Doubly cuspidal cohomology for principal congruence subgroups of $\textrm {GL}(3,\textbf {Z})$

Abstract:

The cohomology of arithmetic groups is made up of two pieces, the cuspidal and noncuspidal parts. Within the cuspidal cohomology is a subspace— the f-cuspidal cohomology—spanned by the classes that generate representations of the associated finite Lie group which are cuspidal in the sense of finite Lie group theory. Few concrete examples of f-cuspidal cohomology have been computed geometrically, outside the cases of rational rank 1, or where the symmetric space has a Hermitian structure. This paper presents new computations of the f-cuspidal cohomology of principal congruence subgroups $\Gamma (p)$ of ${\text {GL}}(3,\mathbb {Z})$ of prime level p. We show that the f-cuspidal cohomology of $\Gamma (p)$ vanishes for all $p \leqslant 19$ with $p \ne 11$, but that it is nonzero for $p = 11$. We give a precise description of the f-cuspidal cohomology for $\Gamma (11)$ in terms of the f-cuspidal representations of the finite Lie group ${\text {GL}}(3,\mathbb {Z}/11)$. We obtained the result, ultimately, by proving that a certain large complex matrix M is rank-deficient. Computation with the SVD algorithm gave strong evidence that M was rank-deficient; but to prove it, we mixed ideas from numerical analysis with exact computation in algebraic number fields and finite fields.