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arXiv · 2409.01360

Exceptional zeros for Heegner points and $p$-converse to the theorem of Gross-Zagier and Kolyvagin

Francesc Castella
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Abstract

We prove a $p$-converse to the theorem of Gross-Zagier and Kolyvagin for elliptic curves $E/\mathbf{Q}$ at primes $p>3$ of multiplicative reduction. Two key ingredients in the argument are an extension to this setting of a $p$-adic formula of Bertolini-Darmon-Prasanna obtained in our earlier work, and an exceptional zero formula for Heegner points. By independent approaches different from ours, a similar $p$-converse theorem was obtained by Skinner--Zhang under additional ramification hypotheses on $E[p]$, and by Venerucci assuming finiteness of the $p$-primary part of the Tate-Shafarevich group.

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Francesc Castella. 2024-09-02. Exceptional zeros for Heegner points and $p$-converse to the theorem of Gross-Zagier and Kolyvagin. https://arxiv.org/abs/2409.01360

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