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

TL;DR

This work analyzes codimension-one intersections of irreducible components in the reduced Emerton–Gee GL$_2$ stack, linking the geometric intersections to extensions between Serre weights. It first computes GL$_2(\mathcal{O}_K)$-extensions of Serre weights via a spectral-sequence/cocycle approach, and then develops a GK-representation framework to relate the dimensions of families of Galois representations with multiple Serre weights to the dimension of component intersections. The authors classify Type I and Type II codimension-one intersections across all ramification patterns (varying $f$ and $e$) and show that nontrivial Ext groups force a codimension-one intersection, while in the weakly regular case the converse holds. They also quantify the number of top-dimensional components in an intersection by the underlying extension data, revealing a precise dictionary between representation-theoretic extensions and stack geometry. These results advance the geometric understanding of the Emerton–Gee stack and its role in the categorical p-adic Langlands program.

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 $(\varphi, Γ)$-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.

Theorems & Definitions (113)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Proposition 2.4
• proof
• Proposition 2.5
• proof
• Corollary 2.8