The $\mu -$invariant of fine Selmer groups associated to general Drinfeld modules

Abstract

Let $F$ be a global function field over the finite field ${\mathbb{F}}_q$ where $q$ is a prime power and $A$ be the ring of elements in $F$ regular outside $\infty$. Let $\varphi$ be an arbitrary Drinfeld module over $F$ For a fixed non-zero prime ideal $\mathfrak{p}$ of $A$, we show that on the constant ${\mathbb{Z}}_{\text{p}}-$extension $\mathfrak{F}$ of $F$, the Pontryagin dual of the fine Selmer group associated to the $\mathfrak{p}-$primary torsion of $\varphi$ over $\mathfrak{F}$ is a finitely generated Iwasawa module such that its Iwasawa $\mu -$invariant vanishes. This provides a generalization of the results given in arXiv:2311.06499.