Let $E/\mathbb{ Q}$ be an elliptic curve without complex multiplication. By Serre's open image theorem, the mod $\ell$ Galois representation $\overline{ \rho}_{ E, \ell}$ of $E$ is surjective for each prime number $\ell$ that is sufficiently large. Under the generalized Riemann hypothesis, we give an explicit upper bound on the largest prime $\ell$, linear in the logarithm of the conductor of $E$, such that $\overline{ \rho}_{ E, \ell}$ is nonsurjective.
Categories: math.NT