Codimension one intersections between components of the Emerton-Gee stack for $\mathrm{GL}_2$

TL;DR

This paper investigates the structure of the Emerton-Gee stack for GL_2 over p-adic fields, focusing on the intersections of its irreducible components and their relation to Serre weights and extension groups.

Contribution

It provides representation-theoretic criteria for codimension one intersections of irreducible components of the Emerton-Gee stack, linking extensions of Serre weights to geometric intersections.

Findings

Non-trivial extensions of Serre weights imply codimension one intersections.

Sufficiently generic Serre weights ensure the converse.

The number of top-dimensional components in intersections relates to extension groups.

Abstract

Let $p$ be a fixed odd prime, and let $K$ be a finite extension of $\mathbb{Q}_p$ with ring of integers $\mathcal{O}_K$. The Emerton-Gee stack for $\mathrm{GL}_2$ is a stack of $(\varphi, \Gamma)$-modules. The stack, denoted $\mathcal{X}_2$, can be interpreted as a moduli stack of representations of the absolute Galois group of $K$ with $p$-adic coefficients. The reduced part of the Emerton-Gee stack, denoted $\mathcal{X}_{2, \text{red}}$, is an algebraic stack defined over a finite field of characteristic $p$ and can be viewed as a moduli stack of Galois representations with mod $p$ coefficients. The irreducible components of $\mathcal{X}_{2, \text{red}}$ are labelled in a natural way by Serre weights, which are the irreducible mod $p$ representations of $\mathrm{GL}_2(\mathcal{O}_K)$. Each irreducible component of $\mathcal{X}_{2, \text{red}}$ has dimension $[K:\mathbb{Q}_p]$. Motivated by the conjectural categorical $p$-adic Langlands programme, we find representation-theoretic criteria for codimension one intersections of the irreducible components of $\mathcal{X}_{2, \text{red}}$. The methods involve two separate computations and a final comparison between the two. The first of these computations determines extension groups of Serre weights and the second determines all the pairs of irreducible components that intersect in codimension one. We show that a non-trivial extension of a pair of non-isomorphic Serre weights implies a codimension one intersection of the corresponding irreducible components. The converse of this statement is also true when the Serre weights are chosen to be sufficiently generic. Furthermore, we show that the number of top-dimensional components in a codimension one intersection is related to the nature of the extension group of corresponding Serre weights.