{"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/cubic-and-quartic-points-on-modular-curves","title":"Cubic and quartic points on modular curves using generalised symmetric Chabauty","arxiv_id":"2102.08236","date":"2021-02-16","proceeding":null,"authors":["Josha Box","Stevan Gajović","Pip Goodman"],"abstract":"Answering a question of Zureick-Brown, we determine the cubic points on the modular curves $X_0(N)$ for $N \\in \\{53,57,61,65,67,73\\}$ as well as the quartic points on $X_0(65)$. To do so, we develop a \"partially relative\" symmetric Chabauty method. Our results generalise current symmetric Chabauty theorems, and improve upon them by lowering the involved prime bound. For our curves a number of novelties occur. We prove a \"higher order\" Chabauty theorem to deal with these cases. Finally, to study the isolated quartic points on $X_0(65)$, we rigorously compute the full rational Mordell--Weil group of its Jacobian.","url_abs":"https://arxiv.org/abs/2102.08236v2","url_pdf":"https://arxiv.org/pdf/2102.08236v2.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":"cubic-and-quartic-points-on-modular-curves","repo_url":"https://github.com/joshabox/cubicpoints","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}