Weight cycling and Serre-type conjectures for unitary groups

Abstract

We prove that for forms of $U\left(3\right)$ which are compact at infinity and split at places dividing a prime $p$, in generic situations the Serre weights of a mod $p$ modular Galois representation which is irreducible when restricted to each decomposition group above $p$ are exactly those previously predicted by Herzig. We do this by combining explicit computations in $p$-adic Hodge theory (based on a formalism of strongly divisible modules and Breuil modules with descent data which we develop here) with a technique that we call weight cycling.