Joshua Swidinsky

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 article 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.

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 $\barρ_{E,\ell}$ 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 $ρ_{E,2}$ 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.