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

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 $\overline{r}:\text{Gal}(\overline{F^{+}}/F^{+}){\to }^C{U}_2({\overline{\mathbb{𝔽}}}_p)$ 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.