A uniform bound on the smallest surjective prime of an elliptic curve
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| Format: | Preprint |
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2025
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| _version_ | 1866915137976270848 |
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| author | Genao, Tyler Mayle, Jacob Rouse, Jeremy |
| author_facet | Genao, Tyler Mayle, Jacob Rouse, Jeremy |
| contents | 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 $7$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_02345 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A uniform bound on the smallest surjective prime of an elliptic curve Genao, Tyler Mayle, Jacob Rouse, Jeremy Number Theory 11G05 (Primary) 11F80 (Secondary) 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 $7$. |
| title | A uniform bound on the smallest surjective prime of an elliptic curve |
| topic | Number Theory 11G05 (Primary) 11F80 (Secondary) |
| url | https://arxiv.org/abs/2501.02345 |