty \times }$ ) satisfy

(3.1.7) $$ \begin{align}\begin{aligned} a_{m\varpi_{v}^{s}}= O\left(q_{F, v}^{s}\right) \end{aligned} \end{align} $$

in L, uniformly in $m\in \mathbf {A}^{\infty \times }$ . Here for two functions $f, g\colon \mathbf {N}\to L$ , we write

$$ \begin{align*} f=O(g)\Longleftrightarrow \text{there is a constant}\, c>0\, \text{such that}\, \lvert f(s)\rvert\leq c\lvert g(s)\rvert \text{ for all sufficiently large}\ s. \end{align*} $$

The space of p-critical forms is the sum of the spaces of v-critical forms for $v\mid p$ . Any element in those spaces is annihilated by $\ell _{\varphi ^{p},\alpha }$ .

3.2 Analytic kernel

The analytic kernel is a p-adic family of theta-Eisenstein series, related to the p-adic L-function. We review its main properties.

Proposition 3.2.1. There exist p-adic families of q-expansions of modular forms $\mathscr {E}$ over $\mathscr {Y}_{F}$ and $\mathscr {I}$ over $\mathscr {Y}'$ , satisfying the following:

1. 1. For any $\chi _{F}\in \mathscr {Y}_{F}^{\mathrm{l.c.}}(\mathbf {C})$ and any $r=(r_{v})_{v\mid p}$ satisfying $c(\chi _{F})\mid p^{r}$ , we have the identity of q-expansions of twisted modular forms of weight 1:

   $$ \begin{align*} \mathscr{E}\left(u,\phi_{2}^{p\infty};\chi_{F}\right)= \lvert D_{F}\rvert {L^{(p)}(1, \eta\chi_{F})\over L^{(p)}(1, \eta)} \, {}^{\mathbf{q}} E_{r}(u, \phi_{2} ,\chi_{F}), \end{align*} $$
   where
   (3.2.1) $$ \begin{align}\begin{aligned} E_{r}(g,u, \phi_{2} ,\chi_{F}):=\sum_{\gamma\in P^{1}(F)\backslash SL_{2}(F)} \delta_{\chi_{F}, r} (\gamma gw_{r}) r(\gamma g) \phi_{2}(0,u)\end{aligned}\end{align} $$
   is the Eisenstein series defined in [9, §3.2], with respect to $\phi _{2}=\phi _{2}^{p\infty }(\chi _{F})\phi ^{\circ }_{2,p\infty }$ , with $\phi _{2,v}^{\circ }$ as in formula (3.1.4) for $v\mid p$ and $\phi _{2, v}(\chi _{v}) $ for $v\nmid \infty $ and $\phi _{2, v}^{\circ }$ for $v\mid \infty $ as defined in [9, §3.2].

2. 2. For $\phi _{1}\in \overline {\mathscr {S}}({\mathbf {V}}_{1}\times {\mathbf {A}}^{\times })$ and $\chi '\in \mathscr {Y}^{\prime \rm l.c.}$ , consider the twisted modular form of weight $1$ with parameter $t\in E_{\mathbf {A}}^{\times }$ :

   (3.2.2) $$ \begin{align}\begin{aligned} \theta(g,(t, 1) u, \phi_{1}):=\sum_{x_{1}\in E} r_{1}(g, (t, 1))\phi_{1}(x, u). \end{aligned}\end{align} $$
   For any $\chi ' \in \mathscr {Y}^{\prime \rm l.c.}$ , define $\chi _{F} := \omega ^{-1}\chi ^{\prime }_{\rvert \mathbf {A}^{\times } }\in \mathscr {Y}_{F}^{\mathrm{l.c.}}$ . Then for any $r=(r_{v})_{v\mid p}$ satisfying $r_{v}\geq 1$ and $c(\chi _{F})\mid p^{r}$ , we have
   (3.2.3) $$ \begin{align}\begin{aligned} \mathscr{I}(\phi^{p\infty}; \chi')= {c_{U^{p}} \lvert D_{E}\rvert^{1/2} \over \lvert D_{F}\rvert^{1/2}} \int_{[T]}^{*}\chi'(t) \sum_{u\in\mu_{U^{p}}^{2}\backslash F^{\times}} {}^{\mathbf{q}}\theta((t,1),u, \phi_{1}; \chi') \mathscr{E}\left(q(t)u,\phi_{2}^{p\infty};\chi_{F}\right) d^{\circ }t, \end{aligned}\end{align} $$
   where for $v\mid p$ , we have $\phi _{1,v}=\phi _{1, v,r}$ is as in formula (3.1.3).

3. 3. We have

   (3.2.4) $$ \begin{align} \ell_{\varphi^{p},\alpha}(\mathscr{I}(\phi^{p\infty}))= L_{p, \alpha}(\sigma_{E})\cdot \prod_{v\mid p} \lvert d\rvert^{2}_{v}\lvert D\rvert_{v} \prod_{v\nmid p \infty} \mathscr{R}_{v}^{\natural}\left(W_{v}, \phi_{v}, \chi_{v}'\right), \end{align} $$
   where the local terms $\mathscr {R}_{v}^{\natural } $ are as in [9, Propositions 3.5.1, 3.6.1].

Proof. Part 1 is [9, Proposition 3.3.2]. Part 2 summarizes [9, §3.4]. Part 3 is [9, (3.7.1)], with the correction of Appendix B.

Derivative of the analytic kernel

We denote

(3.2.5) $$ \begin{align}\begin{aligned} \mathscr{I}'(\phi^{p\infty};\chi):= {\mathrm{d}}_{F}\mathscr{I}(\phi^{p\infty};\chi), \end{aligned}\end{align} $$

a p-adic modular form with coefficients in $\Gamma _{F}\hat {\otimes } L(\chi )$ .

3.3 Geometric kernel

The geometric kernel function [9, §§5.2, 5.3] is related to the heights of Heegner points. We recall its construction and modularity.

CM divisors

For any $x\in \mathbf {B}^{\infty \times }$ , we have a Hecke translation ${\mathrm{T}}_{x}\colon X\to X$ , and a Hecke correspondence $Z(x)_{U}$ on $X_{U }\times X_{U}$ . Fix any $P\in X^{E^{\times }}\left (E^{\mathrm{ab}}\right )$ , and for $x\in \mathbf {B}^{\infty \times }$ , let $[x]:={\mathrm{T}}_{x}P$ be the Hecke-translate of P by x, and let $[x]_{U}$ be its image in $X_{U}$ . If $H/E$ is any finite extension, the points in $X_{U, H}$ corresponding to Galois orbits of points of the form $[x]_{U}$ are called CM points (for the CM field E).

Let ${\mathrm{Cl}}(X_{U,{\overline {F}}})_{\mathbf {Q}}\supset {\mathrm{Cl}}^{0}(X_{U,{\overline {F}}} )_{\mathbf {Q}}$ be the space of divisor classes with $\mathbf {Q}$ -coefficients and its subspace consisting of classes with degree $0$ on every connected component. Denote by ${(\ )}^{0}\colon {\mathrm{Cl}}(X_{U,{\overline {F}}})_{\mathbf {Q}}\to {\mathrm{Cl}}^{0}(X_{U,{\overline {F}}})_{\mathbf {Q}}$ the linear section of the inclusion whose kernel is spanned by the push-forwards to $X_{U, \overline {F}}$ of the classes of the canonical bundles of the connected components of $X_{U', \overline {F}} $ , for any sufficiently small $U'$ .

We define the $\chi $ -isotypic CM divisors

$$ \begin{align*}\begin{aligned} t_{\chi} &:=\int^{*}_{[T]}\chi(t) \left[t^{-1}\right]_{U}d^{\circ}t \in \mathrm{Div}\left(X_{U, \overline{F}}\right)_{L(\chi)}, \\ t_{\chi}^{0} &:=\int^{*}_{[T]}\chi(t) \left[t^{-1}\right]^{0}_{U}d^{\circ}t \in \mathrm{Div}^{0}\left(X_{U, \overline{F}}\right)_{L(\chi)}, \end{aligned}\end{align*} $$

where the integrations simply reduce to (normalized) finite sums.

Generating series

For $a\in \mathbf {A}^{\infty \times }$ , $\phi ^{\infty }\in \overline {\mathscr {S}}({\mathbf {V}}\times \mathbf {A}^{\times })$ , consider the correspondences

(3.3.1) $$ \begin{align} {\widetilde Z}_{a}(\phi^{\infty}):= c_{U^{p}}w_{U} \lvert a\rvert \sum_{x\in U \backslash \mathbf{B}^{\infty\times}/ U }\phi^{\infty}\left(x, aq(x)^{-1}\right) Z(x)_{U}, \end{align} $$

where $w_{U}=\lvert \{\pm 1\}\cap U\rvert $ and $c_{U^{p}}$ is defined in [9, (3.4.3)]. By [9, Theorem 5.2.1] (due to Yuan, Zhang, and Zhang), there is an automorphic form

(3.3.2) $$ \begin{align}\begin{aligned} {\widetilde Z}(\phi^{\infty}) \in C^{\infty}(\mathrm{GL}_{2}(F)\backslash \mathrm{GL}_{2}(\mathbf{A}), \mathbf{C})\otimes_{\mathbf{Q}} {\mathrm{Pic}}(X_{U}\times X_{U})_{\mathbf{Q}}, \end{aligned}\end{align} $$

whose ath reduced coefficient is the image of ${\widetilde Z}_{a}(\phi ^{\infty })$ for each $a\in \mathbf {A}^{\infty \times }$ .

Let

(3.3.3) $$ \begin{align}\begin{aligned} \langle\ , \ \rangle=\langle\ , \ \rangle_{X}\colon J^{\vee}\left(\overline{F}\right) \times J\left(\overline{F}\right) \to \Gamma_{F}\hat{\otimes} L \end{aligned}\end{align} $$

be the p-adic height pairing defined as in ⁷ [9, Lemma 5.3.1]. (We abusively omit the subscript X, as we will no longer need to use the pairing on $A_{E}(\chi )\otimes A_{E}^{\vee }(\chi )$ .)

We define the geometric kernel to be

(3.3.4) $$ \begin{align}\begin{aligned} {\widetilde Z}(\phi^{\infty},\chi):= \sum_{a\in F^{\times}} \left\langle {\widetilde Z}_{a}(\phi^{\infty})[1]_{U}^{0}, t^{0}_{\chi} \right\rangle {\textbf{q}}^{a}. \end{aligned}\end{align} $$

By [I, Proposition 5.3.2 and the formula after its proof], the series $ {\widetilde Z}(\phi ^{\infty },\chi )$ is (the q-expansion of) a weight $2$ cuspidal Hilbert modular form of central character $\omega ^{-1}$ , with coefficients in $\Gamma _{F}\hat {\otimes } L(\chi )$ .

Geometric kernel and Shimizu lifts

Let

$$ \begin{align*} \theta_{\iota_{\mathfrak{p}}}\colon\left( \sigma^{\infty}\otimes \overline{\mathscr{S}}({\textbf{V}}^{\infty}\times \mathbf{A}^{\infty,\times})\right)\otimes_{M}L\to (\pi\otimes \pi^{\vee})\otimes_{M, \iota_{\mathfrak{p}}} L \end{align*} $$

be Shimizu’s theta lifting defined in [9, §5.1]. Let

$$ \begin{align*} {\mathrm{T}}_{\mathrm{alg}}\colon \pi^{U}\otimes_{M} \pi^{\vee, U} \to \mathrm{Hom}\left(J_{U}, J_{U}^{\vee}\right)\otimes {M} \end{align*} $$

be defined by ${\mathrm{T}}_{\mathrm{alg}}(f_{1}, f_{2}):= f_{2}^{\vee }\circ f_{1}$ .

