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IHARA'S LEMMA FOR IMAGINARY QUADRATIC FIELDS

An analogue over imaginary quadratic elds 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 eld F , the kernel of the sum of the two standard p-degeneracy maps between the cuspidal sheaf cohomology H 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.