$N-$Bundles for $N$ an extension of a finite group by an abelian group

TL;DR

This paper studies the structure of moduli spaces of principal bundles for a group extension, describing fibers as torsors over abelian varieties and exploring properties of Mumford groups.

Contribution

It provides a detailed description of the fibers of the quotient map in the moduli space of principal bundles for an extension of a finite group by an abelian group, including new insights into Mumford groups.

Findings

Fibers of the quotient map are torsors over abelian varieties.

Explicit description of the moduli space structure for group extensions.

Results on properties of Mumford groups.

Abstract

Let $W$ be a finite group and $T$ be an abelian group. Consider an extension $0 \ra T \ra N \ra W \ra 0$. For a smooth projective curve $X$, we give a precise description of the fiber of the quotient by $T$ map $q_T: \cM_X(N) \ra \cM_X(W)$ as a torsor over an abelian variety. We also prove a result on Mumford groups.