Proposition 3.3.1. If $\phi _{v}=\phi _{r,v}$ is as in formula (3.1.4) for all $v\mid p$ , then for any sufficiently large $r'$ , the geometric kernel

$$ \begin{align*} {\widetilde Z}(\phi^{\infty},\chi) \end{align*} $$

is invariant under $\prod _{v\vert p}K_{1}^{1}(\varpi _{v}^{r'})$ , and it satisfies

(3.3.5) $$ \begin{align} & \ell_{\varphi^{p},\alpha}\left(\widetilde{Z}(\phi^{\infty}, \chi)\right) \nonumber \\&\quad =2 \lvert D_{F}\rvert^{1/2} \lvert D_{E}\rvert^{1/2} L(1, \eta) \cdot \left\langle {\mathrm{T}}_{\mathrm{alg}}, \iota_{\mathfrak{p}}\left(\theta_{\iota_{\mathfrak{p}}}\left(\varphi,\alpha_{p} \lvert\cdot \rvert_{p}(\varpi)^{-r'}w_{r'}^{-1}\phi\right)\right) P_{\chi}, P_{\chi}^{-1}\right\rangle_{X}. \end{align} $$

Proof. The invariance under $\prod _{v\vert p}K_{1}^{1} (\varpi _{v}^{r'} )$ follows from the invariance of $\phi _{r}$ under the action of $(\begin {smallmatrix}1&\mathscr {O}_{F, p}\\&1\end {smallmatrix})$ and the continuity of the Weil representation. The proof of equation (3.3.5) is indicated in [9, proof of Proposition 5.4.3] (with the correction of Appendix B).

3.4 Kernel identity

We state our kernel identity and recall how it implies the main theorem.

Assumptions on the data

Consider the following local assumptions on the data at primes above $ p$ :

Assumption 3.4.1. Let $U_{F,v}^{\circ }:=1+\varpi _{v}^{n}\mathscr {O}_{F,v}$ with $n\geq 1$ be such that $\omega _{v}$ is invariant under $U_{F, v}^{\circ }$ . The character $\chi _{p}$ is sufficiently ramified in the sense that it is nontrivial on

$$ \begin{align*} V_{p}^{\circ}:=\prod_{v\mid p}q_{v\mid\mathscr{O}_{E,v}}^{-1}\left(U_{F,v}^{\circ}\right)\subset \mathscr{O}_{E,p}^{\times}. \end{align*} $$

(Recall from §3.1 that $q_{v}\colon E_{v}\to F_{v}$ is the norm map.)

Under this assumption, we have $t_{\chi }=t_{\chi }^{0}$ ; see [I, Proposition 8.1.1.3], where $\xi _{U}\in {\mathrm{Cl}}(X_{U})_{\mathbf {Q}}$ denotes the Hodge class defining the section ${\mathrm{Cl}}(X_{U})_{\mathbf {Q}}\to {\mathrm{Cl}}^{0}(X_{U})_{\mathbf {Q}}$ . The technical advantage gained, which is the same as in [I] and is implicitly reaped in Theorem 3.6.1, is that one may analyze the height-generating series purely in terms of pairs of CM divisors of degree $0$ , thus avoiding a study of $\xi _{U}$ and the recourse to p-adic Arakelov theory made in [8].

Assumption 3.4.2. For each $v \mid p$ , the open compact $U_{v}\subset \mathbf {B}_{v}^{\times }$ satisfies the following:

1. – $U_{v}= U_{v,r}=1+\varpi ^{r}M_{2}\left (\mathscr {O}_{F,{v}}\right )$ for some $r\geq 1$ .

2. – The integer $r\geq n$ is sufficiently large that the characters $\chi _{v}$ and $\alpha _{v}\circ q_{v}$ of $E_{v}^{\times }$ are invariant under $U_{v,r}\cap \mathscr {O}_{E,v}^{\times }$ .

Convention on citations from [9]

In [9], we denoted by $S_{\text {nonsplit}}$ the set of places of F nonsplit in E, and by $S_{p}$ the set of places of F above p. When referring to results from [9], we henceforth stipulate that one should read any assumption such as ‘set $v\in S_{\text {nonsplit}}$ ’ or ‘let v be a place in F nonsplit in E’ as ‘set $v\in S_{\text {nonsplit}} -S_{p}$ ’. Similarly, the set $S_{1}$ fixed in [9, §6.1] should be understood to consist only of places not above p.

Theorem 3.4.3 kernel identity

Assume the hypotheses of Theorem B, and that $U, \varphi ^{p}, \phi ^{p\infty }, \chi , r$ satisfy the assumptions of [9, §6.1] as well as Assumptions 3.4.1 and 3.4.2. Define $\phi _{p}:= \otimes _{v\mid p}\phi _{v, r}$ , with $\phi _{v,r}=$ formula (3.1.4). Then

$$ \begin{align*} \ell_{\varphi^{p},\alpha}({\mathrm{d}}_{F}\mathscr{I}(\phi^{p\infty}; \chi))= {2 \lvert D_{F}\rvert L_{(p)}(1, \eta)}\cdot \ell_{\varphi^{p},\alpha}\left(\widetilde{Z}(\phi^{\infty}, \chi)\right). \end{align*} $$

The elements of the proof will be gathered in §3.6.

Lemma 3.4.4. Theorem 3.4.3 implies Theorem 2.2.1.

Proof. As in [9, Proposition 5.4.3] as corrected in Appendix B, we consider the following equivalent (by [9, Lemma 5.3.1]) form of the identity of Theorem 2.2.1:

(3.4.1) $$ \begin{align} & \left\langle {\mathrm{T}}_{\mathrm{alg}}, \iota_{\mathfrak{p}}(f_{1}\otimes f_{2}) P_{\chi}, P_{{\chi}^{-1}}\right\rangle_{J} \nonumber\\&\quad ={ \zeta_{F}^{\infty}(2)\over \left(\pi^{2}/2\right)^{\left[F:\mathbf{Q}\right]} \lvert D_{E}\rvert^{1/2} L(1, \eta)} \prod_{v\mid p}Z_{v}^{\circ}( \alpha_{v}, \chi_{v})^{-1} \cdot{\mathrm{d}}_{F} L_{p, \alpha}(\sigma_{A,E})( \chi)\cdot Q(f_{1}, f_{2}, \chi), \end{align} $$

where $\iota _{\mathfrak {p}}\colon M\hookrightarrow L(\chi )$ , and . By linearity, equation (3.4.1) extends to an identity that makes sense for any element ${\mathbf {f}}\in \pi \otimes \pi ^{\vee }$ . By the multiplicity $1$ result for $E_{\mathbf {A}^{\infty }}^{\times }$ -invariant linear functionals on each of $\pi , \pi ^{\vee }$ , it suffices to prove equation (3.4.1) for one element ${\mathbf {f}}\in \pi \otimes \pi ^{\vee }$ such that $Q({\mathbf {f}} , \chi )\neq 0$ (compare [29, Lemma 3.23]).

We claim that Theorem 3.4.3 gives equation (3.4.1) for ${\mathbf {f}}=\theta (\varphi , \phi )$ , where:

1. – $\varphi _{\infty }$ is standard antiholomoprhic in the sense of [9] and $\phi _{\infty }$ is standard in the sense of [9]; and

2. – for all $v\mid p$ , we have that $\varphi _{v}=$ equation (3.1.2) and $\phi _{v}=\phi _{v,r}=$ formula (3.1.4) for any sufficiently large r.

The claim follows from equations (3.2.4) and (3.3.5), the local comparison between $\mathscr {R}_{v}^{\natural } $ and $Q_{v}\circ \theta _{v}$ for $v\nmid p$ of [9, Lemma 5.1.1], and the local calculation at $v\vert p$ of Proposition A.2.2.

Finally, the existence of $\varphi , \phi $ satisfying both the required assumptions and $Q(\theta (\varphi , \phi ))\neq 0$ follows from [9, Lemma 6.1.6] away from p, and the explicit formula of Proposition A.2.2 at p.

3.5 Derivative of the analytic kernel

We start by studying the incoherent Eisenstein series $\mathscr {E}\left (\phi _{2}^{p\infty }\right )$ . For $a\in F_{v}^{\times }$ , denote by

$$ \begin{align*} W_{a , v}^{\circ} \end{align*} $$

the normalized local Whittaker function of $E_{r}\left (\phi _{2}^{p\infty }(\chi _{F}){\phi _{2,p}^{\circ }}, \chi _{F}\right )=\text {formula}~(3.2.1)$ , defined as in [9, Proposition 3.2.1 and the paragraph after its proof].

The following reviews and corrects [9, Proposition 7.1.1]:

Proposition 3.5.1. For each $v\mid p$ , let $\phi _{2, v}=\phi _{2, r,v}$ be as in formula (3.1.4).

1. 1. Let v be a place of F and set $a\in F_{v}^{\times }$ .

   1. (a) If a is not represented by $({\mathbf {V}}_{2,v}, uq)$ , then $W_{a,v}^{\circ }(g,u, \mathbf {1})=0$ .

   2. (b) (Local Siegel–Weil formula.) If $v\nmid p$ and there exists $x_{a}\in {\mathbf {V}}_{2,v}$ such that $uq(x_{a})=a$ , then

      $$ \begin{align*} W_{a,v}^{\circ}\left(\left(\begin{smallmatrix}y&\\&1\end{smallmatrix}\right),u, \mathbf{1}\right)= \int_{E_{v}^{1}}r\left(\left(\begin{smallmatrix}y&\\&1\end{smallmatrix}\right), h\right)\phi_{2,v}(x_{a},u) dh. \end{align*} $$

   3. (c) (Local Siegel–Weil formula at p.) If $v\mid p$ , $a\in -1+\varpi ^{r}\mathscr {O}_{F,v}$ , and $u\in 1+\varpi ^{r}\mathscr {O}_{F,v} $ , let $x_{a}\in {\mathbf {V}}_{2,v}$ be such that $uq(x_{a})=a$ . Then

      (3.5.1) $$ \begin{align}\begin{aligned} W_{a,v}^{\circ}\left(\left(\begin{smallmatrix}y&\\&1\end{smallmatrix}\right),u, \mathbf{1}\right)= \lvert d\rvert_{v}\int_{E_{v}^{1}}r\left(\left(\begin{smallmatrix}y&\\&1\end{smallmatrix}\right), h\right)\phi_{2,v}(x_{a},u) dh. \end{aligned}\end{align} $$

2. 2. For any $a\in F_{v}^{\times }\cap \prod _{v\mid p }\left ( -1+\varpi ^{r}\mathscr {O}_{F,v}\right )$ , $u\in F^{\times }\cap \prod _{v\mid p} \left (1+\varpi ^{r}\mathscr {O}_{F,{v}}\right )$ , there is a place $v\nmid p$ of F such that a is not represented by $({\mathbf {V}}_{2}, uq)$ .

