On the pseudo-nullity of the dual fine Selmer groups

Abstract

In this paper, we will study the pseudo-nullity of the dual fine Selmer group and its related question. We investigate certain situations, where one can deduce the pseudo-nullity of the dual fine Selmer group of a general Galois module over an admissible p-adic Lie extension F∞ from the knowledge of the pseudo-nullity of the Galois group of the maximal abelian unramified pro-p extension of F∞ at which every prime of F∞ above p splits completely. In particular, this gives us a way to construct examples of the pseudo-nullity of the dual fine Selmer group of a Galois module that is unramified outside p. We will illustrate our results with many examples.