Smoothness of components of the Emerton-Gee stack for $\text{GL}_2$

TL;DR

This paper studies the geometric structure of the Emerton-Gee stack for $ ext{GL}_2$, showing that many of its components are smooth quotients and using this to compute global sections, advancing understanding of moduli of Galois representations.

Contribution

It demonstrates that most irreducible components of the Emerton-Gee stack are quotients of smooth affine schemes, including non-generic ones, and computes their global sections.

Findings

Most irreducible components are smooth quotients.

Non-generic components are included in the analysis.

Global sections of these components are explicitly computed.

Abstract

Let $K$ be a finite unramified extension of $\mathbb{Q}_p$, where $p>2$. [CEGS22b] and [EG23] construct a moduli stack of two dimensional mod $p$ representations of the absolute Galois group of $K$. We show that most irreducible components of this stack (including several non-generic components) are isomorphic to quotients of smooth affine schemes. We also use this quotient presentation to compute global sections on these components.