A $p$-Converse Theorem for Real Quadratic Fields

In this talk, I will discuss a $p$-converse theorem for elliptic curves over real quadratic fields. More precisely, let $p \gt 5$ be a rational prime. Let $F$ be a real quadratic field such that $p$ is inert in $F$ . Suppose $E$ is an elliptic curve defined over $F$ having split multiplicative reduction at the unique prime of $F$ lying above $p$. Under suitable technical hypotheses, we prove that if the Mordell-Weil group $E(F )$ has rank one and the p-primary part of the Tate--Shafarevich group is finite, then the complex $L$-function $L(E/F, s)$ vanishes to order exactly one at $s = 1$.