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 $\mathbb{ Q}$.
Categories: math.NT