Strong modularity of reducible Galois representations

Resumen

In this paper, we call strongly modular those reducible semi-simple odd mod l Galois representations for which the conclusion of the strongest form of Serre’s original modularity conjecture holds. Under the assumption that the Serre weight k satisfies l > k+1, we give a precise characterization of strongly modular representations, hence generalizing a classical theorem of Ribet pertaining to the case of conductor 1. When the representation ρ is not strongly modular, we give a necessary and sufficient condition on the primes p not dividing Nl for which it arises in level Np, where N denotes the conductor of ρ. This generalizes a result of Mazur on the case (N,k)=(1,2).