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

Kozioł, Karol; Morra, Stefano

doi:10.2140/ant.2022.16.2005

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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

DOI: 10.2140/ant.2022.16.2005

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 $F ∕ F^{+}$ 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 $\bar{r} : Gal (\bar{F^{+}} ∕ F^{+}) \to^{C} U_{2} ({\bar{𝔽}}_{p})$ be a Galois parameter valued in the $C$ -group of a rank 2 unitary group attached to $F ∕ F^{+}$ . We assume that $\bar{r}$ 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 $\bar{r}$ is modular agrees with the set of Serre weights predicted by Gee, Herzig and Savitt.

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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

Vol. 16, No. 9, 2022