Iwasawa theory of fine Selmer groups associated to Drinfeld modules

Abstract

Let $$ be a prime power and $$ be the rational function field over $$, the field with $$ elements. Let $$ be a Drinfeld module over $$ and $$ be a nonzero prime ideal of $$. Over the constant $$-extension of $$, we introduce the fine Selmer group associated to the $$-primary torsion of $$. We show that it is a cofinitely generated module over $$. This proves an analogue of Iwasawa's $$ conjecture in this setting, and provides context for the further study of the objects that have been introduced in this article.