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.
Kolyvagin's conjecture for modular forms at non-ordinary primes
Enrico Da Ronche
2025-01-01
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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