{"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/a-classification-of-genus-0-modular-curves","title":"A Classification of Genus 0 Modular Curves with Rational Points","arxiv_id":"2105.14623","date":"2021-05-30","proceeding":null,"authors":["Rakvi"],"abstract":"Let $E$ be a non-CM elliptic curve defined over $\\mathbb {Q}$. Fix an algebraic closure $\\overline{\\mathbb {Q}}$ of $\\mathbb {Q}$. We get a Galois representation \\[\\rho_E \\colon Gal(\\overline{\\mathbb {Q}}/\\mathbb {Q}) \\to GL_2(\\hat{\\mathbb {Z}})\\] associated to $E$ by choosing a compatible bases for the $N$-torsion subgroups of $E(\\overline{\\mathbb {Q}}).$ Associated to an open subgroup $G$ of $GL_2(\\hat{\\mathbb {Z}})$ satisfying $-I \\in G$ and $det(G)=\\hat{\\mathbb {Z}}^{\\times}$, we have the modular curve $(X_G,\\pi_G)$ over $\\mathbb {Q}$ which loosely parametrises elliptic curves $E$ such that the image of $\\rho_E$ is conjugate to a subgroup of $G^t.$ In this article we give a complete classification of all such genus $0$ modular curves that have a rational point. This classification is given in finitely many families. Moreover, each such modular curve can be explicitly computed.","url_abs":"https://arxiv.org/abs/2105.14623v2","url_pdf":"https://arxiv.org/pdf/2105.14623v2.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":"a-classification-of-genus-0-modular-curves","repo_url":"https://github.com/Rakvi6893/Classification-of-genus-0-Modular-Curves-that-have-a-Rational-Point","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}