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arXiv · 2312.09301

Non-vanishing of Kolyvagin systems and Iwasawa theory

Ashay Burungale·Francesc Castella·Giada Grossi·Christopher Skinner
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Abstract

Let $E/\mathbb{Q}$ be an elliptic curve and $p$ an odd prime. In 1991 Kolyvagin conjectured that the system of cohomology classes for torsion quotients of the $p$-adic Tate module of $E$ derived from Heegner points over ring class fields of a suitable imaginary quadratic field $K$ (i.e., the Heegner point Kolyvagin system of $E/K$) is non-trivial. In this paper we prove Kolyvagin's conjecture when $p$ is a prime of good ordinary reduction for $E$ that splits in $K$. In particular, our results cover many cases where $p$ is an Eisenstein prime for $E$, complementing Wei Zhang's earlier results on the conjecture by a different approach. Our methods also yield a proof of a refinement of Kolyvagin's conjecture expressing the divisibility index of the Heegner point Kolyvagin system in terms of the Tamagawa numbers of $E$, as conjectured by Wei Zhang in 2014, as well as proofs of analogous results for the Kolyvagin system obtained from Kato's Euler system.

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Ashay Burungale, Francesc Castella, Giada Grossi, Christopher Skinner. 2023-12-14. Non-vanishing of Kolyvagin systems and Iwasawa theory. https://arxiv.org/abs/2312.09301

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