Ihara's lemma for imaginary quadratic fields

Abstract

An 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_!(X_0, M_0)^2 --> H^1_!(X_1, M_1) is Eisenstein. Here X_0 and X_1 are analogues over F of the modular curves X_0(N) and X_0(Np), respectively. To prove our theorem we use the method of modular symbols and the congruence subgroup property for the group SL(2) which is due to Serre.