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.

arXiv comment40 pages. To appear in Math. Annalen

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Cited by in corpus (6)

• A proof of Perrin-Riou's Heegner point main conjecture
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• On the anticyclotomic Iwasawa main conjecture for Hilbert modular forms of parallel weights
• On the Iwasawa invariants of Kato's zeta elements for modular forms
• The universal unramified module for GL(n) and the Ihara conjecture
• Iwasawa Theory of Hilbert modular forms for anticyclotomic extension