2022/10/10 by Kalyani Kansal, Kansal, Kalyani
Mathematics · #11S99 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2210.05002
openalex publication_date 2022/10/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let p be a fixed odd prime, and let K be a finite extension of ℚp with ring of integers OK. The Emerton-Gee stack for GL2 is a stack of (φ, Γ)-modules. The stack, denoted X2, 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 X2, 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 X2, red are labelled in a natural way by Serre weights, which are the irreducible mod p representations of GL2(OK). Each irreducible component of X2, red has dimension [K:ℚp]. Motivated by the conjectural categorical p-adic Langlands programme, we find representation-theoretic criteria for codimension one intersections of the irreducible components of X2, 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.