Proof. Parts 1(a) and (b) are as in [9, Proposition 7.1.1]. Before continuing, observe that under our assumptions, a is always represented by $\left ({\mathbf {V}}_{2,v}, uq_{v}\right )$ for all $v\mid p$ : this is clear if v splits in E, and up to possibly enlarging the integer r, it may be seen by the local constancy of $q_{v}$ and the explicit identity $q_{v}({\frak j}_{v})=-1$ , where ${\frak j}_{v}$ =formula (4.1.2) if v is nonsplit. Then part 1(c) follows by explicit computation of both sides (starting, for example, as in [29, proof of Proposition 6.8] for the left-hand side). Explicitly, we have

$$ \begin{align*} &\text{equation~(3.5.1)}= \nonumber \\&\quad \begin{cases} e_{v}^{-1}\lvert d\rvert_{v}\mathrm{vol}\left(E^{1}_{v}\cap \mathscr{O}_{E,v}^{\times}, dh\right)\lvert y\rvert^{1/2} \mathbf{1}_{\mathscr{O}_{F,v}}(ay)\mathbf{1}_{\mathscr{O}_{F, v}^{\times}}\left(y^{-1}u\right) & \text{if } v(a)\geq 0 \text{ and } v(u)=0, \\ 0 & \text{otherwise.} \end{cases} \end{align*} $$

Finally, part 2 follows from the observation of the previous paragraph and [29, Lemma 6.3].

Lemma 3.5.2. Suppose that for all $v\mid p$ , we have that $\phi _{1,v}=\phi _{1,r,v}$ is as in formula (3.1.3). Then for any $t\in T(\mathbf {A})$ , the ath q-expansion coefficient of the the theta series of formula (3.2.2) vanishes unless

$$ \begin{align*} a\in \bigcap_{v\mid p} 1+\varpi^{r}\mathscr{O}_{F,v}. \end{align*} $$

Proof. This is straightforward.

Denote

$$ \begin{align*} \mathrm{U}_{p, *}^{r}:= \left(\prod_{v\vert p}\mathrm{U}_{v, *}\right)^{r}, \end{align*} $$

an operator on modular forms that extends to an operator on all the spaces of p-adic q-expansions defined so far.

Corollary 3.5.3. Suppose that $\phi ^{p\infty } \in {\mathscr {S}}\left ({\mathbf {V}}_{2}^{p\infty }\times {\mathbf {A}^{p\infty , \times }}\right )$ satisfies the assumptions of [9, §6.1]. Set $\chi \in \mathscr {Y}_{\omega }^{\mathrm{l.c.}}$ . For any sufficiently large r, we have

$$ \begin{align*} \mathrm{U}_{p, *}^{r} \mathscr{I}(\phi^{p\infty}; \chi)=0. \end{align*} $$

Proof. For the first assertion, we need to show that the ath reduced q-expansion coefficient of $\mathscr {I}$ vanishes for all a satisfying $v(a)\geq r$ for all $v\mid p$ . By the defining property (3.2.3) of $\mathscr {I}(\chi ')$ and the choice of $\phi _{p}$ , the group $T\left (F_{p}\right )\subset T(\mathbf {A})$ acts trivially on the q-expansion coefficients of $\mathscr {I}(\chi ')$ . The remaining integration on $T(F)Z(\mathbf {A})V^{p}\backslash T(\mathbf {A})/T\left (F_{p}\right ) $ is a finite sum, so the coefficients a are a sum of products of the coefficients of index $a_{1}$ of $\theta $ and of index $ a_{2}$ of $\mathscr {E}(\mathbf {1})$ , for pairs $(a_{1}, a_{2}) $ with $a_{1}+a_{2}=a$ . When $a=0$ , the vanishing follows from the vanishing of the constant term of $\mathscr {E}$ , which is proved as in [29, Proposition 6.7]. For $a\neq 0$ , by Lemma 3.5.2 only the pairs $(a_{1}, a_{2})$ with $a_{1}\in \bigcap _{v\mid p}1+\varpi ^{r}\mathscr {O}_{F,v}$ contribute. If $v(a)\geq r$ for $v\mid p$ , this forces $a_{2}\in \bigcap _{v\mid p}-1+\varpi _{v}^{r}\mathscr {O}_{F,v}$ . Then the coefficient of index $a_{2}$ of $\mathscr {E}(\mathbf {1})$ vanishes by Proposition 3.5.1.

We can now proceed as in [9, §7.2], except for the insertion of the operator $\mathrm {U}_{p, *}^{r}$ . (This will be innocuous for the purposes of Theorem 3.4.3, since the kernel of $\mathrm {U}_{p, *}^{r}$ is contained in the kernel of $\ell _{\varphi ^{p},\alpha }$ .) We obtain, under the assumptions of [9, §6.1], a decomposition of formula (3.2.5)

(3.5.2) $$ \begin{align}\begin{aligned} \mathrm{U}_{p, *}^{r}\mathscr{I}'(\phi^{p\infty};\chi)= \sum_{v\in S_{\text{nonsplit}}- S_{p}} \mathrm{U}_{p, *}^{r}\mathscr{I}'(\phi^{p\infty};\chi)(v) \end{aligned}\end{align} $$

valid in the space ${\mathbf {S}}$ of p-adic q-expansions with coefficients in $\Gamma _{F}\hat {\otimes } L(\chi )$ (see [9, §6.1] for the definition of $\mathscr {I}'(\phi ^{p\infty };\chi )(v)$ ).

3.6 Decomposition of the geometric kernel and comparison

Suppose that Assumptions 3.4.1 and 3.4.2, as well as the assumptions of [9, §6.1], are satisfied. Then we may decompose [9, §8.2] the generating series (3.3.4) as

(3.6.1) $$ \begin{align}\begin{aligned} {\widetilde Z}(\phi^{\infty},\chi) =\sum_{v}{\widetilde Z}(\phi^{\infty},\chi)(v), \end{aligned}\end{align} $$

according to the decomposition $\langle \ , \ \rangle _{X}=\sum _{v} \langle \ , \ \rangle _{X, v}$ of the height pairing. Here the sum runs over all finite places of F.

The following is the main result of [9] on the local comparison away from p. Let $\overline {\mathbf {S}}{}'= \overline {\mathbf {S}}{}^{\prime }_{S_{1}}$ be the quotient of the space of p-adic q-expansions recalled before formula (3.1.5).

Theorem 3.6.1 [9, Theorem 8.3.2]

Let $\phi ^{\infty }=\phi ^{p\infty }\phi _{p}$ with $\phi _{p}=\prod _{v\mid p} \phi _{v}$ as in formula (3.1.4). Suppose that Assumptions 3.4.1 and 3.4.2, as well as the assumptions of [9, §6.1], are satisfied. Then we have the following identities of reduced q-expansions in $\overline {\mathbf {S}}{}'$ :

1. 1. If $v\in S_{\mathrm{split}} - S_{p}$ , then

   $$ \begin{align*} \widetilde{Z}(\phi^{\infty}, \chi)(v)=0. \end{align*} $$

2. 2. If $v\in S_{\mathrm {nonsplit}}- S_{1}-S_{p}$ , then

   $$ \begin{align*} \mathrm{U}_{p, *}^{r}\mathscr{I}'(\phi^{p\infty}; \chi)(v)=2 \lvert D_{F}\rvert L_{\left(p\right)}(1,\eta) \mathrm{U}_{p, *}^{r}\widetilde{Z}(\phi^{\infty}, \chi)(v). \end{align*} $$

3. 3. If $v\in S_{1}$ , then

   $$ \begin{align*} \mathrm{U}_{p, *}^{r} \mathscr{I}'(\phi^{p\infty}; \chi)(v), \qquad \mathrm{U}_{p, *}^{r} \widetilde{Z}(\phi^{\infty}, \chi)(v) \end{align*} $$
   are theta series attached to a quaternion algebra over F.

4. 4. The sum

   $$ \begin{align*} \mathrm{U}_{p, *}^{r}\widetilde{Z}(\phi^{\infty}, \chi)(p):=\sum_{v\in S_{p}} \mathrm{U}_{p, *}^{r} \widetilde{Z}(\phi^{\infty}, \chi)(v) \end{align*} $$
   belongs to the isomorphic image $\overline {\mathbf {S}}\subset \overline {\mathbf {S}}{}'$ of the space of p-adic modular forms ${\mathbf {S}}$ .

By this theorem and the decompositions of $\mathscr {I}'$ and $\widetilde {Z}$ , the proof of the kernel identity of Theorem 3.4.3 (hence of the main theorem) is now reduced to showing the following proposition. (See [9, §8.3, last paragraph] for the details of the deduction.)

Proposition 3.6.2. Retain the assumptions of Theorem 3.6.1, and further assume that $V_{\mathfrak {p}}A $ is potentially crystalline at all $v\mid p$ . Then the p-adic modular form

$$ \begin{align*} \mathrm{U}_{p, *}^{r}\widetilde{Z}(\phi^{\infty}, \chi)(p) \in \overline{\textbf{S}} \end{align*} $$

is annihilated by $\ell _{\varphi ^{p},\alpha }$ .

Let S be a finite set of non-Archimedean places of F such that for all $v\notin S$ , all the data are unramified, $U_{v}$ is maximal, and $\phi _{v}$ is standard. Let $K=K^{p}K_{p}$ be the level of the modular form $\widetilde {Z}(\phi ^{\infty })$ , and let

$$ \begin{align*} T_{\iota_{\mathfrak{p}}}(\sigma^{\vee})\in \mathscr{H}^{S}({L})=\mathscr{H}^{S}({M})\otimes_{M, {\iota_{\mathfrak{p}}}} {L} \end{align*} $$

be any $\sigma ^{\vee }$ -idempotent in the Hecke algebra as in [9, Proposition 2.4.4]. By that result, in order to establish Proposition 3.6.2 it suffices to prove that

(3.6.2) $$ \begin{align} \ell_{\varphi^{p},\alpha}\left( \mathrm{U}_{p, *}^{r}T_{\iota_{\mathfrak{p}}}(\sigma^{\vee}) \widetilde{Z}(\phi^{\infty}, \chi)(p)\right)=0. \end{align} $$

As in [9], we will in fact prove the following, which implies equation (3.6.2):

Proposition 3.6.3. Let $v\mid p$ . Under the assumptions of Proposition 3.6.2, for all $v\mid p$ , the element

$$ \begin{align*} T_{\iota_{\mathfrak{p}}}(\sigma^{\vee})\widetilde{Z}(\phi^{\infty}, \chi)(v)\in \overline{\textbf{S}}{}' \end{align*} $$

is v-critical in the sense of equation (3.1.7).

The proof will occupy the following section.

4 Local heights at p

The goal of this section is to prove Proposition 3.6.3, whose assumptions we retain throughout except for an innocuous modification to the data at the places $v\mid p$ . Namely, let $\widetilde {\phi }_{v}$ be the Schwartz function denoted by $\phi _{r}$ in formula (3.1.4), and let $\widetilde {U}_{v}\subset \mathbf {B}_{v}^{\times }$ be the open compact subgroup denoted by $U_{v, r}$ in Assumption 3.4.2. Then we define

Since $\chi _{v|F_{v}^{\times }}=\omega _{v}^{-1}$ is by construction invariant under $U_{F, v}^{\circ }$ , the geometric kernels $\widetilde {Z}\left (\phi ^{p\infty }\widetilde {\phi }_{p}, \chi \right ) $ and $ \widetilde {Z}\left (\phi ^{p\infty }\phi _{p}, \chi \right )$ are equal. Therefore we may work on the curve $X_{U}$ .

We fix a place $v\mid p$ . If v splits in E, then the desired result is proven in [9, §9] with the correction in Appendix B. Therefore we may and do assume that v is nonsplit. We denote by w the place of E above v.

