On the effective version of Serre's open image theorem

TL;DR

This paper provides an explicit upper bound, linear in the logarithm of the conductor, for the largest prime where the mod $\\ell$ Galois representation of an elliptic curve over $\mathbb{Q}$ is not surjective, assuming GRH.

Contribution

It offers a new explicit bound on the largest prime with non-surjective Galois representation, improving understanding under GRH.

Findings

Bound is linear in the logarithm of the conductor

Assumes generalized Riemann hypothesis for explicitness

Enhances previous qualitative results with quantitative bounds

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve without complex multiplication. By Serre's open image theorem, the mod $\ell$ Galois representation $\overline{\rho}_{E, \ell}$ of $E$ is surjective for each prime number $\ell$ that is sufficiently large. Under the generalized Riemann hypothesis, we give an explicit upper bound on the largest prime $\ell$, linear in the logarithm of the conductor of $E$, such that $\overline{\rho}_{E, \ell}$ is nonsurjective.