On the $p$-part of the Birch-Swinnerton-Dyer formula for multiplicative primes

TL;DR

This paper proves the $p$-part of the Birch-Swinnerton-Dyer formula for semistable elliptic curves with multiplicative reduction at primes greater than 3, using Iwasawa theory and extending previous results.

Contribution

It extends the $p$-part of the BSD formula to primes of multiplicative reduction for semistable elliptic curves, building on and modifying existing methods.

Findings

Established the $p$-part of BSD for multiplicative primes

Extended previous results to a broader class of primes

Utilized Iwasawa theory techniques in the proof

Abstract

Let $E/\mathbf{Q}$ be a semistable elliptic curve of analytic rank one, and let $p>3$ be a prime for which $E[p]$ is irreducible. In this note, following a slight modification of the methods of Jetchev-Skinner-Wan, we use Iwasawa theory to establish the $p$-part of the Birch and Swinnerton-Dyer formula for $E$. In particular, we extend the main result of loc.cit. to primes of multiplicative reduction.

Equations

$${\bar{\rho }}_{E,p}:\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})⟶{\mathrm{Aut}}_{{\mathbf{F}}_p}(E[p])$$

$${\rm ord}_{p}\left(\frac{L^{\prime}(E,1)}{{\rm Reg}(E/\mathbf{Q})\cdot\Omega_{E}}\right)={\rm ord}_{p}\biggl{(}\#\Sha(E/\mathbf{Q})\prod_{\ell\mid N}c_{\ell}(E/\mathbf{Q})\biggr{)},$$

$${\mathrm{ord}}_p(\#\text{Sha}(E/K_1)[p^{\infty }])⩾2\cdot {\mathrm{ord}}_p([E(K_1):\mathbf{Z}.P_{K_1}])-\sum _{\begin{array}{c}w\mid N^{+}\\ \text{wsplit}\end{array}}{\mathrm{ord}}_p(c_w(E/K_1)),$$

$$C{h}_{\Lambda }(X_{\mathrm{ac}}(E[p^{\infty }])){\Lambda }_{R_0}\subseteq (L_p(f))$$

$$L_p(f,\text{mathds}1)=(1-a_p{p}^{-1}+p^{-1}{)}^2\cdot ({\log }_{{\omega }_E}{P}_{K_1}{)}^2$$

$${\mathrm{ord}}_p(\#\text{Sha}(E/K_2)[p^{\infty }])⩽2\cdot \mathrm{ord}([E(K_2):\mathbf{Z}.P_{K_2}]),$$

$$\begin{array}{rl}\bullet & \text{every prime factor of N is either split or ramified in K};\\ \bullet & \text{there is at least one prime q∣N nonsplit in K};\\ \bullet & \text{p splits in K},\end{array}$$

$$C{h}_{\Lambda }(X_{\mathrm{ac}}(E[p^{\infty }])){\Lambda }_{R_0}=(L_p(f)).$$

$$C{h}_{{\Lambda }_{\mathbb{I}}}(X_{\mathrm{ac}}(A_{\mathbf{f}})){\Lambda }_{\mathbb{I},R_0}=(L_p(\mathbf{f})),$$

$$L_p(f,\text{mathds}1)=(1-a_p{p}^{-1}{)}^2\cdot ({\log }_{{\omega }_E}{P}_K{)}^2,$$

$${\bar{\rho }}_{E,p}:\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})⟶{\mathrm{Aut}}_{{\mathbf{F}}_p}(E[p])$$

$${H}^{1}_{{\rm ac}}(K_{w},V):=\left\{\begin{array}[]{ll}{H}^{1}(K_{\overline{\mathfrak{p}}},V)&\textrm{if $w=\overline{\mathfrak{p}}$;}\\ 0&\textrm{if $w=\mathfrak{p}$;}\\ {H}^{1}_{\rm ur}(K_{w},V)&\textrm{if $w\nmid p$},\end{array}\right.$$

$$H^{1}_{\rm ac}(K,E[p^{\infty}]):={\rm ker}\biggl{\{}{H}^{1}(K,E[p^{\infty}])\longrightarrow\prod_{w}\frac{{H}^{1}(K_{w},E[p^{\infty}])}{{H}^{1}_{{\rm ac}}(K_{w},E[p^{\infty}])}\biggr{\}},$$

$$M:=T{\otimes }_{{\mathbf{Z}}_p}{\Lambda }^{\ast },$$

$${\rm Sel}_{\mathfrak{p}}(K_{\infty},E[p^{\infty}]):={\rm ker}\biggl{\{}H^{1}(K,{M})\longrightarrow H^{1}(K_{\mathfrak{p}},{M})\oplus\prod_{w\nmid p}H^{1}(K_{w},{M})\biggr{\}}.$$

$$X_{\mathrm{ac}}^{\Sigma }(E[p^{\infty }]):={\mathrm{Hom}}_{{\mathbf{Z}}_p}({\mathrm{Sel}}_{\mathfrak{p}}^{\Sigma }(K_{\infty },E[p^{\infty }]),{\mathbf{Q}}_p/{\mathbf{Z}}_p),$$

$${\log }_{{\omega }_E}:E({\mathbf{Q}}_p)⟶{\mathbf{Z}}_p$$

$$\#{\mathbf{Z}}_p/f_{\mathrm{ac}}^{\Sigma }(0)$$

$$\times \prod _{\begin{array}{c}w\mid N^{+}\\ w\notin \Sigma \end{array}}{c}_w^{(p)}(E/K)\cdot \prod _{w\in \Sigma }\#H^1(K_w,E[p^{\infty }]),$$

$$E(K)\otimes {\mathbf{Q}}_p/{\mathbf{Z}}_p⟶E(K_w)\otimes {\mathbf{Q}}_p/{\mathbf{Z}}_p$$

$$\#H_{\mathrm{ac}}^1(K,E[p^{\infty }])=\#\text{Sha}(E/K)[p^{\infty }]\cdot \frac{[E(K_{\mathfrak{p}}{)}_{/\mathrm{tors}}\otimes {\mathbf{Z}}_p:{\mathbf{Z}}_p.P{]}^2}{[E(K)\otimes {\mathbf{Z}}_p:{\mathbf{Z}}_p.P{]}^2},$$

$$[E(K_{\mathfrak{p}}{)}_{/\mathrm{tors}}\otimes {\mathbf{Z}}_p:{\mathbf{Z}}_p.P]=\frac{\#{\mathbf{Z}}_p/((\frac{1-a_p+p}{p}){\log }_{{\omega }_E}P)}{\#H^0(K_{\mathfrak{p}},E[p^{\infty }])}$$

$$\#H^{1}_{{\rm ac}}(K,E[p^{\infty}])=\#\Sha(E/K)[p^{\infty}]\cdot\Biggl{(}\frac{\#\mathbf{Z}_{p}/((\frac{1-a_{p}+p}{p})\log_{\omega_{E}}P)}{[E(K)\otimes\mathbf{Z}_{p}:\mathbf{Z}_{p}.P]\cdot\#H^{0}(K_{\mathfrak{p}},E[p^{\infty}])}\Biggr{)}^{2},$$

$$0⟶E_1(K_{\mathfrak{p}})⟶E_0(K_{\mathfrak{p}})⟶{\tilde{E}}_{\mathrm{ns}}({\mathbf{F}}_p)⟶0.$$

$$[E(K_{\mathfrak{p}}{)}_{/\mathrm{tors}}\otimes {\mathbf{Z}}_p:{\mathbf{Z}}_p.P]$$

