Irregular loci in the Emerton-Gee stack for GL_2

TL;DR

This paper investigates the structure of loci of two-dimensional mod p Galois representations with irregular Hodge-Tate weights in the Emerton-Gee stack, proving irreducibility under certain bounded gap conditions.

Contribution

It introduces the concept of irregular loci in the Emerton-Gee stack and proves their irreducibility when weight gaps are bounded by p, extending understanding of Galois representation geometry.

Findings

Irregular loci are irreducible if weight gaps are ≤ p.

Established inclusion relations between different irregular loci.

Extended the analogy of Serre weights to irregular Hodge-Tate weights.

Abstract

Let K/Q_p be unramified. Inside the Emerton-Gee stack X_2, one can consider the locus of two-dimensional mod p representations of the absolute Galois group of K having a crystalline lift with specified Hodge-Tate weights. We study the case where the Hodge-Tate weights are irregular, which is an analogue for Galois representations of the partial weight one condition for Hilbert modular forms. We prove that if the gap between each pair of weights is bounded by p (the irregular analogue of a Serre weight), then this locus is irreducible. We also establish various inclusion relations between these loci.