Let $q$ be a prime power and $F=\mathbb{ F}_q(T)$ be the rational function field over $\mathbb{ F}_q$, the field with $q$ elements. Let $\phi$ be a Drinfeld module over $F$ and $\mathfrak{ p}$ be a non-zero prime ideal of $A:=\mathbb{ F}_q[T]$. Over the constant $\mathbb{ Z}_p$-extension of $F$, we introduce the fine Selmer group associated to the $\mathfrak{ p}$-primary torsion of $\phi$. We show that it is a cofinitely generated module over $A_{ \mathfrak{ p}}$. This proves an analogue of Iwasawa's $\mu=0$ conjecture in this setting, and provides context for the further study of the objects that have been introduced in this article.
Categories: math.NT math.AG