$$=[E(K_{\mathfrak{p}}):E_0(K_{\mathfrak{p}}){]}_p\cdot \frac{\#{\mathbf{Z}}_p/({\log }_{{\omega }_E}P)\cdot \#(E_0(K_{\mathfrak{p}})/E_1(K_{\mathfrak{p}})\otimes {\mathbf{Z}}_p)}{\#{\mathbf{Z}}_p/p{\mathbf{Z}}_p\cdot \#(E(K_{\mathfrak{p}}{)}_{\mathrm{tors}}\otimes {\mathbf{Z}}_p)},$$

$$[E(K_{\mathfrak{p}}{)}_{/\mathrm{tors}}\otimes {\mathbf{Z}}_p:{\mathbf{Z}}_p.P]=[E(K_{\mathfrak{p}}):E_0(K_{\mathfrak{p}}){]}_p\cdot \frac{\#\mathbf{Z}/(\frac{1}{p}{\log }_{{\omega }_E}P)}{\#H^0(K_{\mathfrak{p}},E[p^{\infty }])},$$

$$H_{\mathrm{ac}}^1(K,E[p^{\infty }])=\#\text{Sha}(E/K)[p^{\infty }]\cdot {\left(\frac{[E(K_{\mathfrak{p}}):E_0(K_{\mathfrak{p}}){]}_p\cdot \#{\mathbf{Z}}_p/(\frac{1}{p}{\log }_{{\omega }_E}P)}{[E(K)\otimes {\mathbf{Z}}_p:{\mathbf{Z}}_p.P]\cdot \#H^0(K_{\mathfrak{p}},E[p^{\infty }])}\right)}^2.$$

$$\begin{split}\#\mathbf{Z}_{p}/f_{\rm ac}^{\Sigma}(0)&=\#\Sha(E/K)[p^{\infty}]\cdot\Biggl{(}\frac{\#\mathbf{Z}_{p}/(\frac{1}{p}\log_{\omega_{E}}P)}{[E(K)\otimes\mathbf{Z}_{p}:\mathbf{Z}_{p}.P]}\Biggr{)}^{2}\cdot[E(K_{\mathfrak{p}}):E_{0}(K_{\mathfrak{p}})]_{p}^{2}\\ &\quad\times\prod_{\begin{subarray}{c}w\in S\smallsetminus\Sigma\\ w\nmid p\;{\rm split}\end{subarray}}\#H_{\rm ur}^{1}(K_{w},E[p^{\infty}])\cdot\prod_{w\in\Sigma}\#H^{1}(K_{w},E[p^{\infty}]),\end{split}$$

$$[E(K_{\mathfrak{p}}):E_0(K_{\mathfrak{p}}){]}_p=c_w^{(p)}(E/K)$$

$$\#H_{\mathrm{ur}}^1(K_w,E[p^{\infty }])=c_w^{(p)}(E/K)$$

Full text

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 682152).

1. 1 Introduction
2. 2 Selmer groups
3. 3 A $p$-adic Waldspurger formula
4. 4 Main Conjectures
5. 5 Proof of Theorem A

1. Introduction

Let $E/\mathbf{Q}$ be a semistable elliptic curve of conductor $N$, and let $L(E,s)$ be the Hasse–Weil $L$-function of $E$. By the celebrated work of Wiles [Wil95] and Taylor–Wiles [TW95], $L(E,s)$ is known to admit analytic continuation to the entire complex plane, and to satisfy a functional equation relating its values at $s$ and $2-s$. The purpose of this note is to prove the following result towards the Birch and Swinnerton-Dyer formula for $E$.

Theorem A**.**

Let $E/\mathbf{Q}$ be a semistable elliptic curve of conductor $N$ with ${\rm ord}_{s=1}L(E,s)=1$, and let $p>3$ be a prime such that the mod $p$ Galois representation

[TABLE]

is irreducible. If $p\mid N$, assume in addition that $E[p]$ is ramified at some prime $q\neq p$. Then

[TABLE]

where

• •

${\rm Reg}(E/\mathbf{Q})$* is the discriminant of the Néron–Tate height pairing on $E(\mathbf{Q})\otimes\mathbf{R}$;*

• •

$\Omega_{E}$* is the Néron period of $E$;*

• •

$\Sha(E/\mathbf{Q})$* is the Tate–Shafarevich group of $E$; and*

• •

$c_{\ell}(E/\mathbf{Q})$* is the Tamagawa number of $E$ at the prime $\ell$.*

In other words, the $p$-part of the Birch and Swinnerton-Dyer formula holds for $E$.

Remark*.*

Having square-free conductor, any elliptic curve $E/\mathbf{Q}$ as in Theorem A is necessarily non-CM. By [Ser72, Thm. 2], if follows that $\bar{\rho}_{E,p}$ is in fact surjective for all but finitely many primes $p$; by [Maz78, Thm. 4], this holds as soon as $p\geqslant 11$.

When $p$ is a prime of good reduction for $E$, Theorem A (in the stated level of generality) was first established by Jetchev–Skinner–Wan [JSW15]. (We should note that [JSW15, Thm. 1.2.1] also allows $p=3$ provided $E$ has good supersingular reduction at $p$, the assumption $a_{3}(E)=0$ having been removed in a recent work by Sprung; see [Spr16, Cor. 1.3].) For primes $p\mid N$, some particular cases of Theorem A are contained in the work of Skinner–Zhang (see [SZ14, Thm. 1.1]) under further hypotheses on $N$ and, in the case of split multiplicative reduction, on the $L$-invariant of $E$. Thus the main novelty in Theorem A is for primes $p\mid N$.

Similarly as in [JSW15], our proof of Theorem A uses anticyclotomic Iwasawa theory. In order to clarify the relation between the arguments in loc.cit. and the arguments in this paper, let us recall that the proof of [JSW15, Thm. 1.2.1] (for primes $p\nmid N$) is naturally divided into two steps:

1. (1)

Exact lower bound on the predicted order of $\Sha(E/\mathbf{Q})[p^{\infty}].$ For this part of the argument, in [JSW15] one chooses a suitable imaginary quadratic field $K_{1}=\mathbf{Q}(\sqrt{D_{1}})$ with $L(E^{D_{1}},1)\neq 0$; combined with the hypothesis that $E$ has analytic rank one, it follows that $E(K_{1})$ has rank one and that $\#\Sha(E/K_{1})<\infty$ by the work of Gross–Zagier and Kolyvagin. The lower bound

[TABLE]

where $P_{K_{1}}\in E(K_{1})$ is a Heegner point, $c_{w}(E/K_{1})$ is the Tamagawa number of $E/K_{1}$ at $w$, and $N^{+}$ is the product of the prime factors of $N$ that are either split or ramified in $K_{1}$, is then established by combining:

1. (1.a)

A Mazur control theorem proved “à la Greenberg” [Gre99] for an anticyclotomic Selmer group $X_{\rm ac}(E[p^{\infty}])$ attached to $E/K_{1}$ ([JSW15, Thm. 3.3.1]); 2. (1.b)

The proof by Xin Wan [Wan14a], [Wan14b] of one of the divisibilities predicted by the Iwasawa–Greenberg Main Conjecture for $X_{\rm ac}^{\Sigma}(E[p^{\infty}])$, namely the divisibility

[TABLE]

where $f=\sum_{n=1}^{\infty}a_{n}q^{n}$ is the weight 2 newform associated with $E$, $\Lambda_{R_{0}}$ is a scalar extension of the anticyclotomic Iwasawa algebra $\Lambda$ for $K_{1}$, and $L_{p}(f)\in\Lambda_{R}$ is a certain anticyclotomic $p$-adic $L$-function; 3. (1.c)

The “$p$-adic Waldspurger formula” of Bertolini–Darmon–Prasanna [BDP13] (as extended by Brooks [HB15] to indefinite Shimura curves):

[TABLE]

relating the value of $L_{p}(f)$ at the trivial character to the formal group logarithm of the Heegner point $P_{K_{1}}$.

