Abstract
We prove Ihara’s lemma for the mod l cohomology of Shimura curves, localized 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.
Highlights
Let = 0(N ) be the usual congruence subgroup of S L2(Z), for some N ≥ 1, and let p be a prime not dividing N
We refer to this as “Ihara’s Lemma at p for X K, localized at m”; it depends on K and on a maximal ideal m of the Hecke algebra acting on H 1(X K, Fl ), to which is associated a Galois representation ρm : G F → G L2(Fl )
In [13], Diamond and Taylor overcame this difficulty for Shimura curves over Q using the good reduction of Shimura curves at l and comparison of mod l de Rham and étale cohomology
Summary
For K ⊂ (D ⊗ AF, f )× a compact open subgroup, p a finite place of F at which K and D are unramified, and l a prime, there is an obvious (conjectural) generalisation of Theorem 1 with X replaced by the Shimura curve X K We refer to this as “Ihara’s Lemma at p for X K , localized at m”; it depends on K and on a maximal ideal m of the Hecke algebra acting on H 1(X K , Fl ), to which is associated a Galois representation ρm : G F → G L2(Fl ). Our method proves that the “patched” module corresponding to H 1(X K0(q), Fl ) is flat over some specific local deformation ring at the prime q Using this and some commutative algebra we are able to deduce Ihara’s Lemma for X K from the corresponding result for YK q. This is the reason we do not need to impose any of the restrictions on the prime l appearing in earlier results
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