{"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/serre-s-uniformity-question-and-proper","title":"Serre's uniformity question and proper subgroups of $C_{ns}^+(p)$","arxiv_id":"2305.17780","date":"2023-05-28","proceeding":null,"authors":["Lorenzo Furio","Davide Lombardo"],"abstract":"Serre's uniformity question asks whether there exists a bound $N>0$ such that, for every non-CM elliptic curve $E$ over $\\mathbb{Q}$ and every prime $p>N$, the residual Galois representation $\\rho_{E,p}:\\operatorname{Gal}(\\overline{\\mathbb{Q}}/\\mathbb{Q}) \\to \\operatorname{Aut}(E[p])$ is surjective. The work of many authors has shown that, for $p>37$, this representation is either surjective or has image contained in the normaliser of a non-split Cartan subgroup $C_{ns}^+(p)$. Zywina has further proved that, whenever $\\rho_{E,p}$ is not surjective for $p>37$, its image is either $C_{ns}^+(p)$ or a certain subgroup $G(p)$ of $C_{ns}^+(p)$ of index $3$. Recently, Le Fourn and Lemos showed that the index-$3$ case cannot arise for $p>1.4 \\cdot 10^7$. We strengthen this result by proving that the image of $\\rho_{E, p}$ is not conjugate to $G(p)$ for any prime larger than $5$.","url_abs":"https://arxiv.org/abs/2305.17780v1","url_pdf":"https://arxiv.org/pdf/2305.17780v1.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":"serre-s-uniformity-question-and-proper","repo_url":"https://github.com/davidelombardomath/cartan-cubes","is_official":1,"mentioned_in_paper":1,"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}