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

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.

Final version. To appear in Transactions of the AMS