On the $\mu$ equals zero conjecture for the fine Selmer group in Iwasawa theory

TL;DR

This paper investigates the conditions under which the $$-invariant vanishes in the Iwasawa theory of the fine Selmer group, linking it to properties of Galois deformation rings.

Contribution

It establishes a connection between the vanishing of the $$-invariant and natural properties of Galois deformation rings, providing new cases where the $=0$ conjecture holds.

Findings

Shows $f=0$ in some cases from deformation ring properties

Outlines conditions for $f=0$ for various Galois representations

Links the vanishing of $f$ to deformation theory properties

Abstract

We study the Iwasawa theory of the fine Selmer group associated to certain Galois representations. The vanishing of the $\mu$-invariant is shown to follow in some cases from a natural property satisfied by Galois deformation rings. We outline conditions under which the $\mu=0$ conjecture is shown to hold for various Galois representations of interest.