Finiteness and cofiniteness of fine Selmer groups over function fields

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.

Accepted for publication in the Proceedings of the American Mathematical Society. No changes in the text from v4. 14 pages