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arXiv · 1608.06570

Serre weights and Breuil's lattice conjecture in dimension three

Daniel Le·Bao V. Le Hung·Brandon Levin·Stefano Morra
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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-M\'ezard 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 $\mathrm{GL}_3(\mathbb{F}_q)$.

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Daniel Le, Bao V. Le Hung, Brandon Levin, Stefano Morra. 2016-08-23. Serre weights and Breuil's lattice conjecture in dimension three. https://arxiv.org/abs/1608.06570

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