A $p$-Converse theorem for Real Quadratic 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 $ran{k}_{\mathbb{Z}}E(F)=1$ and $\#(\underline{III}(E/F{)}_{p^{\infty }})<\infty$, then $or{d}_{s=1}L(E/F,s)=1$. Further, we give an application to a $p$-converse theorem over $\mathbb{Q}$.

To appear in Journal of the London Mathematical Society