Finiteness and cofiniteness of fine Selmer groups over function fields

TL;DR

This paper proves finiteness and cofiniteness properties of fine Selmer groups of abelian varieties over certain function field extensions, confirming conjectures and extending results to ramified p-adic Lie extensions.

Contribution

It establishes the finiteness and cofiniteness of fine Selmer groups over unramified and ramified extensions, confirming a conjecture of Coates--Sujatha in the function field setting.

Findings

Dual fine Selmer group is finitely generated over 

Fine Selmer group is finite or zero depending on torsion properties

Results extend to ramified p-adic Lie extensions

Abstract

We prove that the dual fine Selmer group of an abelian variety over the unramified $\mathbb{Z}_{p}$-extension of a function field is finitely generated over $\mathbb{Z}_{p}$. This is a function field version of a conjecture of Coates--Sujatha. We further prove that the fine Selmer group is finite (respectively zero) if the separable $p$-primary torsion of the abelian variety is finite (respectively zero). These results are then generalized to certain ramified $p$-adic Lie extensions.