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

Rodolfo Venerucci

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arXiv · 1407.1816

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

Rodolfo Venerucci

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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.

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Rodolfo Venerucci. 2014-07-07. On the p-converse of the Kolyvagin-Gross-Zagier theorem. https://arxiv.org/abs/1407.1816

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