Finiteness and cofiniteness of fine Selmer groups over function fields

Key Points

• The fine Selmer group is finitely generated over the p-adic integers when considering the dual fine Selmer group.
• When the separable p-primary torsion of the abelian variety is finite, the fine Selmer group is shown to be finite as well.
• In contrast, if the separable p-primary torsion of the abelian variety is zero, the fine Selmer group is determined to be zero too, reflecting its structure accurately under these conditions in function fields for any case reported here related to the conjecture. Including these aspects leads toward a greater understanding of the conjectures surrounding fine Selmer groups and function fields, yet requires more comprehensive analysis in broader contexts of number theory.