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, two-dimensional mod p representations rhobar of the absolute Galois group of F that are reducible locally at v. Let W be the set of predicted Serre weights for the semisimplification of rhobar restricted to the decomposition group at v. We prove that when the local representation is generic, the Serre weights in W for which rhobar is modular are exactly the ones that are predicted (assuming that rhobar is modular). We also determine precisely which subsets of W arise as predicted weights when the local representation varies with fixed generic semisimplification.