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arXiv · 2504.21799

A $p$-Converse theorem for Real Quadratic Fields

Muskan Bansal·Somnath Jha·Aprameyo Pal·Guhan Venkat
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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 application to a $p$-converse theorem over $\mathbb{Q}$.

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Muskan Bansal, Somnath Jha, Aprameyo Pal, Guhan Venkat. 2025-04-30. A $p$-Converse theorem for Real Quadratic Fields. https://arxiv.org/abs/2504.21799

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