Improved bounds for Serre's open image theorem

Abstract

Let $E$ be an elliptic curve over the rationals which does not have complex multiplication. Serre showed that the adelic representation attached to $E/\mathbb{Q}$ has open image, and in particular there is a minimal natural number $C_E$ such that the mod $\ell$ representation $\bar{\rho}_{E,\ell}$ is surjective for any prime $\ell > C_E$. Assuming the Generalized Riemann Hypothesis, Mayle-Wang gave explicit bounds for $C_E$ which are logarithmic in the conductor of $E$ and have explicit constants. The method is based on using effective forms of the Chebotarev density theorem together with the Faltings-Serre method, in particular, using the `deviation group' of the $2$-adic representations attached to two elliptic curves. By considering quotients of the deviation group and a characterization of the images of the $2$-adic representation $\rho_{E,2}$ by Rouse and Zureick-Brown, we show in this paper how to further reduce the constants in Mayle-Wang's results. Another result of independent interest are improved effective isogeny theorems for elliptic curves over the rationals.