Finiteness and cofiniteness of fine Selmer groups over function fields

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.