The geometry of the unipotent component of the moduli space of Weil-Deligne representations

TL;DR

This paper investigates the geometric structure of the unipotent part of the moduli space of Weil-Deligne representations, identifying smooth components and linking these to automorphic forms and deformation rings.

Contribution

It characterizes smooth irreducible components of the moduli space of unipotent Weil-Deligne representations and relates these to automorphic forms and deformation theory.

Findings

Identified which components of the moduli space are smooth.

Established a connection between smooth components and automorphic forms.

Proved that certain automorphic form spaces are locally free modules over deformation rings.

Abstract

In this paper, we study the moduli space of unipotent Weil-Deligne representations valued in a split reductive group $G$ and characterise which irreducible components are smooth. We apply the smoothness results proved to show that a certain space of ordinary automorphic forms is a locally generically free module over the corresponding global deformation ring.