Kolyvagin's conjecture for modular forms at non-ordinary primes

Enrico Da Ronche

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arXiv · 2503.09955

Kolyvagin's conjecture for modular forms at non-ordinary primes

Enrico Da Ronche

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Abstract

In this article we prove a version of Kolyvagin's conjecture for modular forms at non-ordinary primes. In particular, we generalize the work of Wang on a converse to a higher weight Gross-Zagier-Kolyvagin theorem in order to prove the conjecture under the hypothesis that some Selmer group has rank one. The main ingredients that we use in non-ordinary setting are the signed Selmer groups introduced by Lei, Loeffler and Zerbes. We will also use a result of Wan, i.e., the $p$-part of the Tamagawa number conjecture for non-ordinary modular forms with analytic rank zero. Starting from the rank one case we will show how to prove the full version of the conjecture.

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BibTeX RIS

Enrico Da Ronche. 2025-03-13. Kolyvagin's conjecture for modular forms at non-ordinary primes. https://arxiv.org/abs/2503.09955

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