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

2025/01/04 by Genao, Tyler, Mayle, Jacob, Rouse, Jeremy · 1 citation
#11G05 (Primary) 11F80 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2501.02345

Abstract

Let E/ℚ be an elliptic curve without complex multiplication. A well-known theorem of Serre asserts that the ℓ-adic Galois representation ρ_E,ℓ^∞ is surjective for all but finitely many prime numbers 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 ℓ such that ρ_E,ℓ^∞ is surjective is at most 7. Moreover, we completely classify all elliptic curves E/ℚ for which the smallest surjective prime is exactly 7.

Cited by

• The possible adelic indices for elliptic curves admitting a rational cyclic isogeny

Related

• An Improved Bound Towards a Conjecture of Serre on Surjective Galois Representations
• On the surjectivity of mod \ℓ representations associated to elliptic\n curves
• On the surjectivity of mod ℓℓ representations associated to elliptic curves
• Elliptic mod \ℓ Galois representations which are not minimally\n elliptic
• Galois representations and composite moduli