We refer the reader to §1.1 for a general sketch of our argument. It is developed here as follows. In §4.1, we prove that after acting by a high power of $\mathrm {U}_{v,*}$ , the coefficients of the generating series are height pairings with CM points of high v-conductor (norm relation). After some general background in §4.2, we use the norm relation to prove the decay property of arithmetic intersection multiplicities in §4.3. Finally, in §4.4 we use again the norm relation and some p-adic Hodge theory to prove the decay property of local heights.

4.1 Norm relation for the generating series

The goal of this subsection will be to show that for s large enough, each q-expansion coefficient of $\mathrm {U}_{p,*}^{s}\widetilde {Z}(\phi ^{\infty }, \chi )$ is a height pairing of CM divisors, one of which is supported on Galois orbits of CM points of the ‘pseudo-conductor’ s (as defined later).

We start by considering the $\mathrm {U}_{v, *}$ -action on the generating series $\widetilde {Z}(\phi ^{\infty })=\text {formula}~(3.3.2)$ . Recall that $d_{v}$ (§3.1) is a generator of the different ideal of $F_{v}$ .

Lemma 4.1.1. If $a\in \mathbf {A}^{\infty \times }$ satisfies $v(a)\geq -v(d_{v})$ , the ath reduced q-expansion coefficient of $\mathrm {U}_{v,*}{\widetilde Z}(\phi ^{\infty })$ equals

$$ \begin{align*} {\widetilde Z}_{a\varpi_{v}^{}}(\phi^{\infty}), \end{align*} $$

where $ {\widetilde Z}_{a'}(\phi ^{\infty })$ is defined in formula (3.3.1).

Proof. After computing the Weil action of $\mathrm {U}_{v, *}$ on $\phi $ , this is a simple change of variables.

We can factor

$$ \begin{align*} {\widetilde Z}_{a}(\phi^{\infty}) = {\widetilde Z}_{a^{v}}(\phi^{v\infty}) Z_{a_{v}}(\phi_{v}) \end{align*} $$

as the composition of the commuting correspondences

(4.1.1) $$ \begin{align} \nonumber \widetilde{Z}_{a}^{v}(\phi^{v\infty}) &:=c_{U^{p}}\sum_{x^{v}\in U^{v}\backslash \mathbf{B}^{v\infty\times}/U^{v}}\phi^{v\infty}\left(x^{v}, aq(x^{v})^{-1}\right)Z(x^{v})_{{U}}, \\ Z_{a_{v}}(\phi_{v}) &:= \sum_{x_{v}\in U_{v}\backslash \mathbf{B}_{v}^{\times}/U_{v} } \phi_{v} \left(x_{v}, aq(x_{v})^{-1}\right)Z(x_{v})_{{U}}. \end{align} $$

From here until after the proof of Lemma 4.1.2, we work in a local situation and drop v from the notation. Let $\theta \in \mathscr {O}_{E}$ be such that $\mathscr {O}_{E}= \mathscr {O}_{F}+\theta \mathscr {O}_{F}$ , and write $\mathrm {T}={\mathrm {Tr}_{E/F}}(\theta _{})$ , $\mathrm {N}_{} =\mathrm {N}_{E_{}/F_{}}(\theta _{})$ . Fix the embedding $E_{}\to \mathbf {B}_{}=M_{2}(F_{})$ to be

$$ \begin{align*}\begin{aligned} t=a+\theta b\mapsto \left(\begin{array}{cc}a+b\mathrm{T}& b {\mathrm{N}}\\ -b & a\end{array}\right). \end{aligned}\end{align*} $$

Set

(4.1.2) $$ \begin{align}\begin{aligned}\mathrm{j}:= \left(\begin{array}{cc} 1& \mathrm{T}\\ {} & {-1}\end{array}\right). \end{aligned}\end{align} $$

Then $\mathrm {j}^{2}=1$ , $q({\mathrm {j}})=-1$ , and for all $t\in E$ , $\mathrm {j} t =t^{c}\mathrm {j}$ ; and in the orthogonal decomposition ${\mathbf {V}}={\mathbf {V}}_{1}\oplus {\mathbf {V}}_{2}$ , we have

$$ \begin{align*} \mathscr{O}_{{\textbf{V}}_{2}}= {\textbf{V}}_{2}\cap M_{2}(\mathscr{O}_{F})=\mathrm{j} \mathscr{O}_{E}. \end{align*} $$

Let $\Xi (\varpi ^{r}) =1+\varpi ^{r}\mathscr {O}_{E}+ \mathrm {j} \left (\mathscr {O}_{E}\cap q^{-1}(1+\varpi ^{r}\mathscr {O}_{F})\right )$ ; then $\phi $ is a fixed multiple of the characteristic function of $\Xi (\varpi ^{r}) \times (1+\varpi ^{r}\mathscr {O}_{F})\subset \mathbf {B} \times F^{\times }$ . For $a\in F^{\times }$ , let

$$ \begin{align*} \Xi(\varpi^{r})_{a} =\{x\in \Xi(\varpi^{r}) \ |\ q(x)\in a(1 +\varpi^{r}\mathscr{O}_{F})\}. \end{align*} $$

Then the local component (4.1.1) of the generating series equals

$$ \begin{align*} Z_{a}(\phi) = \sum_{x \in U\backslash \Xi\left(\varpi^{r}\right)_{a} U_{F}^{\circ}/U } Z(x)_{{U}}, \end{align*} $$

up to a constant that is independent of a.

Lemma 4.1.2. Let $a\in F^{\times }$ satisfy $v(a)=s\geq r$ . The natural map $\Xi (\varpi ^{r})_{a} U_{F}^{\circ }/U \to U\backslash \Xi (\varpi ^{r})_{a} U_{F}^{\circ } /U$ is a bijection. For either quotient set, a complete set of representatives is given by the elements

$$ \begin{align*} x(b):= 1+\mathrm{j} b \end{align*} $$

as b ranges through a complete set of representatives for

$$ \begin{align*} q^{-1}( 1-a(1+\varpi^{r}\mathscr{O}_{F})) /(1+\varpi^{r+s}\mathscr{O}_{E}) \subset (\mathscr{O}_{E}/\varpi^{r+s}\mathscr{O}_{E})^{\times}. \end{align*} $$

Proof. It is equivalent to prove the same statement for the quotients of $\Xi (\varpi ^{r})_{a}$ by the group $\widetilde {U}=1+\varpi ^{r} M_{2}(\mathscr {O}_{F})$ . By acting on the right with elements of $\mathscr {O}_{E}\cap U$ , we can bring any element of $\Xi $ to one of the form $x(b)$ . Write any $ \gamma \in \widetilde {U}$ as $\gamma =1+\varpi ^{r}u_{1} +{\mathrm{j}} \varpi ^{r}u_{2}$ with $u_{1}$ , $u_{2}\in \mathscr {O}_{E}$ . Then

$$ \begin{align*} x(b)\gamma = (1+\mathrm{j} b)(1+\varpi^{r}u_{1} +\mathrm{j} \varpi^{ r} u_{2}) = 1+ \varpi^{r}(u_{1}+b^{c}u_{2}) + \mathrm{j} (b+ \varpi^{r}(bu_{1}+u_{2}) \end{align*} $$

is another element of the form $x(b')$ if and only if $u_{1}=-b^{c}u_{2}$ . In this case,

$$ \begin{align*} b'= b+\varpi^{r}(1-q(b))u_{2} \in b(1+\varpi^{r+s}\mathscr{O}_{E}). \end{align*} $$

Thus the class of b modulo $\varpi ^{r+s}$ is the only invariant of the quotient $\Xi /\widetilde {U}$ . The $\widetilde {U}$ -action on the left similarly preserves this invariant.

We go back to a global setting and notation, restoring the subscripts v and w. Denote by ${\mathrm{rec}}_{E_{w}}\colon E_{w}^{\times }\to \mathscr {G}_{E,w}^{\mathrm{ab}}$ the reciprocity map of class field theory. Recall that for $x\in \mathbf {B}^{\infty \times }$ , we have a point $[x]_{U}\in X_{U}$ ; for a subset $\Xi '\subset \mathbf {B}^{\infty \times }$ , we similarly denote $[\Xi ']_{U}=\{[x]_{U} \mid x\in \Xi '\}$ .

Lemma 4.1.3. Fix $a\in \mathscr {O}_{F,v}$ with $v(a)=s\geq r$ .

1. 1. Set $b\in \left ( \mathscr {O}_{E,w}/\varpi _{v}^{r+s}\mathscr {O}_{E,w}\right )^{\times }$ , $t\in 1+\varpi _{v}^{r}\mathscr {O}_{E,w}$ . Then

   $$ \begin{align*} {\mathrm{rec}}_{E_{w}}(t)[x(b)]_{U}= [x(b t^{c}/t)]_{U}. \end{align*} $$

2. 2. We have

   $$ \begin{align*} \left[\Xi\left(\varpi_{v}^{r}\right)_{a}U_{F, v}^{\circ}\right]_{U}= \bigsqcup_{\overline{b}} {\mathrm{rec}}_{E_{w}}\left(\left(1+\varpi_{v}^{r}\mathscr{O}_{E,w}\right)/\left(1+\varpi_{v}^{r}\mathscr{O}_{F,v}\right)\left(1+\varpi_{v}^{r+s}\mathscr{O}_{E,w}\right)\right) \left[x\left(\overline{b}\right)\right]_{U}, \end{align*} $$
   where the Galois action is faithful, and $\overline {b}$ ranges through a set of representatives for
   (4.1.3) $$ \begin{align}\begin{aligned} q_{v}^{-1}\left( 1-a\left(1+\varpi_{v}^{r}\mathscr{O}_{F,v}\right)\right) /\left(1+\varpi_{v}^{r+s}\mathscr{O}_{E,w}\right)\cdot \left(1+\varpi_{v}^{r+v\left(\theta-\theta^{c}\right) }\mathscr{O}_{E,w}\cap q_{v}^{-1}(1)\right). \end{aligned}\end{align} $$
   The size of this set is bounded uniformly in a.

Proof. For part 1, we have

$$ \begin{align*} {\mathrm{rec}}_{E_{w}}(t)[x(b)]_{U}= [t+t\mathrm{j} b]_{U}= [t+\mathrm{j} t^{c} b]_{U} = [1+\mathrm{j} b t^{c}/t]_{U}. \end{align*} $$

Part 2 follows Lemma 4.1.2 and part 1, noting that the group $\left (1+\varpi _{v}^{r+v\left (\theta -\theta ^{c}\right ) }\mathscr {O}_{E,{w}}\right )\cap q_{v}^{-1}(1)$ is the image of the map $t\mapsto t^{c}/t$ on $1+\varpi _{v}^{r}\mathscr {O}_{E,w}$ . Finally, the map $({\mathrm{projection}}, q_{v})$ gives an injection from

$$ \begin{align*} q_{v}^{-1}\left( 1-a\left(1+\varpi_{v}^{r}\mathscr{O}_{F,v}\right)\right) /\left(1+\varpi_{v}^{r+s}\mathscr{O}_{E,w}\right)\cdot \left(1+\varpi_{v}^{r+v\left(\theta-\theta^{c}\right) }\mathscr{O}_{E,w}\cap q_{v}^{-1}(1)\right) \end{align*} $$

to

$$ \begin{align*} \left(\mathscr{O}_{E,w}/\varpi_{v}^{r+v\left(\theta -\theta^{c}\right)}\mathscr{O}_{E,w}\right)^{\times} \times \left(1-a\left(1+\varpi_{v}^{r}\mathscr{O}_{F,v}\right)\right)/q\left(1+\varpi_{v}^{r+s}\mathscr{O}_{E,w}\right), \end{align*} $$

whose size is bounded uniformly in a (more precisely, the second factor is isomorphic to $\mathscr {O}_{F,v}/{\mathrm{Tr}}\left (\mathscr {O}_{E,v}\right )$ via the map $1-a\left (1+\varpi _{v}^{r}x\right )\mapsto x$ ).