When combined with the known $p$-part of the Birch and Swinnerton-Dyer formula for the quadratic twist $E^{D_{1}}/\mathbf{Q}$ (being of rank analytic zero, this follows from [SU14] and [Wan14c]), inequality $(\ref{eq:lower})$ easily yields the exact lower bound for $\#\Sha(E/\mathbf{Q})[p^{\infty}]$ predicted by the BSD conjecture. 2. (2)

Exact upper bound on the predicted order of $\Sha(E/\mathbf{Q})[p^{\infty}]$. For this second part of the argument, in [JSW15] one chooses another imaginary quadratic field $K_{2}=\mathbf{Q}(\sqrt{D_{2}})$ (in general different from $K_{1}$) such that $L(E^{D_{2}},1)\neq 0$. Crucially, $K_{2}$ is chosen so that the associated $N^{+}$ (the product of the prime factors of $N$ that are split or ramified in $K_{2}$) is as small as possible in a certain sense; this ensures optimality of the upper bound provided by Kolyvagin’s methods:

[TABLE]

where $P_{K_{2}}\in E(K_{2})$ is a Heegner point coming from a parametrization of $E$ by a Shimura curve attached to an indefinite quaternion algebra (which is nonsplit unless $N$ is prime). Combined with the general Gross–Zagier formula [YZZ13] and the $p$-part of the Birch and Swinnerton-Dyer formula for $E^{D_{2}}/\mathbf{Q}$, inequality $(\ref{eq:upper-bound})$ then yields the predicted optimal upper bound for $\#\Sha(E/\mathbf{Q})[p^{\infty}]$.

Our proof of Theorem A dispenses with part (2) of the above argument; in particular, it only requires the use of classical modular parametrizations of $E$. Indeed, if $K$ is an imaginary quadratic field satisfying the following hypotheses relative to the square-free integer $N$:

[TABLE]

in [Cas17b] (for good ordinary $p$) and [CW16] (for good supersingular $p$) we have completed under mild hypotheses the proof of the Iwasawa–Greenberg main conjecture for the associated $X_{\rm ac}(E[p^{\infty}])$:

[TABLE]

With this result at hand, a simplified form (since $N^{-}=1$ here) of the arguments from [JSW15] in part (1) above lead to an equality in $(\ref{eq:lower})$ taking $K_{1}=K$, and so to the predicted order of $\Sha(E/\mathbf{Q})[p^{\infty}]$ when $p\nmid N$.

To treat the primes $p\mid N$ of multiplicative reduction for $E$ (which, as already noted, is the only new content of Theorem A), we use Hida theory. Indeed, if $a_{p}$ is the $U_{p}$-eigenvalue of $f$ for such $p$, we know that $a_{p}\in\{\pm 1\}$, so in particular $f$ is ordinary at $p$. Let $\mathbf{f}\in\mathbb{I}[[q]]$ be the Hida family associated with $f$, where $\mathbb{I}$ is a certain finite flat extension of the one-variable Iwasawa algebra. In Section 4, we deduce from [Cas17b] and [Wan14a] a proof of a two-variable analog of the Iwasawa–Greenberg main conjecture $(\ref{eq:IMC})$ over the Hida family:

[TABLE]

where $L_{p}(\mathbf{f})\in\Lambda_{\mathbb{I},R_{0}}$ is the two-variable anticyclotomic $p$-adic $L$-function introduced in [Cas14]. By construction, $L_{p}(\mathbf{f})$ specializes to $L_{p}(f)$ in weight $2$, and by a control theorem for the Hida variable, the characteristic ideal of $X_{\rm ac}(A_{\mathbf{f}})$ similarly specializes to $Ch_{\Lambda}(X_{\rm ac}(E[p^{\infty}]))$, yielding a proof of the Iwasawa–Greenberg main conjecture $(\ref{eq:IMC})$ in the multiplicative reduction case. Combined with the anticyclotomic control theorem of (1.a) and the natural generalization (contained in [Cas17a]) of the $p$-adic Waldspurger formula in (1.c) to this case:

[TABLE]

we arrive at the predicted formula for $\#\Sha(E/\mathbf{Q})[p^{\infty}]$ just as in the good reduction case.

Acknowledgements. As will be clear to the reader, this note borrows many ideas and arguments from [JSW15]. It is a pleasure to thank Chris Skinner for several useful conversations.

2. Selmer groups

2.1. Definitions

Let $E/\mathbf{Q}$ be a semistable elliptic curve of conductor $N$, and let $p\geqslant 5$ be a prime such that the mod $p$ Galois representations

[TABLE]

is irreducible. Let $T=T_{p}(E)$ be the $p$-adic Tate module of $E$, and set $V=T\otimes_{\mathbf{Z}_{p}}\mathbf{Q}_{p}$.

Let $K$ be an imaginary quadratic field in which $p=\mathfrak{p}\overline{\mathfrak{p}}$ splits, and for every place $w$ of $K$ define the anticyclotomic local condition ${H}^{1}_{{\rm ac}}(K_{w},V)\subseteq{H}^{1}(K_{w},V)$ by

[TABLE]

where ${H}^{1}_{\rm ur}(K_{w},V):={\rm ker}\{{H}^{1}(K_{w},V)\rightarrow{H}^{1}(I_{w},V)\}$ is the unramified part of cohomology.

Definition 2.1**.**

The anticyclotomic Selmer group for $E$ is

[TABLE]

where ${H}^{1}_{{\rm ac}}(K_{w},E[p^{\infty}])\subseteq{H}^{1}(K_{w},E[p^{\infty}])$ is the image of ${H}^{1}_{{\rm ac}}(K_{w},V)$ under the natural map ${H}^{1}(K_{w},V)\rightarrow{H}^{1}(K_{w},V/T)\simeq{H}^{1}(K_{w},E[p^{\infty}])$.

Let $\Gamma={\rm Gal}(K_{\infty}/K)$ be the Galois group of the anticyclotomic $\mathbf{Z}_{p}$-extension of $K$, and let $\Lambda=\mathbf{Z}_{p}[[\Gamma]]$ be the anticyclotomic Iwasawa algebra. Consider the $\Lambda$-module

[TABLE]

where $\Lambda^{*}={\rm Hom}_{\rm cont}(\Lambda,\mathbf{Q}_{p}/\mathbf{Z}_{p})$ is the Pontryagin dual of $\Lambda^{\rm ac}$. Letting $\rho_{E,p}$ denote the natural action of $G_{K}:={\rm Gal}(\overline{\mathbf{Q}}/K)$ on $T$, the $G_{K}$-action on ${M}$ is given by $\rho_{E,p}\otimes\Psi^{-1}$, where $\Psi$ is the composite character $G_{K}\twoheadrightarrow\Gamma\hookrightarrow\Lambda^{\times}$.

Definition 2.2**.**

The anticyclotomic Selmer group for $E$ over $K_{\infty}^{\rm ac}/K$ is defined by

[TABLE]

More generally, for any given finite set $\Sigma$ of places $w\nmid p$ of $K$, define the “$\Sigma$-imprimitive” Selmer group ${\rm Sel}_{\mathfrak{p}}^{\Sigma}(K_{\infty},E[p^{\infty}])$ by dropping the summands $H^{1}(K_{w},{M})$ for the places $w\in\Sigma$ in the above definition. Set

[TABLE]

which is easily shown to be a finitely generated $\Lambda$-module.

2.2. Control theorems

Let $E$, $p$, and $K$ be an in the preceding section, and let $N^{+}$ denote the product of the prime factors of $N$ which are split in $K$.

