$\mathfrak{M}_H(G)$-property and congruence of Galois representations

Meng Fai Lim

Published 2016-02-08Version 1

In this paper, we study the Selmer groups of two congruent Galois representations over an admissible $p$-adic Lie extension. We will show that under appropriate congruence condition, if the dual Selmer group of one satisfies the $\mathfrak{M}_H(G)$-property, so will the other In the event, the $\mathfrak{M}_H(G)$-property holds and assuming certain further hypothesis on the decomposition of primes in the $p$-adic Lie extension, we compare the ranks of the $\pi$-free quotient of the two dual Selmer groups. We then apply our results to study the variation of the dual Selmer groups of specialization of a big Galois representation. We emphasis that our results \textit{do not} assume the vanishing of the $\mu$-invariant.