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

## TL;DR

This paper proves Ihara's lemma for Shimura curves over totally real fields using a novel approach based on the Taylor--Wiles method and integral models, removing previous restrictions on the prime l.

## Contribution

It extends Ihara's lemma to Shimura curves over totally real fields without the earlier assumptions on l, employing patching techniques and geometric analysis.

## Key findings

- Ihara's lemma established for a broader class of Shimura curves.
- Method avoids previous restrictions on the prime l.
- Utilizes advanced patching and geometric methods.

## 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.