{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-uniform-bound-on-the-smallest-surjective","title":"A uniform bound on the smallest surjective prime of an elliptic curve","arxiv_id":"2501.02345","date":"2025-01-04","proceeding":null,"authors":["Tyler Genao","Jacob Mayle","Jeremy Rouse"],"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 $7$.","url_abs":"https://arxiv.org/abs/2501.02345v2","url_pdf":"https://arxiv.org/pdf/2501.02345v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-uniform-bound-on-the-smallest-surjective","repo_url":"https://github.com/maylejacobj/smallestsurjectiveprime","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}