Papers › Improved bounds for Serre's open image theorem
Improved bounds for Serre's open image theorem
Imin Chen, Joshua Swidinsky
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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 C_E such that the mod ℓ representation ρ̅_(E,ℓ) is surjective for any prime ℓ> 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.
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