Ihara's lemma for imaginary quadratic fields

TL;DR

This paper extends Ihara's lemma to imaginary quadratic fields, demonstrating that the kernel of certain degeneracy maps in cuspidal sheaf cohomology is Eisenstein, using modular symbols and Serre's congruence subgroup property.

Contribution

It establishes an analogue of Ihara's lemma over imaginary quadratic fields, a significant generalization from the classical setting.

Findings

Kernel of degeneracy maps is Eisenstein

Uses modular symbols and Serre's property

Generalizes Ihara's lemma to 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.