We denote by $\overline {w}$ an extension of the place w to $E^{\mathrm{ab}}$ . For $s\geq 0$ , let

$$ \begin{align*} H_{s}\subset E^{\mathrm{ab}} \end{align*} $$

be the finite abelian extension of E with norm group $U_{F}^{\circ } U_{T}^{v}\left (1+ \varpi _{v}^{r_{v}+s}\mathscr {O}_{E,v}\right )$ , where $U_{T}^{v}= U^{v} \cap E_{\mathbf {A}^{\infty }}^{\times }$ . Let $H_{\infty }= \bigcup _{s\geq 0} H_{s}$ . If $r_{v}$ is sufficiently large, for all $s\geq 0$ the extension $H_{s}/H_{0}$ is totally ramified at $\overline {w}$ of degree $q_{F, v}^{s}$ , and

$$ \begin{align*} \mathrm{Gal}(H_{s}/H_{0})\cong \mathrm{Gal}\left(H_{s, \overline{w}}/H_{0, \overline{w}}\right) \cong \left(1+{\varpi}_{v}^{r_{v}}\mathscr{O}_{E,v} \right)/\left(1+\varpi_{v}^{r_{v}}\mathscr{O}_{F,v}\right) \left(1+\varpi_{v}^{r_{v}+s}\mathscr{O}_{E, v}\right). \end{align*} $$

In particular,

$$ \begin{align*} [H_{s}:H_{0}]= \left[H_{s, \overline{w}} : H_{0, \overline{w}}\right] = q_{F, v}^{s}. \end{align*} $$

We will say that a CM point $z\in X_{H_{0}}$ has pseudo-conductor $s\geq 0$ (at $\overline {w}$ ) if $H_{0, \overline {w}}(z)=H_{s, \overline {w}}$ .

Proposition 4.1.4. There exists an integer $d>0$ such that for all $a\in \mathbf {A}^{\infty ,x}$ with $v(a)=s\geq r_{v}$ , there exists a degree $0$ divisor $D_{a}\in d^{-1} \mathrm {Div}^{0}\left (X_{U , H_{s}}\right )$ , supported on CM points, such that

$$ \begin{align*} \widetilde{Z}_{a}(\phi^{\infty})[1]_{U} = {\mathrm{Tr}}_{H_{s}/H_{0}} (D_{a}) \end{align*} $$

in ${\mathrm{Div}}^{0}\left (X_{U, H_{0}}\right ).$ All prime divisor components of $D_{a}$ are CM points of pseudo-conductor s.

Proof. This follows from Lemma 4.1.3(2), by taking $D_{a}$ to be a fixed rational multiple (independent of a) of

$$ \begin{align*} \widetilde{Z}_{a}^{v}(\phi^{v\infty}) \sum_{\overline{b}\in \text{formula~(4.1.3)}} \left[x\left(\overline{b}\right)\right]_{U}. \end{align*} $$

The divisor is of degree $0$ by [9, Proposition 8.1.1]. Its prime components are not defined over proper subfields $H_{s'}\subset H_{s}$ , because of the faithfulness statement of Lemma 4.1.3(2).

4.2 Intersection multiplicities on arithmetic surfaces

Before continuing, we gather some definitions and a key result.

Ultrametricity of intersections on surfaces

Let $\mathscr {X}$ be a $2$ -dimensional regular Noetherian scheme, finite flat over a field $\kappa $ or a discrete valuation ring $\mathscr {O}$ with residue field $\kappa $ . We denote by $(\ \cdot \ )_{\mathscr {X}}$ the usual $\mathbf {Z}$ -bilinear intersection-multiplicity pairing of divisors intersecting properly on $\mathscr {X}$ ; for effective divisors $D_{j}$ ( $j=1, 2$ ) with $\mathscr {O}_{D_{j}}=\mathscr {O}_{\mathscr {X}}/ \mathscr {I}_{j}$ , it is defined by

$$ \begin{align*} (D_{1}\cdot D_{2})_{\mathscr{X}}= {\mathrm{length}}_{\kappa}\mathscr{O}_{\mathscr{X}}/(\mathscr{I}_{1}+\mathscr{I}_{2}). \end{align*} $$

The subscript $\mathscr {X}$ will be omitted when it is clear from context.

We will need the following result of García Barroso, González Pérez, and Popescu-Pampu:

Proposition 4.2.1. Let R be a Noetherian regular local ring of dimension $2$ , which is a flat module over a field or a discrete valuation ring. Let $\Delta $ be any irreducible curve in $\mathrm {Spec}\, R$ . Then the function

$$ \begin{align*}\begin{aligned} d_{\Delta}(D_{1}, D_{2}):= \begin{cases} (D_{1} \cdot \Delta) (D_{2} \cdot \Delta)/(D_{1}\cdot D_{2}) & \text{if}\ D_{1}\neq D_{2},\\ 0 & \text{if}\ D_{1}=D_{2} \end{cases} \end{aligned}\end{align*} $$

is an ultrametric distance on the space of irreducible curves in $\mathrm {Spec} R$ different from $\Delta $ .

Proof. For those rings R that further satisfy the property of containing $\mathbf {C}$ , this is proved in [15]. The proof only relies on (i) the existence of embedded resolutions of divisors in the spectra of such rings and (ii) the negativity of the intersection matrix of the exceptional divisor of a projective birational morphism between spectra of such rings. Both results still hold under our weaker assumptions: see [20, Theorem 9.2.26] for (i) and [20, Theorem 9.1.27, Remark 9.1.28] for (ii).

Arithmetic intersection multiplicities

Suppose now that $\mathscr {X}$ is a $2$ -dimensional regular Noetherian scheme, proper flat over a discrete valuation ring $\mathscr {O}$ with residue field $\kappa $ . A divisor on $\mathscr {X}$ is called horizontal (resp., vertical) if each of the irreducible components of the support $\lvert D\rvert $ is flat over $\mathscr {O}$ (resp., contained in the special fiber $\mathscr {X}_{\kappa }$ ). We extend $(\ \cdot \ ):=(\ \cdot \ )_{\mathscr {X}}$ to a bilinear form $(\ \bullet \ )$ on pairs of divisors on $\mathscr {X}$ sharing no common horizontal irreducible component of the support by

$$ \begin{align*} (\mathscr{X}_{\kappa} \bullet V ):= 0 \end{align*} $$

if V is any vertical divisor.

Denote by X the generic fiber of $\mathscr {X}$ . If $D\in \mathrm {Div}^{0}(X)$ with Zariski closure $\overline {D}$ in $\mathscr {X}$ , a flat extension of D is a divisor $\widehat {D}\in \mathrm {Div}(\mathscr {X})_{\mathbf {Q}}$ such that $\widehat {D}-\overline {D}$ is vertical and

$$ \begin{align*} \left(\widehat{D}\bullet V\right)=0 \end{align*} $$

for any vertical divisor V on $\mathscr {X}$ . A flat extension of D exists and is unique up to addition of rational linear combinations of the connected components of $\mathscr {X}_{\kappa }$ .

The arithmetic intersection multiplicity on divisors with disjoint supports in ${\mathrm{Div}}^{0}(X)$ is then defined by

$$ \begin{align*} m_{X}(D_{1}, D_{2}):=\left(\overline{D}_{1}\bullet \widehat{D}_{2}\right) = \left(\widehat{D}_{1}\bullet \overline{D}_{2}\right) \in \mathbf{Q}. \end{align*} $$

4.3 Decay of intersection multiplicities

We continue using the notation introduced in §4.1. Let $m_{\overline {w}}:= m_{X_{H_{0, \overline {w}}}}$ . Developing the approximation argument sketched in the introduction, we will show that for any degree- $0$ divisor D on $X_{H_{0}}$ , we have

$$ \begin{align*} m_{\overline{w}}\left(\widetilde{Z}_{a\varpi^{s}}(\phi^{\infty}) [1]_{U}, D\right)= O\left(q_{F,v}^{s}\right) \end{align*} $$

in L, uniformly in a.

Set $U_{0,v}:= \mathrm {GL}_{2}\left (\mathscr {O}_{F, v}\right )\subset \mathbf {B}_{v}^{\times }$ and $X_{0}:= X_{U^{v}U_{0,v}}$ , let $\mathscr {X}_{0}$ be the canonical model of $X_{0, F_{v}}$ over $\mathscr {O}_{F, v}$ , which is smooth [6], and let $\mathscr {X}_{0}'$ be its base change to $\mathscr {O}_{H_{0, \overline {w}}}$ . Let $\mathscr {X}$ be the integral closure of $\mathscr {X}_{0}'$ in $X_{H_{0, \overline {w}}}$ , which is a regular model of $X_{H_{0, \overline {w}}} $ over $\mathscr {O}_{H_{0, \overline {w}}}$ , and let ${\mathrm{p}}\colon \mathscr {X}\to \mathscr {X}_{0}'$ be the natural map. Thus we have a diagram

of curves and regular integral models. (The bottom row will be used in proving Lemma 4.3.2.)

Some intersection multiplicities

As a preliminary, we first compute the intersection multiplicities of Zariski closures of CM points with the special fiber of $\mathscr {X}$ , then bound their intersections with horizontal divisors.

We denote by $\kappa $ the residue field of $H_{0, \overline {w}}$ and by k the algebraic closure of $\kappa $ . For a scheme $\mathscr {C}$ and a point $y\in \mathscr {C}$ , we denote $\mathscr {C}_{y}:=\mathrm {Spec}\, \mathscr {O}_{\mathscr {C}, y}$ .

Lemma 4.3.1. Let $z_{s}\in X_{H_{0}}$ be a CM point with pseudo-conductor s, let $\overline {z}_{s}$ be its closure in $\mathscr {X}$ , and let $y\in \mathscr {X}_{\kappa }$ be its reduction modulo $\overline {w}$ . Then

$$ \begin{align*} \left(\overline{z}_{s} \cdot \left[\mathscr{X}_{y,\kappa}\right]\right)_{\mathscr{X}_{y}} =[\kappa(y):\kappa] q_{F,v}^{s}. \end{align*} $$

(Recall that, following §3.1, $\mathscr {X}_{y, \kappa }$ denotes the special fiber of $\mathscr {X}_{y}$ .)

Proof. We will deduce this from Gross’s theory of quasicanonical liftings [18], which we recall. The situation is purely local, and we drop all subscripts v, w, and $\overline {w}$ . For a finite extension $K\supset E$ contained in $E^{\mathrm{ab}}$ , let $K^{\mathrm{un}}$ be the maximal unramified extension of K contained in $E^{\mathrm{ab}}$ (thus the residue field of $K^{\mathrm{un}}$ is identified with k).

