Relaxation of strict parity for reducible Galois representations attached to the homology of GL(3, ℤ)

Abstract

In this paper, we prove the following theorem: Let [Formula: see text] be an algebraic closure of a finite field of characteristic [Formula: see text]. Let [Formula: see text] be a continuous homomorphism from the absolute Galois group of [Formula: see text] to [Formula: see text]) which is isomorphic to a direct sum of a character and a two-dimensional odd irreducible representation. We assume that the image of [Formula: see text] is contained in the intersection of the stabilizers of the line spanned by [Formula: see text] and the plane spanned by [Formula: see text], where [Formula: see text] denotes the standard basis. Such [Formula: see text] will not satisfy a certain strict parity condition. Under the conditions that the Serre conductor of [Formula: see text] is squarefree, that the predicted weight [Formula: see text] lies in the lowest alcove, and that [Formula: see text], we prove that [Formula: see text] is attached to a Hecke eigenclass in [Formula: see text], where [Formula: see text] is a subgroup of finite index in [Formula: see text] and [Formula: see text] is an [Formula: see text]-module. The particular [Formula: see text] and [Formula: see text] are as predicted by the main conjecture of the 2002 paper of the authors and David Pollack, minus the requirement for strict parity.