A uniform bound on the smallest surjective prime of an elliptic curve

TL;DR

This paper proves that for any elliptic curve over rationals without complex multiplication, the smallest prime with a surjective Galois representation is at most 7, and classifies cases where it is exactly 7.

Contribution

It establishes a uniform bound of 7 on the smallest surjective prime for such elliptic curves and classifies all curves with the smallest surjective prime equal to 7.

Findings

The smallest surjective prime is at most 7 for all non-CM elliptic curves over Q.

Complete classification of elliptic curves with smallest surjective prime exactly 7.

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve without complex multiplication. A well-known theorem of Serre asserts that the $\ell$-adic Galois representation $\rho_{E,\ell^\infty}$ is surjective for all but finitely many prime numbers $\ell$. Considerable work has gone into bounding the largest possible nonsurjective prime; a uniform bound of $37$ has been proposed but is yet unproven. We consider an opposing direction, proving that the smallest prime $\ell$ such that $\rho_{E,\ell^\infty}$ is surjective is at most $7$. Moreover, we completely classify all elliptic curves $E/\mathbb{Q}$ for which the smallest surjective prime is exactly $7$.