Anticlotomic Control Theorem

Denote by $\hat{E}$ the formal group of $E$, and let

[TABLE]

the formal group logarithm attached to a fixed invariant differential $\omega_{E}$ on $\hat{E}$. Letting $\gamma\in\Gamma$ be a fixed topological generator, we identify the one-variable power series ring $\mathbf{Z}_{p}[[T]]$ with the Iwasawa algebra $\Lambda=\mathbf{Z}_{p}[[\Gamma]]$ by sending $1+T\mapsto\gamma$.

Theorem 2.3**.**

Let $\Sigma$ be any set of places of $K$ not dividing $p$, and assume that ${\rm rank}_{\mathbf{Z}}(E(K))=1$ and that $\#\Sha(E/K)[p^{\infty}]<\infty$. Then $X_{\rm ac}^{\Sigma}(E[p^{\infty}])$ is $\Lambda$-torsion, and letting $f_{\rm ac}^{\Sigma}(T)\in\Lambda$ be a generator of $Ch_{\Lambda}(X_{\rm ac}^{\Sigma}(E[p^{\infty}]))$, we have

[TABLE]

where $\varepsilon_{p}=p^{-1}$ if $p\nmid N$ and $\varepsilon_{p}=0$ otherwise, $P\in E(K)$ is any point of infinite order, and $c_{w}^{(p)}(E/K)$ is the $p$-part of the Tamagawa number of $E/K$ at $w$.

Proof.

As we are going to show, this follows easily from the “Anticyclotomic Control Theorem” established in [JSW15, §3.3]. The hypotheses imply that ${\rm corank}_{\mathbf{Z}_{p}}{\rm Sel}(K,E[p^{\infty}])=1$ and that the natural map

[TABLE]

is surjective for all $w\mid p$. By [JSW15, Prop. 3.2.1] (see also the discussion in [loc.cit., p. 22]) it follows that $H^{1}_{{\rm ac}}(K,E[p^{\infty}])$ is finite with

[TABLE]

where $E(K_{\mathfrak{p}})_{/{\rm tors}}$ is the quotient $E(K_{\mathfrak{p}})$ by its maximal torsion submodule, and $P\in E(K)$ is any point of infinite order. If $p\nmid N$, then

[TABLE]

as shown in [JSW15, p. 23], and substituting $(\ref{eq:p-notdiv})$ into $(\ref{eq:321})$ we arrive at

[TABLE]

from where the result follows immediately by [JSW15, Thm. 3.3.1].

Suppose now that $p\mid N$. Let $\tilde{E}_{\rm ns}(\mathbf{F}_{p})$ be the group on nonsingular points on the reduction of $E$ modulo $p$, $E_{0}(K_{\mathfrak{p}})$ be the inverse image of $\tilde{E}_{\rm ns}(\mathbf{F}_{p})$ under the reduction map, and $E_{1}(K_{\mathfrak{p}})$ be defined by the exactness of the sequence

[TABLE]

The formal group logarithm defines an injective homomorphism $\log_{\omega_{E}}:E(K_{\mathfrak{p}})_{/{\rm tor}}\otimes\mathbf{Z}_{p}\rightarrow\mathbf{Z}_{p}$ mapping $E_{1}(K_{\mathfrak{p}})$ isomorphically onto $p\mathbf{Z}_{p}$, and hence we see that

[TABLE]

where the first equality follows from the same immediate calculation as in [JSW15, p. 23], and in the second equality $[E(K_{\mathfrak{p}}):E_{0}(K_{\mathfrak{p}})]_{p}$ denotes the $p$-part of the index $[E(K_{\mathfrak{p}}):E_{0}(K_{\mathfrak{p}})]$. By $(\ref{eq:red-p})$, we have $E_{1}(K_{\mathfrak{p}})/E_{0}(K_{\mathfrak{p}})\otimes\mathbf{Z}_{p}\simeq\tilde{E}_{\rm ns}(\mathbf{F}_{p})\otimes\mathbf{Z}_{p}$, which is trivial by e.g. [Sil94, Prop. 5.1] (and $p>2$). Since clearly $E(K_{\mathfrak{p}})_{\rm tors}\otimes\mathbf{Z}_{p}=H^{0}(K_{\mathfrak{p}},E[p^{\infty}])$, we thus conclude that

[TABLE]

and substituting $(\ref{eq:calcul})$ into $(\ref{eq:321})$ we arrive at

[TABLE]

Plugging this formula for $H^{1}_{\rm ac}(K,E[p^{\infty}])$ into [JSW15, Thm. 3.3.1] yields the equality

[TABLE]

where $S$ is any finite set of places of $K$ containing $\Sigma$ and the primes above $N$. Now, if $w\mid p$, then

[TABLE]

by definition, while if $w\nmid p$, then

[TABLE]

by [SZ14, Lem. 9.1]. Since $c^{(p)}_{w}(E/K)=1$ unless $w\mid N$, substituting $(\ref{eq:tam-p})$ and $(\ref{eq:tam-not-p})$ into $(\ref{eq:control-p})$, the proof of Theorem 2.3 follows. ∎

Control Theorem for Greenberg Selmer groups

Let $\Lambda_{W}=\mathbf{Z}_{p}[[W]]$ be a one-variable power series ring. Let $M$ be an integer prime to $p$, let $\chi$ be a Dirichlet character modulo $pM$, and let $\mathbf{f}=\sum_{n=1}^{\infty}\mathbf{a}_{n}q^{n}\in\mathbb{I}[[q]]$ be an ordinary $\mathbb{I}$-adic cusp eigenform of tame level $M$ and character $\chi$ (as defined in [SU14, §3.3.9]) defined over a local reduced finite integral extension $\mathbb{I}/\Lambda_{W}$.

Let $\mathcal{X}^{a}_{\mathbb{I}}$ the set of continuous $\mathbf{Z}_{p}$-algebra homomorphisms $\phi:\mathbb{I}\rightarrow\overline{\mathbf{Q}}_{p}$ whose composition with the structural map $\Lambda_{W}\rightarrow\mathbb{I}$ is given by $\phi(1+W)=(1+p)^{k_{\phi}-2}$ for some integer $k_{\phi}\in\mathbf{Z}_{\geqslant 2}$ called the weight of $\phi$. Then for all $\phi\in\mathcal{X}^{a}_{\mathbb{I}}$ we have

[TABLE]

where $\omega$ is the Teichmüller character. In this paper will only consider the case where $\chi$ is the trivial character, in which case for all $\phi\in\mathcal{X}^{a}_{\mathbb{I}}$ of weight $k_{\phi}\equiv 2\pmod{p-1}$, either

1. (1)

$\mathbf{f}_{\phi}$ is a newform on $\Gamma_{0}(pM)$; 2. (2)

$\mathbf{f}_{\phi}$ is the $p$-stabilization of a $p$-ordinary newform on $\Gamma_{0}(M)$.

As is well-known, for weights $k_{\phi}>2$ only case (2) is possible; for $k_{\phi}=2$ both cases occur.

Let $k_{\mathbb{I}}=\mathbb{I}/\mathfrak{m}_{\mathbb{I}}$ be the residue field of $\mathbb{I}$, and assume that the residual Galois representation

[TABLE]

attached to $\mathbf{f}$ is irreducible. Then there exists a free $\mathbb{I}$-module $T_{\mathbf{f}}$ of rank two equipped with a continuous $\mathbb{I}$-linear action of $G_{\mathbf{Q}}$ such that, for all $\phi\in\mathcal{X}_{\mathbb{I}}^{a}$, there is a canonical $G_{\mathbf{Q}}$-isomorphism

[TABLE]

where $T_{\mathbf{f}_{\phi}}$ is a $G_{\mathbf{Q}}$-stable lattice in the Galois representation $V_{\mathbf{f}_{\phi}}$ associated with $\mathbf{f}_{\phi}$. (Here, $T_{\mathbf{f}}$ corresponds to the Galois representation denoted $M(\mathbf{f})^{*}$ in [KLZ14, Def. 7.2.5]; in particular, ${\rm det}(V_{\mathbf{f}_{\phi}})=\epsilon^{k_{\phi}-1}$, where $\epsilon:G_{\mathbf{Q}}\rightarrow\mathbf{Z}_{p}^{\times}$ is the $p$-adic cyclotomic character.)

