On the pseudo-nullity of the dual fine Selmer groups

Abstract: In this paper, we will study the pseudo-nullity of the fine Selmer group and its related question. Namely, we investigate certain situations, where one can deduce the pseudo-nullity of the dual fine Selmer group of a general Galois module over a admissible $p$-adic Lie extension $F_{\infty}$ from the knowledge that pseudo-nullity of the Galois group of the maximal abelian unramified pro-$p$ extension of $F_{\infty}$ at which every prime of $F_{\infty}$ 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.