Galois Representations, Hecke Operators, and the mod-p Cohomology of GL(3, Z) with Twisted Coefficients

Abstract

We compute the degree 3 homology of Gl(3, Z) with coefficients in the module of homogeneous polynomials in three variables of degree g over F_p, for g ≤ 200 and p ≤ 541. The homology has a “boundary part” and a “quasicuspidal” part which we determine.

By conjecture a Hecke eigenclass in the homology has an attached Galois representation into Gl(3, _p). The conjecture is proved for the boundary part and explored experimentally for the quasicuspidal part.

Keywords:

• Galois representations
• Heeke operators
• mod-p cohomology
• modular symbols