On the p-converse of the Kolyvagin-Gross-Zagier theorem

Let $A\mathrm{/}\mathbb{Q}$ be an elliptic curve having split multiplicative reduction at an odd prime $p$. Under some mild technical assumptions, we prove the statement: $$ran{k}_{\mathbb{Z}}A(\mathbb{Q})=1and\mathrm{\#}(\underset{‾}{III}(A\mathrm{/}\mathbb{Q})[p^{\mathrm{\infty }}])<\mathrm{\infty }⟹{\mathrm{o}\mathrm{r}\mathrm{d}}_{s=1}L(A\mathrm{/}\mathbb{Q},s)=1,$$ thus providing a “$p$-converse” to a celebrated theorem of Kolyvagin-Gross-Zagier.