Serre weights for locally reducible two-dimensional galois representations

Let F be a totally real field, and v a place of F dividing an odd prime p. We study the weight part of Serre's conjecture for continuous totally odd representations -ρ:G_F →GL₂-F_p) that are reducible locally at v. Let W be the set of predicted Serre weights for the semisimplification of -ρ|G_Fv}. We prove that, when |ρ|G_Fv} is generic, the Serre weights in W for which -ρ is modular are exactly the ones that are predicted (assuming that |ρ is modular). We also determine precisely which subsets of W arise as predicted weights when ρ|_GF_v} varies with fixed generic semisimplification.