Lattices in the cohomology of Shimura curves

By M. Emerton, T. Gee, David Savitt. 2013

64 citations recorded by the source corpus.

We prove the main conjectures of Breuil (J Reine Angew Math, 2012) (including a generalisation from the principal series to the cuspidal case) and Dembélé (J Reine Angew Math, 2012), subject to a mild global hypothesis that we make in order to apply certain $$R=\mathbb {T}$$R=T theorems. More precisely, we prove a multiplicity one result for the mod $$p$$p 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 $$p$$p-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–Mézard 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 $$p$$p representations, generalising a result of Breuil and Mézard (Bull Soc Math de France, 2012) in the principal series case.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/115d30d8f4a4bad733b0de9e0fb168f87b698ec2

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

General Serre weight conjectures (2015, methodology): “ ratherto explain what we believe should be true, we avoid making speciﬁc Taylor–Wiles patching arguments, and instead use the abstract language of patching functors, originally introduced for GL2 in [EGS15]. Our patching functors will be for GLn, and will satisfy slightly diﬀerent axioms from those”
Patching and the p-adic local Langlands correspondence (2013, methodology): “ching construction, by applying the Taylor– Wiles–Kisin method. Before doing so, we provide a brief comparison and contrast with the patching constructions in some previous papers, such as [EG13] and [EGS13]. In the latter paper, we employ Taylor–Wiles–Kisin patching to construct what we call patchi”