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arXiv · 2501.02345

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

Tyler Genao·Jacob Mayle·Jeremy Rouse
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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 $7$.

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Tyler Genao, Jacob Mayle, Jeremy Rouse. 2025-01-04. A uniform bound on the smallest surjective prime of an elliptic curve. https://arxiv.org/abs/2501.02345

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