Ihara's lemma for imaginary quadratic fields

AbstractAn analogue over imaginary quadratic fields of a result in algebraic number theory known as Ihara's lemma is established. More precisely, we show that for a prime ideal p of the ring of integers of an imaginary quadratic field F, the kernel of the sum of the two standard p-degeneracy maps between the cuspidal sheaf cohomology H!1(Y0,M˜0)2→H!1(Y1,M˜1) is Eisenstein. Here Y0 and Y1 are analogues over F of the modular curves Y0(N) and Y0(Np), respectively. To prove our theorem we use the method of modular symbols and the congruence subgroup property for the group SL2(Z[1/p]) which is due to Mennicke [J. Mennicke, On Ihara's modular group, Invent. Math. 4 (1967) 202–228] and Serre [J.-P. Serre, Le problème des groupes de congruence pour SL2, Ann. of Math. (2) 92 (1970) 489–527]