Iwasawa theory of fine Selmer groups associated to Drinfeld modules

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