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

Rodolfo Venerucci

doi:10.4171/cmh/390

JournalscmhVol. 91, No. 3pp. 397–444

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

Rodolfo Venerucci
Universität Duisburg-Essen, Germany

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Abstract

Let $A / Q$ be an elliptic curve having split multiplicative reduction at an odd prime $p$ . Under some mild technical assumptions, we prove the statement:

rank_{Z} A (Q) = 1 and # (Ш (A / Q)_{p^{\infty}}) < \infty ⟹ ord_{s = 1} L (A / Q, s) = 1,

thus providing a ‘ $p$ -converse’ to a celebrated theorem of Kolyvagin–Gross–Zagier.

Cite this article

Rodolfo Venerucci, On the $p$ -converse of the Kolyvagin–Gross–Zagier theorem. Comment. Math. Helv. 91 (2016), no. 3, pp. 397–444

DOI 10.4171/CMH/390