Reducible Galois representations and the homology of GL(3,Z)
Abstract
We prove the following theorem: Let $\bar\F_p$ 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 $\GL(3,\bar\F_p)$ which is isomorphic to a direct sum of a character and a two-dimensional odd irreducible representation. Under the condition that the 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 a conjecture of Ash, Doud, Pollack, and Sinnott.