{"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/primitive-algebraic-points-on-curves","title":"Primitive algebraic points on curves","arxiv_id":"2306.17772","date":"2023-06-30","proceeding":null,"authors":["Maleeha Khawaja","Samir Siksek"],"abstract":"A number field $K$ is primitive if $K$ and $\\mathbb{Q}$ are the only subextensions of $K$. Let $C$ be a curve defined over $\\mathbb{Q}$. We call an algebraic point $P\\in C(\\overline{\\mathbb{Q}})$ primitive if the number field $\\mathbb{Q}(P)$ is primitive. We present several sets of sufficient conditions for a curve $C$ to have finitely many primitive points of a given degree $d$. For example, let $C/\\mathbb{Q}$ be a hyperelliptic curve of genus $g$, and let $3 \\le d \\le g-1$. Suppose that the Jacobian $J$ of $C$ is simple. We show that $C$ has only finitely many primitive degree $d$ points, and in particular it has only finitely many degree $d$ points with Galois group $S_d$ or $A_d$. However, for any even $d \\ge 4$, a hyperelliptic curve $C/\\mathbb{Q}$ has infinitely many imprimitive degree $d$ points whose Galois group is a subgroup of $S_2 \\wr S_{d/2}$.","url_abs":"https://arxiv.org/abs/2306.17772v4","url_pdf":"https://arxiv.org/pdf/2306.17772v4.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":"primitive-algebraic-points-on-curves","repo_url":"https://github.com/maleehakhawaja/primitive","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}