On the Cohomology of Congruence Subgroups of GL3 over the Eisenstein Integers

Abstract

Let F be the imaginary quadratic field of discriminant −3 and its ring of integers. Let Γ be the arithmetic group and for any ideal let be the congruence subgroup of level consisting of matrices with bottom row In this paper we compute the cohomology spaces as a Hecke module for various levels where ν is the virtual cohomological dimension of Γ. This represents the first attempt at such computations for GL3 over an imaginary quadratic field, and complements work of Grunewald–Helling–Mennicke and Cremona, who computed the cohomology of over imaginary quadratic fields. In our results we observe a variety of phenomena, including cohomology classes that apparently correspond to nonselfdual cuspforms on