@article{ece13178d0114a88be448b186fb768e8,
title = "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 -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-M{\'e}zard 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.",
author = "Daniel Le and {Le Hung}, {Bao V.} and Brandon Levin and Stefano Morra",
note = "Funding Information: We would like to thank Christophe Breuil, James Humphreys, and Cornelius Pillen for many helpful conversations. We would also like to thank Matthew Emerton, Toby Gee, and Florian Herzig for their support, guidance, and for comments on an earlier draft of this paper. Part of this work was carried out while the authors were visiting the Institut Henri Poincar{\'e} and the Mathematisches Forschungsinstitut Oberwolfach, and we would like to thank these institutions for their hospitality. BLH acknowledges support from the National Science Foundation under grant Nos. DMS-1128155, DMS-1802037 and the Alfred P. Sloan Foundation. DL was supported by the National Science Foundation under agreement Nos. DMS-1128155 and DMS-1703182 and an AMS-Simons travel grant. Finally, the authors express their utmost gratitude to the anonymous referee for his or her meticulous and patient reading of several versions of this paper. The reports were invaluable in helping us to improve the quality, precision, and clarity of this paper. Publisher Copyright: {\textcopyright} 2020 Journal of Materials Research. All rights reserved.",
year = "2020",
doi = "10.1017/fmp.2020.1",
language = "English (US)",
volume = "8",
journal = "Forum of Mathematics, Pi",
issn = "2050-5086",
publisher = "Cambridge University Press",
}