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

Authors: Avner Ash and Mark McConnell
Journal: Math. Comp. 59 (1992), 673-688
MSC: Primary 11F75; Secondary 11F70
MathSciNet review: 1134711
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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.

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