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Serre weights for three-dimensional wildly ramified Galois representations

Daniel Le, Bao V. Le Hung, Brandon Levin and Stefano Morra

Vol. 18 (2024), No. 7, 1221–1274
DOI: 10.2140/ant.2024.18.1221
Abstract

We formulate and prove the weight part of Serre’s conjecture for three-dimensional mod p Galois representations under a genericity condition when the field is unramified at p. This removes the assumption made previously that the representation be tamely ramified at p. We also prove a version of Breuil’s lattice conjecture and a mod p multiplicity one result for the cohomology of U(3)-arithmetic manifolds. The key input is a study of the geometry of the Emerton–Gee stacks using the local models we introduced previously (2023).

Keywords
weight part of Serre's conjectures, local models for Galois representations, mod $p$ cohomology of arithmetic manifolds, congruences of automorphic forms
Mathematical Subject Classification
Primary: 11F33, 11F80, 22E50
Milestones
Received: 9 February 2022
Revised: 21 April 2023
Accepted: 3 September 2023
Published: 13 June 2024
Authors
Daniel Le
Department of Mathematics
Purdue University
West Lafayette, IN
United States
Bao V. Le Hung
Department of Mathematics
Northwestern University
Evanston, IL
United States
Brandon Levin
Department of Mathematics
Rice University
Houston, TX
United States
Stefano Morra
Laboratoire Analyse Géométrie Applications
Université Paris 8
Villetaneuse
France

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