Hecke operators for non-congruence subgroups of Bianchi groups

Hamzeh Zarghani, Saeid

doi:10.1090/S0002-9939-2011-10807-4

Hecke operators for non-congruence subgroups of Bianchi groups
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Proc. Amer. Math. Soc. 139 (2011), 3853-3865 Request permission

Abstract:

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

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