Potentially crystalline deformation rings and Serre weight conjectures

We prove the weight part of Serre's conjecture in generic situations for forms of \(U(3)\) which are compact at infinity and split at places dividing \(p\) as conjectured by Herzig. We also prove automorphy lifting theorems in dimension three. The key input is an explicit description of tamely...

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Bibliographic Details
Published in:arXiv.org 2017-10
Main Authors: Le, Daniel, Le Hung, Bao V, Levin, Brandon, Morra, Stefano
Format: Article
Language:English
Subjects:
Crystal structure
Crystallinity
Deformation
Patching
Weight
Online Access:Get full text
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Summary:We prove the weight part of Serre's conjecture in generic situations for forms of \(U(3)\) which are compact at infinity and split at places dividing \(p\) as conjectured by Herzig. We also prove automorphy lifting theorems in dimension three. The key input is an explicit description of tamely potentially crystalline deformation rings with Hodge-Tate weights \((2,1,0)\) for \(K/\mathbb{Q}_p\) unramified combined with patching techniques. Our results show that the (geometric) Breuil-Mézard conjectures hold for these deformation rings.
ISSN:2331-8422