Explicit Serre weights for modular Galois representations of $U(2)$

Abstract

Let $p$ be an odd prime. Let $F$ be a totally imaginary CM field and $F^+$ be the maximal totally real subfield such that $F/F^+$ is unramified at all finite places, and every place $v | p$ of $F^+$ is split in $F$. We allow arbitrary ramification of $p$ in $F^+$. Let $G$ be a rank two definite unitary group over $F^+$ as defined in [2]. Fix a place $\mathfrak{p} | p$ in $F$. In this article, we determine explicit local Serre weights at $\mathfrak{p}$ attached to a modular mod $p$ Galois representation $\barρ_h$ of the absolute Galois group of $F$, associated to an eigenform $h$ of $G$. The eigenform $h$ has arbitrary weights and small slope at $\mathfrak{p}$. We also show analogous results for certain Hilbert modular eigenforms with arbitrary ramification of $p$ in the totally real field. The eigenforms come from totally definite and indefinite quaternion algebras over totally real fields, split at places above $p$. In proving our results, we identify the non-singular submodules in quotients of certain symmetric power $\mathbb{F}_q[\text{M}_2(\mathbb{F}_q)]$-modules by Dickson polynomials and their twisted versions.