Relaxation of Strict Parity for Reducible Galois Representations Attached to the Homology of Gl(3,z) Avner Ash and Darrin Doud

We prove the following theorem: Let F be an algebraic closure of a finite field of characteristic p > 3. Let ρ be a continuous homomorphism from the absolute Galois group of Q to GL(3,F) which is isomorphic to a direct sum of a character and a two-dimensional odd irreducible representation. We assume that the image of ρ is contained in the intersection of the stabilizers of the line spanned by e2 and the plane spanned by e1, e3, where ei denotes the standard basis. Such ρ will not satisfy the strict parity conditions of [4]. Under the conditions that the Serre conductor of ρ is squarefree, that the predicted weight (a, b, c) lies in the lowest alcove, and that c 6≡ b+1 (mod p−1), we prove that ρ is attached to a Hecke eigenclass in H2(Γ,M), where Γ is a subgroup of finite index in SL(3,Z) and M is an FΓ-module. The particular Γ and M are as predicted by the main conjecture of [4], minus the requirement for strict parity.