A $p$-Converse theorem for Real Quadratic Fields

TL;DR

This work proves a $p$-adic converse-type result for real quadratic fields: if $p>5$ is inert in a real quadratic field $F$, $E/F$ has split multiplicative reduction at the prime above $p$, $ ext{rank}_{Z}E(F)=1$, and $ ext{Sha}(E/F)_{p^ fty}$ is finite, then $ord_{s=1} L(E/F,s)=1$. The authors extend Rodolfo’s approach over $Q$ by leveraging Hida theory to construct a central-critical four-variable $p$-adic $L$-function $L_p^{cc}(f_ abla/K,k)$, Wan’s main conjecture in a three-/four-variable Iwasawa framework, Nekovář’s Selmer complexes, and Mok’s Heegner-point results to relate analytic order of vanishing to Selmer-length and Mordell–Weil data. A pivotal novelty is the four-variable central-critical construction and a $p$-adic weight pairing on Nekovář’s Selmer complexes, which together enable a $p$-converse result over a real quadratic base by a careful choice of the auxiliary imaginary quadratic field $K$. As a corollary, the paper also obtains a $p$-converse theorem for elliptic curves over $Q$, thereby extending the Gross–Zagier–Kolyvagin paradigm beyond $Q$ and providing new tools for connecting $L$-values, Selmer groups, and Mordell–Weil groups in the setting of totally real fields.

Abstract

Let $E$ be an elliptic curve defined over a real quadratic field $F$. Let $p > 5$ be a rational prime that is inert in $F$ and assume that $E$ has split multiplicative reduction at the prime $\mathfrak{p}$ of $F$ dividing $p$. Let $\underline{III}(E/F)$ denote the Tate-Shafarevich group of $E$ over $F$ and $ L(E/F,s) $ be the Hasse-Weil complex $L$-function of $E$ over $F$. Under some technical assumptions, we show that when $rank_{\mathbb{Z}} \hspace{0.01mm} \hspace{1mm} E(F) = 1$ and $\#\Big(\underline{III}(E/F)_ {p^\infty}\Big) < \infty$, then $ord_{s=1} \ L(E/F,s) = 1$. Further, we give an applictaion to a $p$-converse theorem over $\mathbb{Q}$.

Theorems & Definitions (54)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 2.1
• proof
• Theorem 2.2
• Definition 1