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 $U(3)$-arithmetic manifold is purely local, that is, only depends on the Galois representation at places above $p$. This is a generalization to $\text{GL}_{3}$ of the lattice conjecture of Breuil. In the process, we also prove the geometric Breuil-Mezard conjecture for (tamely) potentially crystalline deformation rings with Hodge-Tate weights $(2,1,0)$ as well as the Serre weight conjectures of Herzig ['The weight in a Serre-type conjecture for tame $n$-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 GL(3)(F-q).Open access journalThis item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at repository@u.library.arizona.edu