2024/12/31 by Chen, Imin, Swidinsky, Joshua
#11F80 #11G05 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2501.00202
Let E be an elliptic curve over the rationals which does not have complex multiplication. Serre showed that the adelic representation attached to E/ℚ has open image, and in particular there is a minimal natural number CE such that the mod ℓ representation ρE,ℓ is surjective for any prime ℓ > CE. Assuming the Generalized Riemann Hypothesis, Mayle-Wang gave explicit bounds for CE 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 ρE,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.