Hecke operators for non-congruence subgroups of Bianchi groups

Abstract

We prove that the action of the Hecke operators on the cohomology of a finite index non-congruence subgroup Γ \Gamma of a Bianchi group is essentially the same as the action of Hecke operators on the cohomology groups of Γ ^ \hat {\Gamma } , the congruence closure of Γ \Gamma . This is a generalization of Atkin’s conjecture, first confirmed in a special case by Serre in 1987 1987 and proved in general by Berger in 1994 1994 .