Abstract

Algorithms are presented which find a basis of the vector space of cuspidal cohomology of certain congruence subgroups of SL(3, Z ) and which determine the action of the Hecke operators on this space. These algorithms were implemented on a computer. Four pairs of cuspidal classes were found with prime level less than 100. Tables are given of the eigenvalues of the first few Hecke operators on these classes.

Highlights

  • We report in this paper on some computations of automorphic forms for congruence subgroups of SL(3, Z)

  • The rest of the cohomology of H/T comes from antiholomorphic automorphic cusp forms and Eisenstein automorphic forms in a similar way

  • First we present the relation between the topology of X/T and the theory of automorphic forms for GL(3)

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Summary

INTRODUCTION

We report in this paper on some computations of automorphic forms for congruence subgroups of SL(3, Z). We can define Hecke operators that act directly on the cohomology of X/r’ and study those automorphic forms in a geometrical setting. We followed this out for G = GL(3) to the extent that we could write algorithms for computing spaces of “cuspidal cohomology” and the matrices through which the Hecke operators act. By fcuspvw, C) we mean Hf,,, (H/r(P), C) as defined in the Introduction This proposition, which is not too hard to prove directly, may be derived from Theorem 2.4 of [6] by taking r/r(m)-invariants in the formula for H3@M) there presented, where r(m) is any full congruence subgroup of SL(3, Z) contained in r

LINEAR EQUATIONS FOR COHOMOLOGY AND MODULAR SYMBOLS
HECKE OPERATORS
11 SW char poly
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