Ihara’s lemma for imaginary quadratic fields. (English) Zbl 1213.11101
Summary: 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 \(\mathfrak p\) of the ring of integers of an imaginary quadratic field \(F\), the kernel of the sum of the two standard \( \mathfrak p\)-degeneracy maps between the cuspidal sheaf cohomology \(H_!^1(Y_0\tilde M_0)^2\to H_!^1(Y_1,\tilde M_1)\) is Eisenstein. Here \(Y_{0}\) and \(Y_{1}\) are analogues over \(F\) of the modular curves \(Y_{0}(N)\) and \(Y_{0}(Np)\), respectively. To prove our theorem we use the method of modular symbols and the congruence subgroup property for the group SL\(_{2}(\mathbf Z[1/p])\) which is due to J. Mennicke [Invent. Math. 4, 202–228 (1967; Zbl 0189.02504)] and J.-P. Serre [Ann. Math. (2) 92, 489–527 (1970; Zbl 0239.20063)].
MSC:
| 11F32 | Modular correspondences, etc. |
| 11F25 | Hecke-Petersson operators, differential operators (one variable) |
Keywords:
modular forms over imaginary quadratic fields; Hecke operators; group cohomology; modular symbolsReferences:
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