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

TL;DR

This paper proves that for a broad class of Drinfeld modules over function fields, the associated fine Selmer groups have a vanishing Iwasawa $$-invariant, extending previous specific cases.

Contribution

It generalizes the vanishing of the Iwasawa $$-invariant for fine Selmer groups to arbitrary Drinfeld modules over function fields.

Findings

The Pontryagin dual of the fine Selmer group is finitely generated over the Iwasawa algebra.

The Iwasawa $$-invariant of this dual group is zero.

Results extend previous specific cases to a more general setting.

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 $\phi$ be an arbitrary Drinfeld module over $F$ For a fixed non-zero prime ideal $\mathfrak{p}$ of $A$, we show that on the constant $\mathbb{Z}_{\textit{p}}-$extension $\mathfrak{F}$ of $F$, the Pontryagin dual of the fine Selmer group associated to the $\mathfrak{p}-$primary torsion of $\phi$ 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.