{"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/ternary-and-quaternary-curves-of-small-fixed","title":"Ternary and Quaternary Curves of Small Fixed Genus and Gonality with Many Rational Points","arxiv_id":"2010.07992","date":"2020-10-15","proceeding":null,"authors":["Xander Faber","Jon Grantham"],"abstract":"We extend the computations from our previous paper arXiv:2005.07054 to determine the maximum number of rational points on a curve over $\\mathbb{F}_3$ and $\\mathbb{F}_4$ with fixed gonality and small genus. We find, for example, that there is no curve of genus 5 and gonality 6 over a finite field. We propose two conjectures based on our data. First, an optimal curve of genus $g$ has gonality at most $\\lfloor \\frac{g+3}{2} \\rfloor$. Second, a curve of gonality $\\gamma$ and large genus over $\\mathbb{F}_q$ has $\\gamma(q+1)$ rational points.","url_abs":"https://arxiv.org/abs/2010.07992v2","url_pdf":"https://arxiv.org/pdf/2010.07992v2.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":"ternary-and-quaternary-curves-of-small-fixed","repo_url":"https://github.com/RationalPoint/gonality","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"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}