Cuspidal Cohomology for Principal Congruence Subgroups of Gl(3, Z)

The cohomology of arithmetic groups is made up of two pieces, the cuspidal and noncuspidal parts. Within the cuspidal cohomology is a subspace— the /-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 /-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 /-cuspidal cohomology of principal congruence subgroups T(p) of GL(3,Z) of prime level p. We show that the /-cuspidal cohomology of F(p) vanishes for all p < 19 with p ,¿ 11 , but that it is nonzero for p = 11 . We give a precise description of the /-cuspidal cohomology for T( 11 ) in terms of the /-cuspidal representations of the finite Lie group GL(3, Z/l 1). 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.