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SERRE WEIGHTS AND BREUIL’S LATTICE CONJECTURE IN DIMENSION THREE

Published online by Cambridge University Press:  25 March 2020

DANIEL LE
Affiliation:
Department of Mathematics, University of Toronto, 40 St. George Street,Toronto, ON M5S 2E4, Canada; le@math.toronto.edu
BAO V. LE HUNG
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA; lhvietbao@googlemail.com
BRANDON LEVIN
Affiliation:
Department of Mathematics, University of Arizona, 617 N. Santa Rita Avenue, P.O. Box 210089, Tucson, Arizona 85721, USA; bwlevin@math.arizona.edu
STEFANO MORRA
Affiliation:
Université Paris 8, Laboratoire Analyse, Géométrie et Applications, Université Sorbonne Paris Nord, CNRS, UMR 7539, F-93430, Villetaneuse, France; morra@math.univ-paris13.fr

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, 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–Mézard 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 $\text{GL}_{3}(\mathbb{F}_{q})$ .

Type
Number Theory
Creative Commons
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

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