Lattices in the cohomology of Shimura curves

TL;DR

This paper proves conjectures relating the mod p cohomology of Shimura curves to local Galois representations, generalizes key results, and introduces new methods in the study of p-adic Hodge theory and deformation rings.

Contribution

It extends the Breuil-Dembele conjecture to the cuspidal case, establishes a multiplicity one result, and offers a new perspective on the Taylor-Wiles-Kisin patching method.

Findings

Proved conjectures of Breuil and Breuil-Dembele.

Determined tamely potentially Barsotti-Tate deformation rings.

Provided a new proof of the Buzzard-Diamond-Jarvis conjecture in generic cases.

Abstract

We prove conjectures of Breuil and Breuil-Dembele (C. Breuil, "Sur un probleme de compatibilite local-global modulo p pour GL(2)"), including a generalisation from the principal series to the cuspidal case, subject to a mild global hypothesis that we make in order to apply certain R=T theorems. More precisely, we prove a multiplicity one result for the mod 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-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-Mezard philosophy developed by two of the authors, 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 representations, generalising a result of Breuil-Mezard in the principal series case.