A $p$-Converse theorem for Real Quadratic Fields

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 $\underset{‾}{III}(E\mathrm{/}F)$ denote the Tate-Shafarevich group of $E$ over $F$ and $L(E\mathrm{/}F,s)$ 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 $\mathrm{\#}(\underset{‾}{III}(E\mathrm{/}F{)}_{p^{\mathrm{\infty }}})<\mathrm{\infty }$, then $or{d}_{s=1}L(E\mathrm{/}F,s)=1$. Further, we give an applictaion to a $p$-converse theorem over $\mathbb{Q}$.