Ihara's lemma for Shimura curves over totally real fields via patching

Abstract

We prove Ihara's lemma for the mod $l$ cohomology of Shimura curves, localised at a maximal ideal of the Hecke algebra, under a large image hypothesis on the associated Galois representation. This was proved by Diamond and Taylor, for Shimura curves over $\mathbb{Q}$, under various assumptions on $l$. Our method is totally different and can avoid these assumptions, at the cost of imposing the large image hypothesis. It uses the Taylor--Wiles method, as improved by Diamond and Kisin, and the geometry of integral models of Shimura curves at an auxiliary prime.