Serre weights for three-dimensional wildly ramified Galois representations

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 in [LLHLM18, LLHLM20] 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 [EG] using the local models introduced in [LLHLMa].