Experimental Indications of Three-dimensional Galois Representations from the Cohomology of SL(3, Z)

Abstract

Conjecturally, any “algebraic” automorphic represent ation on GL(n) should have an n-d imensional Galois representation attached. Many examples of algebraic automorphic representations come from the cohomology over C of congruence subgroups of GL(n, 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 In attached n-dimensional Galois representation. By computer, we found Heeke eigenclasses in the mod p co-homology of certain congruence subgroups of SL(3, Z). In a range of examples, we then found a Galois representation (uniquely determined up to isomorphism by our data) that seemed to be attac hed to the Hecke eigenclass.