Serre weights for three-dimensional wildly ramified Galois representations

Abstract

We formulate and prove the weight part of Serre’s conjecture for three-dimensional mod $p$ Galois representations under a genericity condition when the field is unramified at $p$. This removes the assumption made previously that the representation be tamely ramified at $p$. We also prove a version of Breuil’s lattice conjecture and a mod $p$ multiplicity one result for the cohomology of $U(3)$-arithmetic manifolds. The key input is a study of the geometry of the Emerton–Gee stacks using the local models we introduced previously (2023).