Doubly cuspidal cohomology for principal congruence subgroups of 𝐺𝐿(3,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 Γ ( p ) \Gamma (p) of GL ( 3 , Z ) {\text {GL}}(3,\mathbb {Z}) of prime level p. We show that the f-cuspidal cohomology of Γ ( p ) \Gamma (p) vanishes for all p ⩽ 19 p \leqslant 19 with p ≠ 11 p \ne 11 , but that it is nonzero for p = 11 p = 11 . We give a precise description of the f-cuspidal cohomology for Γ ( 11 ) \Gamma (11) in terms of the f-cuspidal representations of the finite Lie group GL ( 3 , Z / 11 ) {\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.