Let $\Lambda_{\mathbb{I}}:=\mathbb{I}[[\Gamma]]$ be the anticylotomic Iwasawa algebra over $\mathbb{I}$, and consider the $\Lambda_{\mathbb{I}}$-module

[TABLE]

where $\Lambda_{\mathbb{I}}^{*}={\rm Hom}_{\rm cont}(\Lambda_{\mathbb{I}},\mathbf{Q}_{p}/\mathbf{Z}_{p})$ is the Pontrjagin dual of $\Lambda_{\mathbb{I}}$. This is equipped with a natural $G_{K}$-action defined similarly as for the $\Lambda$-module $M=T\otimes_{\mathbf{Z}_{p}}\Lambda^{*}$ introduced in $\S\ref{sec:defs}$.

Definition 2.4**.**

The Greenberg Selmer group of $E$ over $K_{\infty}/K$ is

[TABLE]

The Greenberg Selmer group $\mathfrak{Sel}_{\rm Gr}(K_{\infty},A_{\mathbf{f}})$ for $\mathbf{f}$ over $K_{\infty}/K$, where $A_{\mathbf{f}}:=T_{\mathbf{f}}\otimes_{\mathbb{I}}\mathbb{I}^{*}$, is defined by replacing $M$ by $M_{\mathbf{f}}$ in the above definition.

Similarly as for the anticyclotomic Selmer groups in $\S\ref{sec:defs}$, for any given finite set $\Sigma$ of places $w\nmid p$ of $K$, we define $\Sigma$-imprimitive Selmer groups $\mathfrak{Sel}_{\rm Gr}^{\Sigma}(K_{\infty},E[p^{\infty}])$ and $\mathfrak{Sel}_{\rm Gr}^{\Sigma}(K_{\infty},A_{\mathbf{f}})$ by dropping the summands $H^{1}(I_{w},{M})$ and $H^{1}(I_{w},{M}_{\mathbf{f}})$, respectively, for the places $w\in\Sigma$ in the above definition. Let

[TABLE]

be the Pontrjagin dual of $\mathfrak{Sel}^{\Sigma}_{\rm Gr}(K_{\infty},E[p^{\infty}])$, and define $X_{\rm Gr}^{\Sigma}(A_{\mathbf{f}})$ similarly.

We will have use for the following comparison between the Selmer groups $\mathfrak{Sel}_{\rm Gr}(K_{\infty},E[p^{\infty}])$ and ${\rm Sel}_{\mathfrak{p}}(K_{\infty},E[p^{\infty}])$. Note that directly from the definition we have an exact sequence

[TABLE]

where $\mathcal{H}_{v}^{\rm ur}={\rm ker}\{H^{1}(K_{v},M)\rightarrow H^{1}(I_{v},M)\}$ is the set of unramified cocycles.

For a torsion $\Lambda$-module $X$, let $\lambda(X)$ (resp. $\mu(X)$) denote the $\lambda$-invariant (resp. $\mu$-invariant) of a generator of $Ch_{\Lambda}(X)$.

Proposition 2.5**.**

Assume that $X_{\rm Gr}^{\Sigma}(E[p^{\infty}])$ is $\Lambda$-torsion. Then $X_{\rm ac}^{\Sigma}(E[p^{\infty}])$ is $\Lambda$-torsion, and we have the relations

[TABLE]

and

[TABLE]

Proof.

Since $X_{\rm ac}^{\Sigma}(E[p^{\infty}])$ is a quotient of $X_{\rm Gr}^{\Sigma}(E[p^{\infty}])$, the first claim of the proposition is clear. Also, note that $X_{\rm Gr}^{\Sigma}(E[p^{\infty}])$ is $\Lambda$-torsion for some $\Sigma$ if and only if it is $\Lambda$-torsion for any finite set of primes $\Sigma$. Therefore to establish the claimed relations between Iwasawa invariants, it suffices to consider primitive Selmer groups, i.e. $\Sigma=\emptyset$.

For primes $v\nmid p$ which are split in $K$, it is easy to see that the restriction map $H^{1}(K_{v},M)\rightarrow H^{1}(I_{v},M)$ is injective (see [PW11, Rem. 3.1]), and so $\mathcal{H}_{v}^{\rm ur}$ vanishes. Since $M^{I_{\mathfrak{p}}}=\{0\}$, the term $\mathcal{H}_{\mathfrak{p}}^{\rm ur}$ also vanishes, and the exact sequence $(\ref{eq:defs})$ thus reduces to

[TABLE]

Now, a straightforward modification of the argument in [PW11, Lem. 3.4] shows that

[TABLE]

where $t_{E}(w):={\rm ord}_{p}(c_{w}(E/K))$ is the $p$-exponent of the Tamagawa number of $E$ at $w$, and $\Lambda^{*}$ is the Pontrjagin dual of $\Lambda$. In particular, $\mathcal{H}_{w}^{\rm ur}$ is $\Lambda$-torsion, with $\lambda(\mathcal{H}_{w}^{\rm ur})=0$ and $\mu(\mathcal{H}_{w}^{\rm ur})={\rm ord}_{p}(c_{w}(E/K))$. Since the rightmost arrow in $(\ref{eq:defs-})$ is surjective by [PW11, Prop. A.2], taking characteristic ideals in $(\ref{eq:defs-})$ the result follows. ∎

For the rest of this section, assume that $E$ has ordinary reduction at $p$, so that the associated newform $f\in S_{2}(\Gamma_{0}(N))$ is $p$-ordinary. Let $\mathbf{f}\in\mathbb{I}[[q]]$ be the Hida family associated with $f$, let $\wp\subseteq\mathbb{I}$ be the kernel of the arithmetic map $\phi\in\mathcal{X}_{\mathbb{I}}^{a}$ such that $\mathbf{f}_{\phi}$ is either $f$ itself (if $p\mid N$) or the ordinary $p$-stabilization of $f$ (if $p\nmid N$), and set $\widetilde{\wp}:=\wp\Lambda_{\mathbb{I}}\subseteq\Lambda_{\mathbb{I}}$. Since we assume that $\bar{\rho}_{E,p}$ is irreducible, so is $\bar{\rho}_{\mathbf{f}}$.

Theorem 2.6**.**

Let $S_{p}$ be the places of $K$ above $p$, and assume that $\Sigma\cup S_{p}$ contains all places of $K$ at which $T$ is ramified. Then there is a canonical isomorphism

[TABLE]

Proof.

