Serre weights and Breuil's lattice conjecture in dimension three

Abstract

We prove in generic situations that the lattice in a tame type induced by the completed cohomology of a $U(3)$-arithmetic manifold is purely local, i.e., only depends on the Galois representation at places above $p$. This is a generalization to ${\mathrm{GL}}_3$ of the lattice conjecture of Breuil. In the process, we also prove the geometric Breuil-Mézard conjecture for (tamely) potentially crystalline deformation rings with Hodge-Tate weights $(0,1,2)$ as well as the Serre weight conjectures over an unramified field extending our previous results. We also prove results in modular representation theory about lattices in Deligne-Luzstig representations for the group ${\mathrm{GL}}_3({\mathbb{F}}_q)$.

102 pages, major revision, includes addendum to arxiv:1512.06380

References in corpus (1)

• On mod p local-global compatibility for GL_3 in the ordinary case

Cited by in corpus (1)

• Non-admissible irreducible representations of $p$-adic ${\mathrm{GL}}_n$ in characteristic $p$