Exploration of a mod pq variant of Serre's conjecture

Consider a continuous odd Galois representation

$$\rho: G_{\mathbf{Q}}\rightarrow \mbox{\rm GL}_2(\mathbf{Z}/... ...{Z}/p\mathbf{Z})\times\mbox{\rm GL}_2(\mathbf{Z}/q\mathbf{Z})$$ (1)

that is irreducible when reduced modulo p and modulo q. We say that $\rho$ is modular if there exists a newform f such that $\rho\sim \rho_{f,\lambda_p}\times\rho_{f,\lambda_q}$, with $\lambda_p$ and $\lambda_q$ primes lying over p and q, respectively. Is every mod pq representation modular? When $\rho$ is modular, what is the minimal level $N(\rho)$ and weight $k(\rho)$ of a newform f giving rise to $\rho$? In this section we report on a first investigation, conducted jointly with B. Mazur, into this question.