Potentially semi-stable deformation rings for representations with values in $PGL_n$

TL;DR

This paper investigates the structure of potentially semi-stable deformation rings for Galois representations valued in PGL_n, relating them to GL_n cases and proposing an analogue of the Breuil-Mézard conjecture.

Contribution

It establishes a comparison between PGL_n and GL_n deformation rings and extends the Breuil-Mézard conjecture to the PGL_n setting.

Findings

Deformation rings for PGL_n can be understood via those for GL_n.

An analogue of the Breuil-Mézard conjecture is formulated for PGL_n.

The PGL_n case reduces to the GL_n case in the deformation theory context.

Abstract

We study the potentially semi-stable deformation rings for Galois representations taking their values in $PGL_n$, by comparing them to the deformation rings for $GL_n$. As an application, we state an analogue of the Breuil-M\'ezard conjecture, and we show that the case of $PGL_n$ follows from the case of $GL_n$.