## Codimension one intersections between components of the Emerton-Gee stack for GL2

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2023-04-04

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Johns Hopkins University

##### Abstract

Let p be a fixed odd prime, and let K be a finite extension of Q_p with ring of integers O_K. The Emerton-Gee stack for GL_2 is a stack of (phi, Gamma)-modules. The stack, denoted 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 X_{2, 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 X_{2, red} are labelled in a natural way by Serre weights, which are the irreducible mod p representations of GL_2(O_K). Each irreducible component of X_{2, red} has dimension [K:Q_p].
Motivated by the conjectural categorical p-adic Langlands programme, the objective of this thesis is to find representation-theoretic criteria for codimension one intersections of the irreducible components of X_{2, 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.
This thesis shows 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.

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Langlands, number theory, Emerton-Gee stack