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Higher Fitting ideals and the structure of anticyclotomic Shafarevich-Tate groups

Enrico Da Ronche, Matteo Longo, Stefano Vigni

Abstract

Let $p$ be a prime number. We investigate a refined version of the Iwasawa main conjectures for rational elliptic curves (and more general Galois representations) over anticyclotomic $\mathbb Z_p$-extensions of imaginary quadratic fields, both in the definite and in the indefinite settings. In order to do this, we describe (under mild arithmetic assumptions) all the higher Fitting ideals of Pontryagin duals of Selmer and Shafarevich-Tate groups over anticyclotomic $\mathbb Z_p$-extensions in terms of the bipartite Euler systems introduced by Bertolini and Darmon. As an application of our work on Fitting ideals, we offer new results on the structure of (Pontryagin duals of) anticyclotomic Selmer and Shafarevich-Tate groups of elliptic curves.

Higher Fitting ideals and the structure of anticyclotomic Shafarevich-Tate groups

Abstract

Let be a prime number. We investigate a refined version of the Iwasawa main conjectures for rational elliptic curves (and more general Galois representations) over anticyclotomic -extensions of imaginary quadratic fields, both in the definite and in the indefinite settings. In order to do this, we describe (under mild arithmetic assumptions) all the higher Fitting ideals of Pontryagin duals of Selmer and Shafarevich-Tate groups over anticyclotomic -extensions in terms of the bipartite Euler systems introduced by Bertolini and Darmon. As an application of our work on Fitting ideals, we offer new results on the structure of (Pontryagin duals of) anticyclotomic Selmer and Shafarevich-Tate groups of elliptic curves.
Paper Structure (18 sections, 45 theorems, 240 equations)

This paper contains 18 sections, 45 theorems, 240 equations.

Table of Contents

  1. Introduction
  2. Background on Fitting ideals
  3. Definition and basic properties
  4. Fitting ideals of modules over a DVR
  5. Fitting ideals of $\Lambda$-modules
  6. Bipartite Euler systems
  7. Bipartite Euler systems
  8. Bipartite Euler systems over a principal artinian local ring
  9. Bipartite Euler systems over a DVR
  10. Bipartite Euler systems over $\Lambda$
  11. Higher Fitting ideals of $X$
  12. Construction of the ideals $\mathfrak{C}_i$
  13. Main result
  14. Elliptic curves
  15. Selmer groups
  16. ...and 3 more sections

Key Result

Theorem 1

Let $i\geq0$ be an integer. In the definite case, there is an equivalence

Theorems & Definitions (103)

  • Theorem 1
  • Theorem 2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • ...and 93 more