Doubly cuspidal cohomology for principal congruence subgroups of

Authors:
Avner Ash and Mark McConnell

Journal:
Math. Comp. **59** (1992), 673-688

MSC:
Primary 11F75; Secondary 11F70

DOI:
https://doi.org/10.1090/S0025-5718-1992-1134711-3

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 of of prime level *p*. We show that the *f*-cuspidal cohomology of vanishes for all with , but that it is nonzero for . We give a precise description of the *f*-cuspidal cohomology for in terms of the *f*-cuspidal representations of the finite Lie group .

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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DOI:
https://doi.org/10.1090/S0025-5718-1992-1134711-3

Article copyright:
© Copyright 1992
American Mathematical Society