By [6, §7.4], for any supersingular point $y_{0} \in \mathscr {X}_{0,\mathscr {O}_{E}^{\mathrm{un}}}$ the completed local ring of $\mathscr {X}_{0, \mathscr {O}_{E}^{\mathrm{un}}}$ at $ y_{0}$ is isomorphic to $ \mathscr {O}_{E^{\mathrm{un}}}[\kern-1pt[ u]\kern-1pt] $ and is the deformation ring of formal modules studied by Gross. The main result of [18] is that for any CM point $z_{0}\in X_{0,{E}^{\mathrm{un}}}$ , there exists a unique integer t (the conductor of $z_{0}$ ) such that the following hold. First, the field $E^{\mathrm{un }}(z_{0})$ is the abelian extension $E^{(t)} $ of $E^{\mathrm{un}}$ with norm group $\left (\mathscr {O}_{F}+\varpi _{v}^{t}\mathscr {O}_{E}\right )^{\times }/\mathscr {O}_{F}^{\times }$ , which is totally ramified of some degree $d_{t}$ . Second, the inclusion of the Zariski closure $\overline {z}_{0}\hookrightarrow \mathscr {X}_{0, \mathscr {O}_{E}^{\mathrm{un}}}$ gives rise to a map of complete local rings

$$ \begin{align*} \mathscr{O}_{E^{\mathrm{un}}}[\kern-1pt[ u]\kern-1pt] \to \mathscr{O}_{E^{\left(t\right)}} =\mathscr{O}\left(\overline{z}_{0}\right), \end{align*} $$

which sends u to a uniformizer $\varpi ^{(t) }$ of $E^{(t)}$ . It follows that if $\mu _{t}$ is the minimal polynomial of $\varpi ^{(t)}$ ,

(4.3.1) $$ \begin{align}\left(\overline{z_{0}} \cdot \mathscr{X}_{0, k}\right)_{\mathscr{X}_{0, \mathscr{O}_{E}^{\mathrm{un}}, y_{0}}} = \dim_{k} \mathscr{O}_{E^{\mathrm{un}}}[\kern-1pt[ u]\kern-1pt] /(\varpi_{E}, \mu_{t}(u)) = \dim_{k}\mathscr{O}_{E^{(t)}}/ \varpi_{E}= d_{t}.\end{align} $$

Consider now the situation of the lemma. By the projection formula,

$$ \begin{align*} \left(\overline{z}_{s} \cdot \left[\mathscr{X}_{y, \kappa}\right] \right)_{\mathscr{X}_{y}} & =\left( \overline{z}_{s} \cdot {\mathrm{p}}^{*}\mathscr{X}_{0,\kappa_{0}} \right)_{\mathscr{X}_{y}} \\ &=\left( {\mathrm{p}}_{*}\overline{z}_{s} \cdot \mathscr{X}_{0,\kappa}\right)_{\mathscr{X}^{\prime}_{0,y'}} = \left[H_{0}(z_{s})\colon H_{0}\left(z_{0,s}\right)\right] \cdot \left(\overline{z}_{0,s} \cdot \mathscr{X}_{0,\kappa }\right)_{\mathscr{X}^{\prime}_{0,y'}}, \end{align*} $$

where $y'={\mathrm{p}}(y)$ and $z_{0, s}= {\mathrm{p}}(z_{s})\in X_{0, H_{0}}$ is a CM point. The last intersection multiplicity is $[\kappa (y):\kappa ]$ times the multiplicities of the base-changed divisors to the ring of integers of $H_{0}^{\mathrm{un}}$ , where $z_{0, s}$ remains an irreducible divisor since $H_{0}\left (z_{0, s}\right )\subset H_{0}(z_{s})$ is totally ramified over $H_{0}$ . We perform such base change to $\mathscr {O}_{H_{0}^{\mathrm{un}}}$ without altering the notation. Let z be the image of $z_{0,s}\in X_{H_{0}^{\mathrm{un}}}$ in $X_{E^{\mathrm{un}}}$ , and let t be the conductor of z; so $E^{(t)}\subset H_{0}^{\mathrm{un}}\left (z_{0,s}\right )$ . The fiber above $z\cong \mathrm {Spec}\, E^{(t)}$ in $X_{H_{0}^{\mathrm{un}}}$ is

$$ \begin{align*} \mathrm{Spec} E^{(t)}\otimes_{E^{\mathrm{un}}} H_{0}^{\mathrm{un}}= \mathrm{Spec} H_{0}^{\mathrm{un}}\left(z_{0,s}\right)^{\oplus c} \end{align*} $$

for $c=\left [E^{(t)}: E^{\mathrm{un}}\right ] \cdot \left [H_{0}^{\mathrm{un}}:E^{\mathrm{un}}\right ]/ \left [H_{0}^{\mathrm{}}\left (z_{0,s}\right ): H_{0}^{\mathrm{}}\right ] $ , and $z_{0,s}$ is one of the factors in the right-hand side. By the projection formula applied to $ \mathscr {X}_{0}\times \mathrm {Spec}\, \mathscr {O}_{H_{0}^{\mathrm{un }}} \to \mathscr {X}_{0}\times \mathrm {Spec}\, \mathscr {O}_{E^{\mathrm{un}}}$ and equation (4.3.1), we have

$$ \begin{align*}\begin{aligned} \ \left[H_{0}(z_{s})\colon H_{0}\left(z_{0,s}\right)\right] \cdot \left(\overline{z}_{0,s} \cdot \mathscr{X}_{0,\kappa }\right)_{\mathscr{X}^{\prime}_{0,y'}} &=[\kappa(y):\kappa] c^{-1} \cdot \left[H_{0}(z_{s})\colon H_{0}\left(z_{0,s}\right)\right] \cdot \left[H_{0}^{\mathrm{un}}:E^{\mathrm{un}}\right] \cdot d_{t} \\ &=[\kappa(y):\kappa] \cdot [H_{0}(z_{s}): H_{0}] \cdot d_{t}^{-1} \cdot d_{t}= [\kappa(y):\kappa] q_{F}^{s}, \end{aligned}\end{align*} $$

as desired.

Lemma 4.3.2. Let $\Delta $ be an irreducible horizontal divisor in $\mathscr {X}$ . The intersection multiplicities

$$ \begin{align*} \left(\Delta\cdot \overline{z}\right) \end{align*} $$

are bounded by an absolute constant as z ranges among CM points of sufficiently large pseudo-conductor reducing to y.

Proof. The intersection multiplicity $\left (\Delta \cdot \overline {z}\right )$ is bounded by the degree of the natural map ${\mathrm{q}}\colon \mathscr {X}\to \mathscr {X}_{0, \mathscr {O}_{E,{v}}}$ near y, times the intersection multiplicities of the push-forward divisors to $\mathscr {X}_{0, \mathscr {O}_{E,v}}$ . Similar to the proof of Lemma 4.3.1, we may estimate this intersection in the base change of $\mathscr {X}_{0}$ to $\mathscr {O}_{E}^{\mathrm{un}}$ . The base change of the divisor ${\mathrm{q}}_{*}z_{}$ equals a sum of CM points of $\mathscr {X}_{0, \mathscr {O}_{E^{\mathrm{un}}}}$ ; because the extensions $H_{s}/H_{0}$ are totally ramified, the number of points in this divisor is bounded by an absolute constant, and the conductors of all those CM points go to infinity with the pseudo-conductor s of z. Thus it suffices to show that if $\Delta _{0}$ is a fixed horizontal divisor on $\mathscr {X}_{0, E^{\mathrm{un}}}$ , its intersection multiplicity with CM points of conductor t is bounded as $t\to \infty $ .

Let $z\in \mathscr {X}_{0, \mathscr {O}_{E}^{\mathrm{un}}}$ be a CM point of conductor t, and let $y_{0}\in \mathscr {X}_{0, k}$ be the image of the reduction of t. Now write the image of $\Delta _{0}$ in the completion of $\mathscr {X}_{0, \mathscr {O}_{E^{\mathrm{un}}}}$ at $y_{0}$ as

$$ \begin{align*} \widehat{\Delta}_{0}=\mathrm{Spec} \mathscr{O}_{E^{\mathrm{un}}}[\kern-1pt[ u]\kern-1pt] /(f)\subset \mathrm{Spec} \mathscr{O}_{E^{\mathrm{un}}}[\kern-1pt[ u]\kern-1pt], \end{align*} $$

with $f=\sum _{i=1}^{d} a_{i}u^{i}$ an integral nonconstant monic polynomial. Let ${\widetilde f}\in k[u]$ be the reduction of f. Then

$$ \begin{align*} \left(\Delta_{0}\cdot \overline{z}\right) &= \dim_{k}\mathscr{O}_{E^{\mathrm{un}}}[\kern-1pt[ u]\kern-1pt] / (f, \mu_{t}) \\&=\dim_{k} \mathscr{O}_{E^{(t)}}/\left(f\left(\varpi^{(t)}\right)\right) =\dim_{k} \mathscr{O}_{E^{(t)}}/\left(\left(\varpi^{(t)}\right)^{\deg \left(\tilde{f}\right)}\right)=\deg \left(\tilde{f}\right)\leq d \end{align*} $$

if t is sufficiently large, since the normalized valuations of $\varpi ^{(t)}$ decrease to $0$ as $t\to \infty $ . This completes the proof of the lemma.

Approximation by vertical divisors

The following proposition contains the essential new ingredient of this work. We denote by

$$ \begin{align*} {\mathrm{CM}}\left(X_{H_{0}}\right)_{\geq s}\supset {\mathrm{CM}}\left(X_{H_{0}}\right)_{s} \end{align*} $$

respectively the set of CM points of $X_{H_{0}}$ that have pseudo-conductor at least and equal to a given integer s. We denote by $\mathscr {V}$ the set of irreducible components of $\mathscr {X}_{\kappa }$ (henceforth: ‘vertical components’), and if $y\in \mathscr {X}_{\kappa }$ is a closed point, we denote by $\mathscr {V}_{y}\subset \mathscr {V}$ the set of vertical components of $\mathscr {X}_{y, \kappa }$ . We still use a bar to denote Zariski closure.

Proposition 4.3.3. There exist an integer $s_{0}\geq 1$ , depending only on $X_{H_{0}}$ , and a function

$$ \begin{align*} (V, \rho)\colon {\mathrm{CM}}\left(X_{H_{0}}\right)_{\geq s_{0}} \longrightarrow \mathscr{V}\times \mathbf{Q}^{\times} \end{align*} $$

satisfying the following property:

For every divisor $D\in \mathrm {Div}(\mathscr {X})_{L}$ , there exists a constant $s_{D}\geq s_{0}$ depending only on the support of D, such that if $z\in X_{H_{0}}$ is a CM point of conductor $s\geq s_{D}$ , then $\left (\overline {z}\cdot D\right ) $ may be computed as follows. Let $V=V(z)$ , $\rho =\rho (z)$ , and write

$$ \begin{align*} D= c\mathscr{X}_{\kappa} +D' \end{align*} $$

with $c\in L$ and $D'\in \mathrm {Div}(\mathscr {X})_{L}$ a divisor whose support does not contain V. Then

(4.3.2) $$ \begin{align}\begin{aligned} \left(\overline{z}\cdot D\right) = c[\kappa(y):\kappa] q_{F,v}^{s}+ \rho (V\bullet D), \end{aligned}\end{align} $$

where $y\in \mathscr {X}_{\kappa }$ denotes the reduction of z modulo $\overline {w}$ .

Remark 4.3.4. The vertical component $V(z)$ will be characterized as the one maximizing the intersection multiplicity with $\overline {z}$ . We refer the reader to [11, §2, Figure 1 in §1] for an equivalent and possibly more vivid geometric description ⁸ of the relation of V to $z_{s}$ : one can define pairwise disjoint open subsets (‘geometric basins’) of the Berkovich analytification of X, labelled by the irreducible components of the special fiber; then $z_{s}$ belongs to the basin corresponding to V.

