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arXiv · 2402.07317

Congruences of modular forms and modularity of Tate-Shafarevich classes

Matteo Tamiozzo
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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 $p$.

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Matteo Tamiozzo. 2024-02-11. Congruences of modular forms and modularity of Tate-Shafarevich classes. https://arxiv.org/abs/2402.07317

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