SERRE WEIGHTS and BREUIL'S LATTICE CONJECTURE in DIMENSION THREE

We prove in generic situations that the lattice in a tame type induced by the completed cohomology of a -arithmetic manifold is purely local, that is, only depends on the Galois representation at places above. This is a generalization to 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 as well as the Serre weight conjectures of Herzig ['The weight in a Serre-type conjecture for tame -dimensional Galois representations', Duke Math. J. 149(1) (2009), 37-116] over an unramified field extending the results of Le et al. ['Potentially crystalline deformation 3985 rings and Serre weight conjectures: shapes and shadows', Invent. Math. 212(1) (2018), 1-107]. We also prove results in modular representation theory about lattices in Deligne-Lusztig representations for the group.