Strong modularity of reducible Galois representations

Let ρ:Gal(Q¯/Q)→GL2(F¯l)\rho \colon \mathrm {Gal}(\overline {\mathbf {Q}}/\mathbf {Q}) \rightarrow \mathrm {GL}_2(\overline {\mathbf {F}}_{l}) be an odd, semi-simple Galois representation. Here, l≥5l\geq 5 is prime and F¯l\overline {\mathbf {F}}_{l} is an algebraic closure of the finite field Z/lZ\mathbf {Z}/l\mathbf {Z}. When the representation is irreducible, the strongest form of Serre’s original modularity conjecture (which is now proved) asserts that ρ\rho arises from a cuspidal eigenform of type (N,k,ε)(N,k,\varepsilon ) over F¯l\overline {\mathbf {F}}_{l}, where NN, kk and ε\varepsilon are, respectively, the level, weight and character attached to ρ\rho by Serre. In this paper we characterize, under the assumption l>k+1l>k+1, reducible semi-simple representations, that we call strongly modular, such that the same result holds. This characterization generalizes a classical theorem of Ribet pertaining to the case N=1N=1. When the representation is not strongly modular, we give a necessary and sufficient condition on primes pp not dividing NlNl for which ρ\rho arises in level NpNp, hence generalizing a classical theorem of Mazur concerning the case (N,k)=(1,2)(N,k)=(1,2). The proofs rely on the classical analytic theory of Eisenstein series and on local properties of automorphic representations attached to newforms.