Colength one deformation rings

2023/04/06 by Le, Daniel, Hung, Bao Le, Morra, Stefano +2
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2304.03061

Abstract

Let K/Q_p be a finite unramified extension, ρ:Gal(Q_p/K)\rightarrowGL_n(F_p) a continuous representation, and τ a tame inertial type of dimension n. We explicitly determine, under mild regularity conditions on τ, the potentially crystalline deformation ring R^η,τ_ρ in parallel Hodge--Tate weights η=(n-1,⋯,1,0) and inertial type τ when the shape of ρ with respect to τ has colength at most one. This has application to the modularity of a class of shadow weights in the weight part of Serre's conjecture. Along the way we make unconditional the local-global compatibility results of \citePQ and further study the geometry of moduli spaces of Fontaine--Laffaille representations in terms of colength one weights.

Related

• Potentially crystalline deformation rings in the ordinary case
• Local models for Galois deformation rings and applications
• Serre weights, Galois deformation rings, and local models
• Potentially crystalline deformation rings and Serre weight conjectures
• Inertial and Hodge--Tate weights of crystalline representations