Fine Selmer groups of congruent Galois representations

Meng Fai Lim

Published 2016-03-29Version 1

In this paper, we study the fine Selmer groups of two congruent Galois representations over an admissible $p$-adic Lie extension. We will show that under appropriate congruence conditions, if the dual fine Selmer group of one is pseudo-null, so will the other. In fact, 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. We also apply our results to compare the structure of the dual fine Selmer groups of various specializations of a big Galois representation.