Potentially crystalline deformation rings and Serre weight conjectures: shapes and shadows

• 成果类型: Article
• 署名作者: Le, Daniel; Le Hung, Bao V.; Levin, Brandon; Morra, Stefano
• 署名单位: University of Toronto; Institute for Advanced Study - USA; University of Arizona; Universite de Montpellier
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-017-0762-0
• 发表日期: 2018
• 页码: 1-107
• 关键词: local-global compatibility; Galois representations

摘要

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 (Duke Math J 149(1):37-116, 2009). 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 unramified combined with patching techniques. Our results show that the (geometric) Breuil-M,zard conjectures hold for these deformation rings.