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Congruences of modular forms and modularity of Tate-Shafarevich classes

Matteo Tamiozzo

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 $p$.

Congruences of modular forms and modularity of Tate-Shafarevich classes

Abstract

We prove, under suitable assumptions, that -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 .
Paper Structure (34 sections, 10 theorems, 16 equations)

This paper contains 34 sections, 10 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Bertolini--Darmon's bipartite Kolyvagin system
  3. A basis of $\mathrm{Sel}(K, E[p])$
  4. Modularity of Tate--Shafarevich classes
  5. Structure of the text
  6. Generating the mod $p$ Selmer group
  7. Bipartite Kolyvagin systems modulo $p$
  8. Setting
  9. Selmer groups
  10. Bipartite Kolyvagin systems for $E[p]$
  11. A basis of $\mathrm{Sel}(K, E[p])$
  12. Bertolini--Darmon's bipartite Kolyvagin system modulo $p$
  13. Level raising
  14. Assumptions
  15. Construction of $Z^{BD}$
  16. ...and 19 more sections

Key Result

Theorem 1.3.1

Let $E/\mathbf{Q}$ be an elliptic curve of squarefree conductor $N$, without complex multiplication. Let $p>3$ be a prime of good ordinary reduction. Assume that $\bar{\rho}: \mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{Q})\rightarrow \mathrm{Aut}_{\mathbf{F}_p}(E[p])$ is surjective and ramified at every p

Theorems & Definitions (22)

  • Theorem 1.3.1
  • Lemma 2.1.4
  • proof
  • Proposition 2.1.5
  • proof
  • Proposition 2.1.6
  • proof
  • Proposition 2.2.1
  • proof
  • Theorem 2.3.6
  • ...and 12 more