Fine Selmer groups of congruent Galois representations

TL;DR

This paper investigates the properties of fine Selmer groups associated with congruent Galois representations over p-adic Lie extensions, establishing conditions under which their pseudo-nullity and structural features are preserved.

Contribution

It introduces new results linking the pseudo-nullity and structural aspects of fine Selmer groups for congruent Galois representations over p-adic Lie extensions.

Findings

Pseudo-nullity of one fine Selmer group implies the same for the other under congruence.

Comparison of $ ext{pi}$-primary submodules of dual fine Selmer groups.

Application to the structure of Galois groups of maximal unramified pro-$p$ extensions.

Abstract

In this paper, we study the fine Selmer groups of two congruent Galois representations over an admissible $p$-adic Lie extension. We show that under appropriate congruence conditions, if the dual fine Selmer group of one is pseudo-null, so is the other. Our results also compare the $\pi$-primary submodules of the two dual fine Selmer groups. We then apply our results to compare the structure of Galois group of the maximal abelian unramified pro-$p$ extension of an admissible $p$-adic Lie extension and the structure of the dual fine Selmer group over the said admissible $p$-adic Lie extension.