Improved bounds for Serre's open image theorem

TL;DR

This paper improves explicit bounds on the primes for which elliptic curves over rationals have surjective mod $\ell$ Galois representations, refining previous results under the Generalized Riemann Hypothesis using advanced number theory techniques.

Contribution

It introduces new methods to further reduce the constants in existing bounds for Serre's open image theorem, utilizing quotient groups and recent characterizations of 2-adic representations.

Findings

Reduced the explicit bounds for surjective mod $\ell$ representations.

Provided improved effective isogeny theorems for elliptic curves over $\mathbb{Q}$.

Enhanced the understanding of the deviation group and 2-adic image characterization.

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.