Lattices in the cohomology of Shimura curves

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.