Sharper explicit constants for the smallest bound C_E beyond which the mod-l Galois representations of a non-CM elliptic curve over the rationals are all surjective. Assuming the Generalized Riemann Hypothesis, the constants in the known logarithmic-in-the-conductor bound are reduced by passing to smaller quotients of the Faltings-Serre deviation group and using the classification of 2-adic images of elliptic curves. The same technique yields improved effective isogeny theorems bounding the least prime distinguishing two non-isogenous curves by their traces of Frobenius.
Concerns Serre's open image theorem: for an elliptic curve E over the rationals without complex multiplication, there is a smallest constant C_E beyond which every mod-l Galois representation attache…