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

TL;DR

The paper proves that for any non-CM elliptic curve $E/\mathbb{Q}$, the smallest prime $\ell$ with surjective $\ell$-adic Galois representation satisfies $\ell\le 7$, and completely classifies curves with $\ell=7$ as the smallest surjective prime. The authors develop a framework using modular curves $X_H$ and fiber products to reduce the problem to finite rational points on a small set of curves, then compute these points with Magma to identify a finite set of exceptional $j$-invariants. They show that, except for six specific $j$-invariants, the smallest surjective prime is at most $5$, and they exhibit explicit curves (including an example with smallest surjective prime $7$) where the bound is sharp. Consequently, at least one of the $2$-, $3$-, or $5$-adic representations is surjective for every non-CM $E/\mathbb{Q}$, providing a precise, computationally verifiable instance of Serre’s open image theme and contributing toward a uniform bound on surjective primes in the non-CM case. The work combines modular-curve techniques with explicit arithmetic geometry computations to yield a concrete, finite classification and an accessible computational roadmap.

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 $ρ_{E,\ell^\infty}$ 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 $ρ_{E,\ell^\infty}$ is surjective is at most $7$. Moreover, we completely classify all elliptic curves $E/\mathbb{Q}$ for which the smallest surjective prime is exactly $7$.

Theorems & Definitions (6)

• Theorem 1: Serre MR387283, 1972
• Theorem 2
• Example 3
• Example 4
• Lemma 5
• proof