Experimental indications of three-dimensional Galois representations from the cohomology of {${\rm SL}(3,{\bf Z})$}

Abstract

Conjecturally, any "algebraic'' automorphic representation on $\GL(n)$ should have an $n$-dimensional Galois representation attached. Many examples of algebraic automorphic representations come from the cohomology over $\bold C$ of congruence subgroups of $\GL(n,\bold Z)$. On the other hand, the first author has conjectured that for any Hecke eigenclass in the mod $p$ cohomology of a congruence subgroup of $\GL(n,\Z)$ there should be an attached $n$-dimensional Galois representation.

By computer, we found Hecke eigenclasses in the mod $p$ cohomology of certain congruence subgroups of $\SL(3,\bold Z)$. In a range of examples, we then found a Galois representation (uniquely determined up to isomorphism by our data) that seemed to be attached to the Hecke eigenclass.