Kolyvagin's conjecture for modular forms at non-ordinary primes
Enrico Da Ronche
TL;DR
The paper develops a non-ordinary analogue of Kolyvagin's conjecture for modular forms by constructing Kolyvagin classes from Heegner-type cycles on Shimura curves and relating them to theta elements via a reciprocity law. The authors adapt signed local conditions in place of Greenberg conditions, use level-raising to transfer to a non-ordinary setting, and prove the rank-one case through a control-splitting-Wan argument, ultimately extending to higher rank via triangulation and parity. A key input is Wan's $p$-part Tamagawa conjecture for non-ordinary modular forms, which connects special $L$-values, Tamagawa data, and Selmer groups to deduce nontriviality of the Kolyvagin classes. The results yield the $p$-part of the Tamagawa number conjecture for modular motives of analytic rank zero and one, linking $L$-values, regulator data, and Selmer structures in a non-ordinary context with explicit valuation identities. Overall, the work broadens the reach of Kolyvagin-type Euler systems to non-ordinary primes and higher weights, offering new tools for the arithmetic of modular forms and their associated motives.
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.