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

Abstract

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