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Serre weight conjectures for $p$-adic unitary groups of rank 2

Karol Kozioł and Stefano Morra

Vol. 16 (2022), No. 9, 2005–2097
Abstract

We prove a version of the weight part of Serre’s conjecture for mod p Galois representations attached to automorphic forms on rank 2 unitary groups which are nonsplit at p. More precisely, let FF+ denote a CM extension of a totally real field such that every place of F+ above p is unramified and inert in F, and let r¯ : Gal (F+¯F+)CU2(𝔽¯p) be a Galois parameter valued in the C-group of a rank 2 unitary group attached to FF+. We assume that r¯ is semisimple and sufficiently generic at all places above p. Using base change techniques and (a strengthened version of) the Taylor–Wiles–Kisin conditions, we prove that the set of Serre weights in which r¯ is modular agrees with the set of Serre weights predicted by Gee, Herzig and Savitt.

Keywords
generalization of Serre weight conjectures, mod-$p$ Langlands, nonsplit unitary groups
Mathematical Subject Classification 2010
Primary: 11F80
Secondary: 11F33, 11F55, 20C33
Milestones
Received: 9 March 2019
Revised: 17 July 2021
Accepted: 27 August 2021
Published: 19 December 2022
Authors
Karol Kozioł
Department of Mathematics
The University of Michigan
Ann Arbor, MI
United States
Stefano Morra
Laboratoire Analyse Géométrie Applications
Université Paris 8
Villetaneuse
France