Reducible Galois Representations and the Homology of GL(3, ℤ)

Abstract

Let Fp be an algebraic closure of a finite field of characteristic p. Let ρ be a continuous homomorphism from the absolute Galois group of Q to GL(3, Fp) which is isomorphic to a direct sum of a character and a two-dimensional odd irreducible representation. Under the condition that the Serre conductor of ρ is squarefree, we prove that ρ is attached to a Hecke eigenclass in the homology of an arithmetic subgroup Γ of GL(3,Z). In addition, we prove that the coefficient module needed is, in fact, predicted by the main conjecture of [3].