This follows from a slight variation of the arguments proving [SU14, Prop. 3.7] (see also [Och06, Prop. 5.1]). Since $M\simeq M_{\mathbf{f}}[\widetilde{\wp}]$, by Pontrjagin duality it suffices to show that the canonical map

[TABLE]

is an isomorphism. Note that our assumption on $S:=\Sigma\cup S_{p}$ implies that

[TABLE]

where $M_{?}=M_{\mathbf{f}}[\widetilde{\wp}]$ or $M_{\mathbf{f}}$, $G_{K,S}$ is the Galois group of the maximal extension of $K$ unramified outside $S$, and

[TABLE]

As shown in the proof of [SU14, Prop. 3.7] (taking $A=\Lambda_{\mathbb{I}}$ and $\mathfrak{a}=\widetilde{\wp}$ in loc.cit.), we have $H^{1}(G_{K,S},M_{\mathbf{f}}[\widetilde{\wp}])=H^{1}(G_{K,S},M_{\mathbf{f}})[\widetilde{\wp}]$. On the other hand, using that $G_{K_{\mathfrak{p}}}/I_{\mathfrak{p}}$ has cohomological dimension one, we immediately see that

[TABLE]

From the long exact sequence in $I_{\mathfrak{p}}$-cohomology associated with $0\rightarrow\Lambda_{\mathbb{I}}^{*}[\widetilde{\wp}]\rightarrow\Lambda_{\mathbb{I}}^{*}\rightarrow\widetilde{\wp}\Lambda_{\mathbb{I}}^{*}\rightarrow 0$ tensored with $T_{\mathbf{f}}$, we obtain

[TABLE]

Since $H^{0}(I_{\mathfrak{p}},M_{\mathbf{f}})=\{0\}$, we thus have a commutative diagram

[TABLE]

in which the right vertical map is injective. In light of $(\ref{eq:useful})$, the result follows. ∎

3. A $p$-adic Waldspurger formula

Let $E$, $p$, and $K$ be an introduced in $\S\ref{sec:defs}$. In this section, we assume in addition that $K$ satisfies the following Heegner hypothesis relative to the square-free integer $N$:

[TABLE]

Anticyclotomic $p$-adic $L$-function

Let $f=\sum_{n=1}^{\infty}a_{n}q^{n}\in S_{2}(\Gamma_{0}(N))$ be the newform associated with $E$. Denote by $R_{0}$ the completion of the ring of integers of the maximal unramified extension of $\mathbf{Q}_{p}$, and set $\Lambda_{R_{0}}:=\Lambda\hat{\otimes}_{\mathbf{Z}_{p}}R_{0}$, where as before $\Lambda=\mathbf{Z}_{p}[[\Gamma]]$ is the anticyclotomic Iwasawa algebra.

Theorem 3.1**.**

There exists a $p$-adic $L$-function $L_{p}(f)\in\Lambda_{R_{0}}$ such that if $\hat{\phi}:\Gamma\rightarrow\mathbf{C}_{p}^{\times}$ is the $p$-adic avatar of an unramified anticyclotomic Hecke character $\phi$ with infinity type $(-n,n)$ with $n>0$, then

[TABLE]

*where $\varepsilon_{p}=p^{-1}$ if $p\nmid N$ and $\varepsilon_{p}=0$ otherwise, and $\Omega_{p}\in R_{0}^{\times}$ and $\Omega_{K}\in\mathbf{C}^{\times}$ are CM periods. *

Proof.

Let $\psi$ be an anticyclotomic Hecke character of $K$ of infinity type $(1,-1)$ and conductor prime to $p$, let $\mathscr{L}_{\mathfrak{p},\psi}(f)\in\Lambda_{R_{0}}$ be as in [CH17, Def. 3.7], and set

[TABLE]

where ${\rm Tw}_{\psi^{-1}}:\Lambda_{R_{0}}\rightarrow\Lambda_{R_{0}}$ is the $R_{0}$-linear isomorphism given by $\gamma\mapsto\psi^{-1}(\gamma)\gamma$ for $\gamma\in\Gamma$. If $p\nmid N$, the interpolation property for $L_{p}(f)$ is a reformulation of [CH17, Thm. 3.8]. Since the construction in [CH17, §3.3] readily extends to the case $p\mid N$, with the $p$-adic multiplier $e_{\mathfrak{p}}(f,\phi)$ in loc.cit. reducing to $1-a_{p}p^{-1}\phi(\mathfrak{p})$ for unramified $\phi$ (cf. [Cas17a, Thm. 2.10]), the result follows. ∎

If $\Sigma$ is any finite set of place of $K$ not lying above $p$, we define the “$\Sigma$-imprimitive” $p$-adic $L$-function $L_{p}^{\Sigma}(f)$ by

[TABLE]

where $P_{w}(X):={\rm det}(1-X\cdot{\rm Frob}_{w}|V^{I_{w}})$, $\epsilon:G_{K}\rightarrow\mathbf{Z}_{p}^{\times}$ is the $p$-adic cyclotomic character, ${\rm Frob}_{w}\in G_{K}$ is a geometric Frobenius element at $w$, and $\gamma_{w}$ is the image of ${\rm Frob}_{w}$ in $\Gamma$.

$p$-adic Waldspurger formula

We will have use for the following formula for the value at the trivial character $\mathds{1}$ of the $p$-adic $L$-function of Theorem 3.1.

Recall that $E/\mathbf{Q}$ is assumed to be semistable. From now on, we shall also assume that $E$ is an optimal quotient of the new part of $J_{0}(N)={\rm Jac}(X_{0}(N))$ in the sense of [Maz78, §2], and fix a corresponding modular parametrization

[TABLE]

sending the cusp $\infty$ to the origin of $E$. If $\omega_{E}$ a Néron differential on $E$, and $\omega_{f}=\sum a_{n}q^{n}\frac{dq}{q}$ is the one-form on $J_{0}(N)$ associated with $f$, then

[TABLE]

for some $c\in\mathbf{Z}_{(p)}^{\times}$ (see [Maz78, Cor. 4.1]).

Theorem 3.2**.**

The following equality holds up to a $p$-adic unit:

[TABLE]

where $\varepsilon_{p}=p^{-1}$ if $p\nmid N$ and $\varepsilon_{p}=0$ otherwise, and $P_{K}\in E(K)$ is a Heegner point.

Proof.

This follows from [BDP13, Thm. 5.13] and [CH17, Thm. 4.9] in the case $p\nmid N$ and [Cas17a, Thm. 2.11] in the case $p\mid N$. Indeed, in our case, the generalized Heegner cycles $\Delta$ constructed in either of these references are of the form

[TABLE]

where $H$ is the Hilbert class field of $K$, and $(A,A[\mathfrak{N}])$ is a CM elliptic curve equipped with a cyclic $N$-isogeny. Letting $F$ denote the $p$-adic completion of $H$, the aforementioned references then yield the equality

[TABLE]

By [BK90, Ex. 3.10.1], the $p$-adic Abel–Jacobi map appearing in $(\ref{eq:BDP-formula})$ is related to the formal group logarithm on $J_{0}(N)$ by the formula

[TABLE]

and by $(\ref{eq:manin})$ we have the equalities up to a $p$-adic unit:

[TABLE]

Thus, taking $P_{K}:=\sum_{\sigma\in{\rm Gal}(H/K)}\pi(\Delta^{\sigma})\in E(K)$, the result follows. ∎

4. Main Conjectures

Let $\mathbf{f}\in\mathbb{I}[[q]]$ be an ordinary $\mathbb{I}$-adic cusp eigenform of tame level $M$ as in Section 2 (so $p\nmid M$), with associated residual representation $\bar{\rho}_{\mathbf{f}}$. Letting $D_{p}\subseteq G_{\mathbf{Q}}$ be a fixed decomposition group at $p$, we say that $\bar{\rho}_{\mathbf{f}}$ is $p$-distinguished if the semisimplication of $\bar{\rho}_{\mathbf{f}}|_{D_{p}}$ is the direct sum of two distinct characters.

Let $K$ be an imaginary quadratic field in which $p=\mathfrak{p}\overline{\mathfrak{p}}$ splits, and which satisfies hypothesis (Heeg) from Section 3 relative to $M$.

For the next statement, note that for any eigenform $f$ defined over a finite extension $L/\mathbf{Q}_{p}$ with associated Galois representation $V_{f}$, we may define the Selmer group $X_{\rm Gr}^{\Sigma}(A_{f})$ as in $\S\ref{sec:control}$, replacing $T=T_{p}E$ by a fixed $G_{\mathbf{Q}}$-stable $\mathcal{O}_{L}$-lattice in $V_{f}$, and setting $A_{f}:=V_{f}/T_{f}$.

