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

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.

51 pages. Fixed minor typos, and added a description of the Steinberg component