Serre weights and Breuil's lattice conjecture in dimension three

2016/08/23 by Daniel Le, Le, Daniel, Bao V. Le Hung +5
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1608.06570

openalex publication_date 2016/08/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove in generic situations that the lattice in a tame type induced by the completed cohomology of a U(3)-arithmetic manifold is purely local, i.e., only depends on the Galois representation at places above p. This is a generalization to GL₃ of the lattice conjecture of Breuil. In the process, we also prove the geometric Breuil-Mézard conjecture for (tamely) potentially crystalline deformation rings with Hodge-Tate weights (0,1,2) as well as the Serre weight conjectures over an unramified field extending our previous results. We also prove results in modular representation theory about lattices in Deligne-Luzstig representations for the group GL₃(\mathbbF_q).

Citations

• On mod local-global compatibility for in the ordinary case

Related

• Breuil's Lattice Conjecture for GL2(K)
• Serre weights, Galois deformation rings, and local models
• Potentially crystalline deformation rings and Serre weight conjectures
• Local models for Galois deformation rings and applications
• Serre weights and the Breuil-Mézard conjecture for modular forms