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

2025/03/13 by Da Ronche, Enrico
#11F11 #14C15 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2503.09955

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.

Related

• Kolyvagin's conjecture for modular forms
• The refined Tamagawa number conjectures for GL₂
• On the Tamagawa number conjecture for newforms at Eisenstein primes
• The Tamagawa number conjecture and Kolyvagin's conjecture for motives of modular forms
• Refined applications of Kato's Euler systems for modular forms