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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 $F∕{F}^{+}$ 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 $\overline{r}:\mathrm{Gal}\left(\overline{{F}^{+}}∕{F}^{+}\right){\to }^{C}\phantom{\rule{-0.17em}{0ex}}{U}_{2}\left({\overline{\mathbb{𝔽}}}_{p}\right)$ be a Galois parameter valued in the $C$-group of a rank 2 unitary group attached to $F∕{F}^{+}$. We assume that $\overline{r}$ 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 $\overline{r}$ 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