Non-generic components of the Emerton-Gee stack for $\mathrm{GL}_2$

TL;DR

This paper analyzes specific non-generic irreducible components of the Emerton-Gee stack for GL2 over unramified extensions of Qp, detailing their geometric properties and implications for the p-adic Langlands program.

Contribution

It precisely characterizes the smoothness, normality, and Gorenstein properties of these components and explores their resolutions, advancing understanding of the stack's structure.

Findings

Identifies which components are smooth or normal.

Shows that normalizations admit smooth covers by resolution-rational schemes.

Determines the singular loci of the components.

Abstract

Let $K$ be a finite unramified extension of $\mathbb{Q}_p$ with $p > 3$. We study the extremely non--generic irreducible components in the reduced part of the Emerton--Gee stack for $\mathrm{GL}_2$. We show precisely which irreducible components are smooth, which are normal, and which have Gorenstein normalizations. We show that the normalizations of the irreducible components admit smooth--local covers by resolution--rational schemes. We also determine the singular loci on the components, and use our results to update expectations about the conjectural categorical $p$--adic Langlands correspondence.