Abstract

Let p be a fixed odd prime, and let K be a finite extension of Qp with ring of integers O_K. The Emerton-Gee stack for GL2 is a stack of (phi, Gamma)-modules and can be interpreted as a moduli stack of representations of the absolute Galois group of K with p-adic coefficients. The irreducible components of the reduced part of the stack, denoted X, are labelled in a natural way by Serre weights, which are the irreducible mod p representations of GL2 .

Motivated by the conjectural categorical p-adic Langlands programme, we find representation-theoretic criteria for codimension one intersections of the irreducible components of X. 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.