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

TL;DR

This paper refines the Iwasawa main conjectures for elliptic curves over anticyclotomic extensions, describing higher Fitting ideals of Selmer and Shafarevich-Tate groups using bipartite Euler systems, leading to new structural insights.

Contribution

It provides a detailed description of higher Fitting ideals of Selmer and Shafarevich-Tate groups in anticyclotomic extensions, advancing understanding of their structure.

Findings

Explicit formulas for higher Fitting ideals of Selmer groups.

New structural results on Shafarevich-Tate groups.

Connections established between Fitting ideals and bipartite Euler systems.

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.