Experimental Indications of Three-dimensional Galois Representations

from the Cohomology of SL(3, Z)

Avner Ash Mark McConnell

Abstract

Conjecturally, any “algebraic” automorphic representation on mathrmGL(n) should have an n-dimensional Galois representation attached. Many examples of algebraic automorphic representations come from the cohomology over C of congruence subgroups of mathrmGL(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 mathrmGL(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 mathrmSL(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 attached to the Hecke eigenclass.