Galois representations, Hecke operators, and the mod-$p$ cohomology of ${\rm GL}(3,\mathbb Z)$ with twisted coefficients

Abstract

We compute the degree 3 homology of GL(3,\,\funnyZ) with coefficients in the module of homogeneous polynomials in three variables of degree $g$ over $\mathbb F_p$, for $g\leq 200$ and $p\leq 541$. 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 $GL(3,\,\bar{\mathbb F}_p$). The conjecture is proved for the boundary part and explored experimentally for the quasicuspidal part.