A uniform bound on the smallest surjective prime of an elliptic curve
Abstract
Abstract Let $$E/\mathbb {Q}$$ E / Q be an elliptic curve without complex multiplication. A well-known theorem of Serre asserts that the $$\ell $$ ℓ -adic Galois representation $$\rho _{E,\ell ^\infty }$$ ρ E , ℓ ∞ is surjective for all but finitely many prime numbers $$\ell $$ ℓ . 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 $$\ell $$ ℓ such that $$\rho _{E,\ell ^\infty }$$ ρ E , ℓ ∞ is surjective is at most 7. Moreover, we completely classify all elliptic curves $$E/\mathbb {Q}$$ E / Q for which the smallest surjective prime is exactly 7.
Fields of Study & Subjects
Algebraic Geometry and Number Theory
Historical and Political Studies
Analytic Number Theory Research
Publication details
- Journal / source
- The Ramanujan Journal
- Publisher
- Springer Science and Business Media LLC
- Volume
- 70
- Issue
- 2
- Publication date
- 12 May 2026
- Language
- EN