Fine Selmer groups of congruent Galois representations

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 π-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.