Congruences of modular forms and modularity of Tate-Shafarevich classes

2024/02/11 by Tamiozzo, Matteo
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2402.07317

Abstract

We prove, under suitable assumptions, that p-torsion Tate-Shafarevich classes for elliptic curves over the rationals are visible in quotients of Jacobians of modular curves, as predicted by a conjecture of Jetchev-Stein. The key ingredient is the non-triviality of the Bertolini-Darmon bipartite Kolyvagin system, which implies that suitable cohomology classes of the system form a basis of the Selmer group modulo p.

Related

• On the structure of Selmer and Shafarevich-Tate groups of higher weight modular forms
• Explicit realization of elements of the Tate-Shafarevich group constructed from Kolyvagin classes
• Kolyvagin's conjecture for modular forms
• Rankin-Selberg Euler systems and p-adic interpolation
• Kolyvagin's Conjecture and patched Euler systems in anticyclotomic Iwasawa theory