Euler systems for Galois deformations and the pseudo-isomorphism class of the dual of fine Selmer groups

TL;DR

This paper investigates the structure of the dual fine Selmer group in the context of $p$-adic Galois deformations, using Euler systems to approximate Fitting ideals and determine pseudo-isomorphism classes.

Contribution

It introduces a method to construct ideals from Euler systems that approximate Fitting ideals, helping classify dual fine Selmer groups for certain Galois deformations.

Findings

Ideals $C_i(c)$ approximate higher Fitting ideals of $X$.

Euler systems from Beilinson-Kato elements determine the pseudo-isomorphism class.

Results apply to ordinary and nearly ordinary Hida deformations.

Abstract

In this article, we study the pseudo-isomorphism class of the dual fine Selmer group $X$ attached to a $p$-adic Galois deformation whose deformation ring $\Lambda$ is isomorphic to the ring of formal power series. By using the "Kolyvagin system" arising from a given Euler system $c$, we shall construct ideals $C_i(c)$ of $\Lambda$, and prove that the ideals $C_i(c)$ approximate the higher Fitting ideals of $X$ under suitable hypothesis. In particular, we shall prove that the ideals $C_i(c)$ arising from the Euler system of Beilinson-Kato elements determine the pseudo-isomorphism classes of the dual fine Selmer groups attached to ordinary and nearly ordinary Hida deformations satisfying certain conditions.