Deformations of Certain Reducible Galois Representations

Let ρ¯ : Gal( ¯Q /Q) = G_Q → GL₂(𝔽_q ) be a continuous Galois representation. In [R2], for ρ¯ absolutely irreducible, we constructed characteristic zero deformations of ρ¯. Here we address the case of ρ¯ reducible and odd. Using Kummer theory Khare has constructed reducible characteristic zero deformations of even and odd ρ¯ in [Kh]. Our aim is to construct irreducible characteristic zero deformations of the reducible ρ¯.