{"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/improved-bounds-for-serre-s-open-image","title":"Improved bounds for Serre's open image theorem","arxiv_id":"2501.00202","date":"2024-12-31","proceeding":null,"authors":["Imin Chen","Joshua Swidinsky"],"abstract":"Let $E$ be an elliptic curve over the rationals which does not have complex multiplication. Serre showed that the adelic representation attached to $E/\\mathbb{Q}$ has open image, and in particular there is a minimal natural number $C_E$ such that the mod $\\ell$ representation $\\bar{\\rho}_{E,\\ell}$ is surjective for any prime $\\ell > C_E$. Assuming the Generalized Riemann Hypothesis, Mayle-Wang gave explicit bounds for $C_E$ which are logarithmic in the conductor of $E$ and have explicit constants. The method is based on using effective forms of the Chebotarev density theorem together with the Faltings-Serre method, in particular, using the `deviation group' of the $2$-adic representations attached to two elliptic curves. By considering quotients of the deviation group and a characterization of the images of the $2$-adic representation $\\rho_{E,2}$ by Rouse and Zureick-Brown, we show in this paper how to further reduce the constants in Mayle-Wang's results. Another result of independent interest are improved effective isogeny theorems for elliptic curves over the rationals.","url_abs":"https://arxiv.org/abs/2501.00202v1","url_pdf":"https://arxiv.org/pdf/2501.00202v1.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":"improved-bounds-for-serre-s-open-image","repo_url":"https://github.com/ichensfuca/ChenSwidinsky","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}