Proof. We will omit all subscripts $v, w, \overline {w}$ and use some of the notation introduced in the proof of Lemma 4.3.1.

Let $y\in \mathscr {X}_{\kappa }$ be a closed point, and write $\left [\mathscr {X}_{y, \kappa }\right ]= \sum _{V' \in \mathscr {V}_{y}} e_{V'}V'$ as divisors. By Lemma 4.3.1, the weighted sum

(4.3.3) $$ \begin{align}\begin{aligned} \sum_{V'} {e_{V'} } \left(\overline{z}_{s}\cdot V'\right)=\left(\overline{z}_{s}\cdot \left[\mathscr{X}_{y, \kappa}\right]\right)= [\kappa(y)\colon \kappa] q_{F}^{s}\end{aligned}\end{align} $$

is independent of the choice of a CM point $z_{s}\in X_{H_{0}}$ of pseudo-conductor s reducing to y. As equation (4.3.3) goes to infinity with s (and the coefficients $e_{V'}$ are independent of s), the quantity

(4.3.4) $$ \begin{align}\begin{aligned} \textstyle{ \max_{V'\in \mathscr{V}_{}} \left(\overline{z}_{s}\cdot V'\right)} \end{aligned}\end{align} $$

(more precisely, the minimum of those maxima as $z_{s}$ varies in ${\mathrm{CM}}(X_{H_{0}})_{s}$ ) goes to infinity with s.

Fix now a CM point $z_{s}\in X_{H_{0}}$ of pseudo-conductor s, let $y\in \mathscr {X}_{\kappa }$ be its reduction, and let $V\in \mathscr {V}_{y}$ be a vertical component realizing the maximum in expression (4.3.4). Let $D\neq V$ be an irreducible divisor in $\mathscr {X}_{y}$ . Pick any irreducible horizontal divisor $\Delta \neq D, \overline {z}_{s}$ in $\mathscr {X}_{y}$ , and consider the ultrametric distance $d_{\Delta }$ of Proposition 4.2.1 for $ R=\mathscr {O}_{\mathscr {X}, y}$ . (Note that $\Delta $ may be drawn from a finite set independent of $z_{s}$ and D; in fact, we may fix any set $\underline {\Delta }$ of at least two irreducible horizontal divisors that are not Zariski closures of CM points, and for given D pick any $\Delta \in \underline {\Delta }-\{D\}$ .)

By the choice of V and Lemma 4.3.2, if s is sufficiently large (a condition depending on D), we have

$$ \begin{align*} d_{\Delta}\left(\overline{z}_{s}, V\right) = {\left(\overline{z}_{s}\cdot \Delta\right) (V\cdot \Delta)\over \left(\overline{z}_{s}\cdot V\right)} < d_{\Delta}( V, D), \end{align*} $$

so that by Proposition 4.2.1,

$$ \begin{align*} d_{\Delta}\left(\overline{z}_{s}, D\right) = d_{\Delta}( V, D). \end{align*} $$

Unwinding the definitions,

(4.3.5) $$ \begin{align}\begin{aligned} \left(\overline{z}_{s}\cdot D\right)= \rho (V\cdot D) \end{aligned}\end{align} $$

for $ \rho := {\left (\overline {z}_{s}\cdot \Delta \right )/ (V\cdot \Delta ) }$ . Applied to a vertical component $D=V'\neq V$ , formula (4.3.5) together with Lemma 4.3.2 shows the uniqueness of the maximizing $V =:V(z_{s})$ for large s; it is clear that $\rho =:\rho (z_{s})$ is then uniquely determined as well. Now the intersection formula (4.3.2) follows by linearity from equation (4.3.5) and Lemma 4.3.1.

Corollary 4.3.5. If $D\in \mathrm {Div}^{0}\left (X_{H_{0}}\right )_{L}$ is any degree- $0$ divisor, then for all sufficiently large s and all a,

$$ \begin{align*} m_{\overline{w}}( \widetilde{Z}_{a\varpi^{s}}(\phi^{\infty}) [1]_{U}, D)=O(q_{F,v}^{s}) \end{align*} $$

in L, where the implied constant can be fixed independently of a and s.

Proof. Let $\widehat {D}$ be a flat extension of D to a divisor on $\mathscr {X}$ (with coefficients in L), and abbreviate $Z_{a,s}:= \widetilde {Z}_{a\varpi ^{s}}(\phi ^{\infty }) [1]_{U}$ . Then by Propositions 4.1.4 and 4.3.3,

$$ \begin{align*} m_{\overline{w}}( Z_{a,s}, D)= (\overline{Z}_{a,s}\cdot \widehat{D}) = A q_{F,v}^{s}+ \sum_{i} \lambda_{i}(V_{i} \bullet \widehat{D}) \end{align*} $$

for some vertical components $V_{i}\subset \mathscr {X} $ and some A, $\lambda _{i}\in L$ . By the definition of flat extension, $ (V_{i} \bullet \widehat {D} )=0$ for all i. The constant A is a linear combination of the constants c in equation (4.3.2) (which depend only on $\widehat {D}$ ), with coefficients whose denominators are bounded by those of $Z_{a,s}$ ; by Proposition 4.1.4, the latter are bounded independently of a and s.

4.4 Decay of local heights

Recall that we need to prove (Proposition 3.6.3) that

$$ \begin{align*} T_{\iota_{\mathfrak{p}}}(\sigma^{\vee})\widetilde{Z}(\phi^{\infty}, \chi)(v) \end{align*} $$

is a v-critical element of $ \overline {\mathbf {S}}{}'$ .

As in [9, §9.2, proof of Proposition 9.2.1], this is reduced to the following. For any $s\geq 0$ , denote by $\langle \ , \ \rangle _{s,\overline {w}}$ the local height pairing on $X_{H_{s, \overline {w}}}$ , which is valued in $H_{s, \overline {w}}^{\times }\hat {\otimes } L$ ; and let

$$ \begin{align*} \langle \ , \ \rangle_{\overline{w}}= \left[H_{s,\overline{w}}:F_{v}\right]^{-1} \cdot N_{H_{s,\overline{w}}/F_{v}} \left( \left\langle \ , \ \right\rangle_{s,\overline{w}}\right), \end{align*} $$

which is valued in $F_{v}^{\times }\hat {\otimes } L\subset \Gamma _{F}\hat {\otimes }L$ and is compatible with varying s by [9, (4.1.6)]. Then we will show that for all $\overline {w}\mid v$ and all $a\in \mathbf {A}^{S_{1}\infty , \times }$ with $v(a)=r_{v}$ , we have

(4.4.1) $$ \begin{align}\begin{aligned} \left\langle \widetilde{Z}_{a\varpi^{s}}(\phi^{\infty})[1]_{U}, {\mathrm{T}}_{\iota_{\mathfrak{p}}}(\sigma^{\vee})_{U}^{\mathrm{t}} t_{\chi}\right\rangle_{\overline{w}} &= O\left(q_{F,v}^{s}\right) & \text{in } {F}_{v}^{\times}\hat{\otimes}L(\chi),\\ \left\langle \widetilde{Z}_{a\varpi^{s}}(\phi^{\infty})[1]_{U}, {\mathrm{T}}_{\iota_{\mathfrak{p}}}(\sigma^{\vee})_{U}^{\mathrm{t}} t_{\chi}\right\rangle_{0, \overline{w}} &= O\left(q_{F,v}^{s}\right) &\text{in } H_{0, \overline{w}}^{\times} \hat{\otimes}L(\chi),\\ \end{aligned}\end{align} $$

where the second statement implies the first one. Until Lemma 4.4.3, the argument follows the lines of previous works [9, 24, 27].

The norm relation and heights

Denote by $N_{s}$ the norm from $H_{s , \overline {w}}$ to $H_{0, \overline {w}}$ , set $L':= L(\chi )$ , and let $\mathfrak {p}'\subset \mathscr {O}_{L'}$ be the maximal ideal. By the norm relation of Proposition 4.1.4, the aforementioned compatibility [9, (4.1.6)], and the integrality result of [9, Proposition 4.3.2], ⁹

(4.4.2) $$ \begin{align}\begin{aligned} \left\langle \widetilde{Z}_{a\varpi^{s}}(\phi^{\infty}) [1]_{U}, {\mathrm{T}}_{\iota_{\mathfrak{p}}}(\sigma^{\vee})_{U}^{\mathrm{t}} t_{\chi}\right\rangle_{0, \overline{w}} &= \left\langle {\mathrm{Tr}}_{H_{s}/H_{0}} \left(D_{a\varpi^{s}}\right), {\mathrm{T}}_{\iota_{\mathfrak{p}}}(\sigma^{\vee})_{U}^{\mathrm{t}} t_{\chi}\right\rangle_{0, \overline{w}} \\ &= N_{s}\left( \left\langle D_{a\varpi^{s}}, {\mathrm{T}}_{\iota_{\mathfrak{p}}}(\sigma^{\vee})_{U}^{\mathrm{t}} t_{\chi}\right\rangle_{s,\overline{w}}\right) \\ & \in \mathfrak{p}^{\prime-\left( d_{00}+ d_{0}+d_{1,s}+d_{2,s}\right)} N_{s}\left(H_{s, \overline{w}}^{\times}\hat{\otimes} \mathscr{O}_{L'}\right) \end{aligned}\end{align} $$

for some integers $d_{i, (s)}\geq 0$ that we now define and study.

Boundedness of denominators

Set

$$ \begin{align*} V':=\pi_{A^{\vee}}^{U}\otimes_{M} V_{\mathfrak{p}}A^{\vee}(\chi^{-1}) \subset V:= V_{p}J_{U} \otimes_{\mathbf{Q}_{p}}L' \end{align*} $$

considered as $\mathscr {G}_{E_{w}}$ -modules; let $V''$ be its direct complement in the decomposition of V in [9, (9.2.4)], and let $0\to V^{\prime +}\to V'\to V^{\prime -}\to 0$ be the ordinary filtration analogous to formula (1.2.6). If $?\in \{ ', '', {}^{\prime +}\}$ , define $T^{?}:= T_{p}J_{U}\otimes _{\mathbf {Z}_{p}}\mathscr {O}_{L'}\cap V^{?}$ , and let $T^{\prime -}=T'/T^{\prime +}$ . Then the integers $d_{i, (s)}$ are defined as follows:

1. – $d_{00}$ accounts for the denominators of the divisors, and it can be taken to be independent of s by Proposition 4.1.4.

2. – $d_{0}$ is such that $\mathfrak {p}^{\prime d_{0}} T\subset T'\oplus T''$ .

3. – $d_{1,s}:={\mathrm{length}}_{\mathscr {O}_{L'}} H^{1}\left ({H}_{s,\overline {w}}, T^{\prime \prime *}(1)\right )_{\mathrm{tors}}$ .

