Strong modularity of reducible Galois representations

Abstract:

Let $\rho \colon \mathrm {Gal}(\overline {\mathbf {Q}}/\mathbf {Q}) \rightarrow \mathrm {GL}_2(\overline {\mathbf {F}}_{l})$ be an odd, semi-simple Galois representation. Here, $l\geq 5$ is prime and $\overline {\mathbf {F}}_{l}$ is an algebraic closure of the finite field $\mathbf {Z}/l\mathbf {Z}$. When the representation is irreducible, the strongest form of Serre’s original modularity conjecture (which is now proved) asserts that $\rho$ arises from a cuspidal eigenform of type $(N,k,\varepsilon )$ over $\overline {\mathbf {F}}_{l}$, where $N$, $k$ and $\varepsilon$ are, respectively, the level, weight and character attached to $\rho$ by Serre.

In this paper we characterize, under the assumption $l>k+1$, 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 $N=1$. When the representation is not strongly modular, we give a necessary and sufficient condition on primes $p$ not dividing $Nl$ for which $\rho$ arises in level $Np$, hence generalizing a classical theorem of Mazur concerning the case $(N,k)=(1,2)$.

The proofs rely on the classical analytic theory of Eisenstein series and on local properties of automorphic representations attached to newforms.