Theorem 4.1**.**

Let $f\in S_{2}(\Gamma_{0}(M))$ be a $p$-ordinary newform of level $M$, with $p\nmid M$, and let $\bar{\rho}_{f}$ be the associated residual representation. Assume that:

• •

$M$* is square-free;*

• •

$\bar{\rho}_{f}$* is ramified at every prime $q\mid M$ which is nonsplit in $K$, and there is at least one such prime;*

• •

$\bar{\rho}_{f}|_{G_{K}}$* is irreducible.*

If $\Sigma$ is any finite set of prime not lying above $p$, then $X_{\rm Gr}^{\Sigma}(A_{f})$ is $\Lambda$-torsion, and

[TABLE]

where $L_{p}^{\Sigma}(f)$ is as in $(\ref{eq:def-imp})$.

Proof.

As in the proof of [JSW15, Thm. 6.1.6], the result for an arbitrary finite set $\Sigma$ follows immediately from the case $\Sigma=\emptyset$, which is the content of [Cas17b, Thm. 3.4]. (In [Cas17b] it is assumed that $f$ has rational Fourier coefficients but the extension of the aforementioned result to the setting considered here is immediate.) ∎

Recall that $\Lambda_{\mathbb{I}}$ denotes the anticyclotomic Iwasawa algebra over $\mathbb{I}$, and set $\Lambda_{\mathbb{I},R_{0}}:=\Lambda_{\mathbb{I}}\hat{\otimes}_{\mathbf{Z}_{p}}R_{0}$. For any $\phi\in\mathcal{X}_{\mathbb{I}}^{a}$, set $\widetilde{\wp}_{\phi}:={\rm ker}(\phi)\Lambda_{\mathbb{I},R_{0}}$.

Theorem 4.2**.**

Let $\Sigma$ be a finite set of places of $K$ not above $p$. Letting $M$ be the tame level of $\mathbf{f}$, assume that:

• •

$M$* is square-free;*

• •

$\bar{\rho}_{\mathbf{f}}$* is ramified at every prime $q\mid M$ which is nonsplit in $K$, and there is at least one such prime;*

• •

$\bar{\rho}_{\mathbf{f}}|_{G_{K}}$* is irreducible;*

• •

$\bar{\rho}_{\mathbf{f}}$* is $p$-distinguished.*

Then $X_{\rm Gr}^{\Sigma}(A_{\mathbf{f}})$ is $\Lambda_{\mathbb{I}}$-torsion, and

[TABLE]

where $L_{p}^{\Sigma}(\mathbf{f})\in\Lambda_{\mathbb{I},R_{0}}$ is such that

[TABLE]

*for all $\phi\in\mathcal{X}_{\mathbb{I}}^{a}$. *

Proof.

Let $\mathscr{L}_{\mathfrak{p},\boldsymbol{\xi}}(\mathbf{f})\in\Lambda_{\mathbb{I},R_{0}}$ be the two-variable anticyclotomic $p$-adic $L$-function constructed in [Cas14, §2.6], and set

[TABLE]

where $\boldsymbol{\xi}$ is the $\mathbb{I}$-adic character constructed in loc.cit. from a Hecke character $\lambda$ of infinity type $(1,0)$ and conductor prime to $p$, and ${\rm Tw}_{\boldsymbol{\xi}^{-1}}:\Lambda_{\mathbb{I},R_{0}}\rightarrow\Lambda_{\mathbb{I},R_{0}}$ is the $R_{0}$-linear isomorphism given by $\gamma\mapsto\boldsymbol{\xi}^{-1}(\gamma)\gamma$ for $\gamma\in\Gamma$. Viewing $\lambda$ as a character on $\mathbb{A}_{K}^{\times}$, let $\lambda^{\tau}$ denote the composition of $\lambda$ with the action of complex conjugation on $\mathbb{A}_{K}^{\times}$. If the character $\psi$ appearing in the proof of Theorem 3.1 is taken to be $\lambda^{1-\tau}:=\lambda/\lambda^{\tau}$, then the proof of [Cas14, Thm. 2.11] shows that $L_{p}(\mathbf{f})$ reduces to $L_{p}(\mathbf{f}_{\phi})$ modulo $\widetilde{\wp}_{\phi}$ for all $\phi\in\mathcal{X}_{\mathbb{I}}^{a}$. Similarly as in $(\ref{eq:def-imp})$, if for any $\Sigma$ as above we set

[TABLE]

where $P_{\mathbf{f},w}(X):={\rm det}(1-X\cdot{\rm Frob}_{w}|(T_{\mathbf{f}}\otimes_{\mathbb{I}}F_{\mathbb{I}})^{I_{w}})$, with $F_{\mathbb{I}}$ the fraction field of $\mathbb{I}$, the specialization property $(\ref{eq:sp-property})$ thus follows.

Let $\phi\in\mathcal{X}_{\mathbb{I}}^{a}$ be such that $\mathbf{f}_{\phi}$ is the $p$-stabilization of a $p$-ordinary newform $f\in S_{2}(\Gamma_{0}(M))$. By Theorem 4.2, the associated $X_{\rm Gr}^{\Sigma}(A_{f})$ is $\Lambda$-torsion, and we have

[TABLE]

In particular, by Theorem 2.6 (with $A_{f}$ in place of $E[p^{\infty}]$) it follows that $X_{\rm Gr}^{\Sigma}(A_{\mathbf{f}})$ is $\Lambda_{\mathbb{I}}$-torsion. On the other hand, from [Wan14a, Thm. 1.1] we have the divisibility

[TABLE]

in $\Lambda_{\mathbb{I},R_{0}}$, where $\mathcal{L}_{p}^{\Sigma}(\mathbf{f})^{-}$ is the projection onto $\Lambda_{\mathbb{I},R_{0}}$ of the $p$-adic $L$-function constructed in [Wan14a, §7.4]. Since a straightforward extension of the calculations in [JSW15, §5.3] shows that

[TABLE]

as ideals in $\Lambda_{\mathbb{I},R_{0}}$, the result follows from an application of [SU14, Lem. 3.2] using $(\ref{eq:IMC-wt2})$, $(\ref{eq:wan-thm})$, and $(\ref{eq:compare})$. (Note that the possible powers of $p$ in [JSW15, Cor. 5.3.1] only arise when there are primes $q\mid M$ inert in $K$, but these are excluded by our hypothesis (Heeg) relative to $M$.) ∎

In order to deduce from Theorem 4.2 the anticyclotomic main conjecture for arithmetic specializations of $\mathbf{f}$ (especially in the cases where the conductor of $\mathbf{f}_{\phi}$ is divisible by $p$, which are not covered by Theorem 4.1), we will require the following technical result.

Lemma 4.3**.**

Let $X_{\rm Gr}^{\Sigma}(A_{\mathbf{f}})_{\rm null}$ be the largest pseudo-null $\Lambda_{\mathbb{I}}$-submodule of $X_{\rm Gr}^{\Sigma}(A_{\mathbf{f}})$, let $\wp\subseteq\mathbb{I}$ be a height one prime, and let $\widetilde{\wp}:=\wp\Lambda_{\mathbb{I}}$. With hypotheses as in Theorem 4.2, the quotient

[TABLE]

is a pseudo-null $\Lambda_{\mathbb{I}}/\widetilde{\wp}$-module.

Proof.

Using $(\ref{eq:useful})$ as in the proof of Theorem 2.6 and considering the obvious commutative diagram obtained by applying the map given by multiplication by $\widetilde{\wp}$, the proof of [Och06, Lem. 7.2] carries through with only small changes. (Note that the argument in loc.cit. requires knowing that $X_{\rm ac}^{\Sigma}(M_{\mathbf{f}}[\widetilde{\wp}])$ is $\Lambda_{\mathbb{I}}/\widetilde{\wp}$-torsion, but this follows immediately from Theorem 4.2 and the isomorphism of Theorem 2.6.) ∎

For the next result, let $E/\mathbf{Q}$ be an elliptic curve of square-free conductor $N$, and assume that $K$ satisfies hypothesis (Heeg) relative to $N$, and that $p=\mathfrak{p}\overline{\mathfrak{p}}$ splits in $K$.