4. – $d_{2, s}:= {\mathrm{length}}_{\mathscr {O}_{L'}} H^{1}_{f}\left (H_{s, \overline {w}}, T'\right )/{N}_{\infty } H^{1}_{f}\left (H_{s, \overline {w}}, T'\right )$ , where $N_{\infty }$ denotes the universal norms ([23, §6], [9, §4.3]) with respect to the infinite abelian extension of $H_{s, \overline {w}}$ cut out by the closure in $\mathscr {G}_{H_{s, \overline {w}}}^{\mathrm{ab}}\supset H_{s, \overline {w}}^{\times } $ of

   $$ \begin{align*} \mathrm{Ker} \left[ H_{s, \overline{w}}^{\times} \stackrel{N_{H_{s, \overline{w}}/F_{v}}}{\longrightarrow} F_{v}^{\times} \to \Gamma_{F}\hat{\otimes} L\right]. \end{align*} $$

Proposition 4.4.1. Suppose that $V_{\mathfrak {p}}A $ is potentially crystalline as a representation of $\mathscr {G}_{F_{v}}$ ; then the sequences of integers $\left (d_{1,s}\right )$ and $\left (d_{2, s}\right )$ are bounded.

We will use the following vanishing result, in which $\overline {L}$ denotes an algebraic closure of $L'$ :

Lemma 4.4.2. Set $\Gamma _{\infty }:=\mathrm {Gal}\left (H_{\infty , \overline {w}}/E_{ {w}}\right )\cong E^{\times }\backslash E^{\times }_{\mathbf {A}^{\infty }}/ U^{v}U_{F,v}^{\circ }$ . For all Hodge–Tate characters $\psi \colon \mathscr {G}_{E_{w}}\to \overline {L}^{\times }$ factoring through $\Gamma _{\infty }$ , and for any

$$ \begin{align*} V^{?}\in \left\{V, V^{\prime\prime*}(1), V^{\prime+, *}(1), V^{\prime-}\right\}, \end{align*} $$

we have

$$ \begin{align*} H^{0}\left(E_{w}, V^{?}(\psi)\right)= 0. \end{align*} $$

Proof. The proof is largely similar to that of [9, Lemma 9.2.4], to which we refer for the background on the p-adic Hodge theory of characters.

We have

$$ \begin{align*} H^{0}\left(E_{w}, V^{?}\right) = {\textbf{D}}_{\mathrm{crys}}\left(V^{?}(\psi)\right)^{\varphi=1}, \end{align*} $$

where $\varphi $ is the crystalline Frobenius, and it suffices to prove that $ {\mathbf {D}}_{\mathrm{crys}}\left (V^{?}(\psi )\right )^{\varphi ^{d}=1}=0$ for $d=\left [E_{w}:E_{w, 0}\right ]$ , where $E_{w,0}$ is the maximal unramified extension of $\mathbf {Q}_{p}$ contained in $E_{w}$ . As $V'$ has been assumed potentially crystalline, it is pure of weight $-1$ , hence so are all the subquotients of $V'$ and $V^{\prime *}(1)$ . In particular, $\varphi ^{d}$ acts with negative weights on ${\mathbf {D}}_{\mathrm{crys}}\left (V^{?}\right )$ for $V^{?}=V^{\prime +, *}(1), V^{\prime -}$ ; by [22, Theorem 5.3], this last assertion is also true of $V^{?}=V\cong V^{*}(1)$ and its subquotients such as $V^{?}=V^{\prime \prime *}(1)$ . Therefore, it suffices to show that $\varphi ^{d}$ acts with weight $0$ on $ {\mathbf {D}}_{\mathrm{crys}}(\psi ^{m})$ for m such that $\psi ^{m}$ is crystalline.

Since $\psi $ is trivial on $U_{F}^{\circ }$ , the Hodge–Tate weights $(n_{\tau })_{\tau \in \mathrm {Hom}\left (E_{w}, \overline {L}\right )}$ satisfy $n_{\tau }+n_{\tau c}=0$ , where c is the complex conjugation of $E_{w}/F_{v}$ . The action of $\varphi ^{d}$ on $ {\mathbf {D}}_{\mathrm{crys}}(\psi ^{m})$ is by

(4.4.3) $$ \begin{align}\begin{aligned} \psi\circ {\mathrm{rec}}_{E, w}(\varpi_{w})^{-m}\cdot \prod_{\tau\in \mathrm{Hom}\left(E_{w}, \overline{L}\right)}\varpi_{w}^{mn_{\tau}},\end{aligned}\end{align} $$

where $\varpi _{w}\in E_{w}$ is any uniformizer. Choose $\varpi _{w}$ so that $\varpi _{w}^{e\left (E_{w}/F_{v}\right )}=\varpi _{v}$ is a uniformizer in $F_{v}$ . Then $\varpi _{w}^{c}=\pm \varpi _{w}$ , so that the second factor in expression (4.4.3) is $\pm 1$ . On the other hand, the subgroup $F^{\times }\backslash F_{\mathbf {A}^{\infty }}^{\times }/U_{F, v}^{\circ }\left (U^{v}\cap F_{\mathbf {A}^{\infty }}^{\times }\right )\subset \Gamma _{\infty }$ is finite, hence $\varpi _{v}$ and $\varpi _{w}$ have finite order in $\Gamma _{\infty }$ . It follows that the first factor in expression (4.4.3) is a root of unity too, hence $\varphi ^{d}$ acts with weight $0$ on $ {\mathbf {D}}_{\mathrm{crys}}(\psi ^{m})$ .

Proof Proof of Proposition 4.4.1

By the long exact sequence attached to

$$ \begin{align*} 0\to T^{\prime\prime*}(1)\to V^{\prime\prime*}(1)\to T^{\prime\prime*}(1)\otimes L'/\mathscr{O}_{L'}\to 0 \end{align*} $$

and the vanishing of $H^{0}\left ({H}_{s,\overline {w}}, V^{\prime *}(1)\right )$ (which follows from Lemma 4.4.2), we have

$$ \begin{align*}\begin{aligned} H^{1}\left({H}_{s,\overline{w}}, T^{\prime\prime*}(1)\right)_{\mathrm{tors}} &\cong H^{0}\left({H}_{s,\overline{w}}, T^{\prime\prime*}(1) \otimes_{\mathscr{O}_{L'}} L'/\mathscr{O}_{L'}\right) \\ &= H^{0}\left(E_{w}, T^{\prime\prime*}(1)\otimes_{\mathscr{O}_{L}} \mathscr{O}_{L}\left[\mathrm{Gal}\left({H}_{s,\overline{w}}/E_{w}\right)\right]\otimes_{\mathscr{O}_{L}} {L'/\mathscr{O}_{L'}} \right). \end{aligned}\end{align*} $$

By [23, Theorems 6.6, 6.9] (or strictly speaking, a slightly generalized form thereof which still holds true by the arguments in [9, proof of Proposition 4.3.2]) and the vanishing of $H^{0}\left ({H}_{s,\overline {w}}, V^{\prime +, *}(1)\oplus V^{\prime -}\right )$ (which follows from Lemma 4.4.2), we have

$$ \begin{align*}\begin{aligned} d_{2, s}&\leq {\mathrm{length}}_{\mathscr{O}_{L'}} H^{0} \left({H}_{s,\overline{w}}, T^{\prime+,*}(1) \otimes_{\mathscr{O}_{L'}} L'/\mathscr{O}_{L'}\right) + {\mathrm{length}}_{\mathscr{O}_{L'}} H^{0} \left({H}_{s,\overline{w}}, T^{\prime-} \otimes_{\mathscr{O}_{L'}} L'/\mathscr{O}_{L'}\right) \\ &={\mathrm{length}}_{\mathscr{O}_{L'}} H^{0}\left(E_{w},\left(T^{\prime+,*}(1) \oplus T^{\prime-}\right) \otimes_{\mathscr{O}_{L}} \mathscr{O}_{L}\left[\mathrm{Gal}\left({H}_{s,\overline{w}}/E_{w}\right)\right]\otimes_{\mathscr{O}_{L}} {L'/\mathscr{O}_{L'}} \right). \end{aligned}\end{align*} $$

Then the boundedness of $d_{1,s}$ and $d_{2,s}$ follows as in [27, proof of Proposition 8.10] from the vanishing of

$$ \begin{align*} H^{0}\left(H_{\infty, \overline{w}}, V^{?}\right) \subset \bigoplus_{\psi\colon \Gamma_{\infty}\to \overline{L}^{\times} \text{ Hodge--Tate}} H^{0}\left(E_{w}, V^{?}(\psi)\right) \end{align*} $$

for $V^{?}\in \left \{ V^{\prime \prime *}(1),V^{\prime +,*}(1), V'\right \}$ , which is a consequence of Lemma 4.4.2.□

Completion of the proofs

We are ready to reduce our decay statement for local heights to the decay statement for intersection multiplicities proved in §4.3.

Lemma 4.4.3. For all $s'\leq s$ , the restriction of the $\overline {w}$ -adic valuation yields an isomorphism of $\mathscr {O}_{L'}$ -modules

$$ \begin{align*} \overline{w}\colon N_{s}\left(H_{s, \overline{w}}^{\times}\hat{\otimes}\mathscr{O}_{L'}\right) / q^{s'}\cdot\left( H_{0, \overline{w}}^{\times}\hat{\otimes}\mathscr{O}_{L'}\right) \to \mathscr{O}_{L'}/q^{s'}\mathscr{O}_{L'}. \end{align*} $$

Proof. We drop all subscripts $\overline {w}$ . Recall that the extension $H_{s}/H_{0}$ is totally ramified of degree $q^{s}$ . Let $\varpi _{s}\in \mathscr {O}_{H_{s}}$ be a uniformizer; then $\omega _{0}:=N_{s}(\varpi _{s})$ is a uniformizer of $H_{0} $ . For $*=0,s$ we have the decompositions

$$ \begin{align*} H_{*}^{\times}\hat{\otimes} \mathscr{O}_{L'} =\mathscr{O}_{H_{*}}^{\times}\hat{\otimes}\mathscr{O}_{L' }\oplus \varpi_{*}\otimes\mathscr{O}_{L'}. \end{align*} $$

The map $N_{s}$ respects the decompositions and, by local class field theory, has image

$$ \begin{align*} q^{s}\cdot\left(\mathscr{O}_{H_{0}}^{\times}\hat{\otimes}\mathscr{O}_{ L'}\right) \oplus \varpi_{0}\otimes\mathscr{O}_{L'}. \end{align*} $$

The valuation map annihilates the first summand and sends the second one isomorphically to $\mathscr {O}_{L'}$ . The result follows.

Proof Proof of Proposition 3.6.3

By the comparison of the valuation component of local heights with arithmetic intersections in [9, Proposition 4.3.1], applied to the curve $X_{U,H_{0}}$ , the image of the left-hand side of equation (4.4.2) under $\overline {w}$ is

(4.4.4) $$ \begin{align}\begin{aligned} m\left( \widetilde{Z}_{a\varpi^{s}}(\phi^{\infty})_{U}[1], {\mathrm{T}}_{\iota_{\mathfrak{p}}}(\sigma^{\vee})_{U}^{\mathrm{t}} t_{\chi}\right). \end{aligned}\end{align} $$

By Corollary 4.3.4 applied to $D= {\mathrm{T}}_{\iota _{\mathfrak {p}}}(\sigma ^{\vee })_{U}^{\mathrm{t}} t_{\chi }$ , the right-hand side of expression (4.4.4) is $O\left (q_{F,v}^{s}\right )$ . By equation (4.4.2), Proposition 4.4.1, and Lemma 4.4.3, we deduce the desired decay statement (4.4.1).

Summary

We have just completed the proof of Proposition 3.6.3. It implies Proposition 3.6.2, which together with Theorem 3.6.1 implies the kernel identity of Theorem 3.4.3. By Lemma 3.4.4, that implies Theorem 2.2.1, which is in turn an equivalent form of Theorem B by Lemma 2.2.2.