GALOIS REPRESENTATIONS, HECKE OPERATORS, AND THE MOD-p COHOMOLOGY OF GL(3, ℤ) WITH

TWISTED COEFFICIENTS

GERALD ALLISON, AVNER ASH, AND ERIC CONRAD

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, 𝔽̄_{p}). The conjecture is proved for the boundary part and explored experimentally for the quasicuspidal part.