Lattices in the cohomology of Shimura curves

• 成果类型: Article
• 署名作者: Emerton, Matthew; Gee, Toby; Savitt, David
• 署名单位: University of Chicago; Imperial College London; University of Arizona
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-014-0517-0
• 发表日期: 2015
• 页码: 1-96
• 关键词: modular-representations; CONSTRUCTION; conjecture

摘要

We prove the main conjectures of Breuil (J Reine Angew Math, 2012) (including a generalisation from the principal series to the cuspidal case) and Demb,l, (J Reine Angew Math, 2012), subject to a mild global hypothesis that we make in order to apply certain theorems. More precisely, we prove a multiplicity one result for the mod cohomology of a Shimura curve at Iwahori level, and we show that certain apparently globally defined lattices in the cohomology of Shimura curves are determined by the corresponding local -adic Galois representations. We also indicate a new proof of the Buzzard-Diamond-Jarvis conjecture in generic cases. Our main tools are the geometric Breuil-M,zard philosophy developed in Emerton and Gee (J Inst Math Jussieu, 2012), and a new and more functorial perspective on the Taylor-Wiles-Kisin patching method. Along the way, we determine the tamely potentially Barsotti-Tate deformation rings of generic two-dimensional mod representations, generalising a result of Breuil and M,zard (Bull Soc Math de France, 2012) in the principal series case.