Notes on the fine Selmer groups

TL;DR

This paper investigates the properties of fine Selmer groups associated with Galois modules over Noetherian rings, focusing on their behavior under residual representations and field extensions, especially in cyclotomic towers.

Contribution

It establishes a connection between the variation of fine Selmer groups and class groups in cyclotomic extensions, and discusses pseudo-nullity examples.

Findings

Variation of fine Selmer groups relates to class group variation in cyclotomic towers.

Shows pseudo-nullity of certain fine Selmer groups.

Provides examples illustrating these properties.

Abstract

In this paper, we study the fine Selmer groups attached to a Galois module defined over a commutative complete Noetherian ring with finite residue field of characteristic p. Namely, we are interested in its properties upon taking residual representation and within field extensions. In particular, we will show that the variation of the fine Selmer group in a cyclotomic $\Zp$-extension is intimately related to the variation of the class groups in the cyclotomic tower. We also discuss some examples of pseudo-nullity of fine Selmer groups.