### Abstract

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.

Original language | English (US) |
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Article number | e5 |

Journal | Forum of Mathematics, Pi |

Volume | 8 |

DOIs | |

State | Published - 2020 |

### ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Statistics and Probability
- Mathematical Physics
- Geometry and Topology
- Discrete Mathematics and Combinatorics

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## Cite this

*Forum of Mathematics, Pi*,

*8*, [e5]. https://doi.org/10.1017/fmp.2020.1