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

TL;DR

This paper proves a p-converse to the Kolyvagin-Gross-Zagier theorem for elliptic curves with split multiplicative reduction at an odd prime p, linking rank and L-function order under certain conditions.

Contribution

It establishes a p-converse result connecting the rank and the order of vanishing of the L-function for specific elliptic curves, under mild assumptions.

Findings

Proves the p-converse implication for elliptic curves with split multiplicative reduction.

Links the rank of the elliptic curve to the order of vanishing of its L-function.

Provides conditions under which the rank-one case implies L-function order one.

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: $$rank_{\mathbb{Z}}A(\mathbb{Q})=1 \ \ and\ \ \ \#\Big(\underline{III}(A/\mathbb{Q})[p^\infty]\Big)<\infty\ \ \implies\ \ \mathrm{ord}_{s=1}L(A/\mathbb{Q},s)=1,$$ thus providing a "$p$-converse" to a celebrated theorem of Kolyvagin-Gross-Zagier.