Weight cycling and Serre-type conjectures for unitary groups

TL;DR

This paper proves the predicted Serre weights for certain mod p Galois representations associated with unitary groups, using p-adic Hodge theory and a novel weight cycling technique.

Contribution

It introduces a new method called weight cycling and combines it with explicit p-adic Hodge theory computations to confirm Serre weight conjectures for U(3).

Findings

Confirmed Serre weights match predictions in generic cases.

Developed a formalism of strongly divisible and Breuil modules with descent data.

Introduced the weight cycling technique for analyzing Galois representations.

Abstract

We prove that for forms of U(3) 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 the third author. 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 in the paper, with a technique that we call "weight cycling".