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

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.