Theorem 4.4**.**

Assume that $\bar{\rho}_{E,p}:G_{\mathbf{Q}}\rightarrow{\rm Aut}_{\mathbf{F}_{p}}(E[p])\simeq{\rm GL}_{2}(\mathbf{F}_{p})$ is irreducible and ramified at every prime $q\mid N$ which is nonsplit in $K$, and assume that there is at least one such prime. Then $Ch_{\Lambda}(X_{{\rm ac}}(E[p^{\infty}]))$ is $\Lambda$-torsion and

[TABLE]

Proof.

If $E$ has good ordinary (resp. supersingular) supersingular reduction at $p$, the result follows from [Cas17b, Thm. 3.4] (resp. [CW16, Thm. 5.1]). (Note that bBy [Ski14, Lem. 2.8.1] the hypotheses in Theorem 4.4 imply that $\bar{\rho}_{E,p}|_{G_{K}}$ is irreducible.) Since the conductor of $N$ is square-free, it remains to consider the case in which $E$ has multiplicative reduction at $p$. The associated newform $f=\sum_{n=1}^{\infty}a_{n}q^{n}\in S_{2}(\Gamma_{0}(N))$ then satisfies $a_{p}=\pm{1}$ (see e.g. [Ski16, Lem. 2.1.2]); in particular, $f$ is $p$-ordinary. Let $\mathbf{f}\in\mathbb{I}[[q]]$ be the ordinary $\mathbb{I}$-adic cusp eigenform of tame level $N_{0}:=N/p$ attached to $f$, so that $\mathbf{f}_{\phi}=f$ for some $\phi\in\mathcal{X}_{\mathbb{I}}^{a}$. Let $\wp:={\rm ker}(\phi)\subseteq\mathbb{I}$ be the associated height one prime, and set

[TABLE]

Let $\Sigma$ be a finite set of places of $K$ not dividing $p$ containing the primes above $N_{0}D$, where $D$ is the discriminant of $K$. As shown in the proof of [JSW15, Thm. 6.1.6], it suffices to show that

[TABLE]

Since $\mathbf{f}$ specializes $f$, which has weight 2 and trivial nebentypus, the residual representation $\bar{\rho}_{\mathbf{f}}\simeq\bar{\rho}_{E,p}$ is automatically $p$-distinguished (see [KLZ17, Rem. 7.2.7]). Thus our assumptions imply that the hypotheses in Theorem 4.2 are satisfied, which combined with Theorem 2.6 show that $X_{\rm Gr}^{\Sigma}(E[p^{\infty}])$ is $\Lambda$-torsion. Moreover, letting $\mathfrak{l}$ be any height one prime of $\Lambda_{\wp,R_{0}}$ and setting $\mathfrak{l}_{0}:=\mathfrak{l}\cap\Lambda_{\wp}$, by Theorem 2.6 we have

[TABLE]

On the other hand, if $\widetilde{\mathfrak{l}}\subseteq\Lambda_{\mathbb{I},R_{0}}$ maps to $\mathfrak{l}$ under the specialization map $\Lambda_{\mathbb{I},R_{0}}\rightarrow\Lambda_{\wp,R_{0}}$ and we set $\widetilde{\mathfrak{l}}_{0}:=\widetilde{\mathfrak{l}}\cap\Lambda_{\mathbb{I}}$, by Theorem 4.2 we have

[TABLE]

Since Lemma 4.3 implies the equality

[TABLE]

combining $(\ref{eq:control})$ and $(\ref{eq:2-var-IMC})$ we conclude that

[TABLE]

for every height one prime $\mathfrak{l}$ of $\Lambda_{\wp,R_{0}}$, and so

[TABLE]

Finally, since our hypothesis on $\bar{\rho}_{E,p}$ implies that $c_{w}(E/K)$ is a $p$-adic unit for every prime $w$ nonsplit in $K$ (see e.g. [PW11, Def. 3.3]), we have $Ch_{\Lambda}(X_{\rm Gr}^{\Sigma}(E[p^{\infty}]))=Ch_{\Lambda}(X_{\rm ac}^{\Sigma}(E[p^{\infty}]))$ by Proposition 2.5. Equality $(\ref{eq:imprim})$ thus reduces to $(\ref{eq:impr})$, and the proof of Theorem 4.4 follows. ∎

5. Proof of Theorem A

Let $E/\mathbf{Q}$ be a semistable elliptic curve of conductor $N$ as in the statement of Theorem A; in particular, we note that there exists a prime $q\neq p$ such that $E[p]$ is ramified at $q$. Indeed, if $p\mid N$ this follows by hypothesis, while if $p\nmid N$ the existence of such $q$ follows from Ribet’s level lowering theorem [Rib90, Thm 1.1], as explained in the first paragraph of [JSW15, §7.4].

Proof of Theorem A.

Choose an imaginary quadratic field $K=\mathbf{Q}(\sqrt{D})$ of discriminant $D<0$ such that

• •

$q$ is ramified in $K$;

• •

every prime factor $\ell\neq q$ of $N$ splits in $K$;

• •

$p$ splits in $K$;

• •

$L(E^{D},1)\neq 0$.

(Of course, when $p\mid N$ the third condition is redundant.) By Theorem 4.4 and Proposition 3.2 we have the equalities

[TABLE]

where $\varepsilon_{p}=p^{-1}$ if $p\nmid N$ and $\varepsilon_{p}=0$ otherwise, and $P_{K}\in E(K)$ is a Heegner point. Since we assume that ${\rm ord}_{s=1}L(E,s)=1$, our last hypothesis on $K$ implies that ${\rm ord}_{s=1}L(E/K,s)=1$, and so $P_{K}$ has infinite order, ${\rm rank}_{\mathbf{Z}}(E(K))=1$ and $\#\Sha(E/K)<\infty$ by the work of Gross–Zagier and Kolyvagin. This verifies the hypotheses in Theorem 2.3, which (taking $\Sigma=\emptyset$ and $P=P_{K}$) yields a formula for $\#\mathbf{Z}_{p}/f_{\rm ac}(0)$ that combined with $(\ref{eq:IMC+BDP})$ immediately leads to

[TABLE]

where $N^{+}$ is the product of the prime factors of $N$ which are split in $K$. Since $E[p]$ is ramified at $q$, we have ${\rm ord}_{p}(c_{w}(E/K))=0$ for every prime $w\mid q$ (see e.g. [Zha14, Lem. 6.3] and the discussion right after it), and since $N^{+}=N/q$ by our choice of $K$, we see that $(\ref{eq:+control})$ can be rewritten as

[TABLE]

On the other hand, as explained in [JSW15, p. 47] the Gross–Zagier formula [GZ86], [YZZ13] (as refined in [CST14]) can be paraphrased as the equality

[TABLE]

up to a $p$-adic unit,111This uses a period relation coming from [SZ14, Lem. 9.6], which assumes that $(D,pN)=1$, but the same argument applies replacing $D$ by $D/(D,pN)$ in the last paragraph of the proof of their result. which combined with (5.3) and the immediate relation

[TABLE]

(see [SZ14, Cor. 7.2]) leads to the equality

[TABLE]

Finally, since $L(E^{D},1)\neq 0$, by the known $p$-part of the Birch and Swinnerton-Dyer formula for $E^{D}$ (as recalled in [JSW15, Thm. 7.2.1]) we arrive at

[TABLE]

concluding the proof of the theorem. ∎