{"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/sporadic-cubic-torsion","title":"Sporadic Cubic Torsion","arxiv_id":"2007.13929","date":"2020-07-28","proceeding":null,"authors":["Maarten Derickx","Anastassia Etropolski","Mark van Hoeij","Jackson S. Morrow","David Zureick-Brown"],"abstract":"Let $K$ be a number field, and let $E/K$ be an elliptic curve over $K$. The Mordell--Weil theorem asserts that the $K$-rational points $E(K)$ of $E$ form a finitely generated abelian group. In this work, we complete the classification of the finite groups which appear as the torsion subgroup of $E(K)$ for $K$ a cubic number field. To do so, we determine the cubic points on the modular curves $X_1(N)$ for \\[N = 21, 22, 24, 25, 26, 28, 30, 32, 33, 35, 36, 39, 45, 65, 121.\\] As part of our analysis, we determine the complete list of $N$ for which $J_0(N)$ (resp., $J_1(N)$, resp., $J_1(2,2N)$) has rank 0. We also provide evidence to a generalized version of a conjecture of Conrad, Edixhoven, and Stein by proving that the torsion on $J_1(N)(\\mathbb{Q})$ is generated by $\\text{Gal}(\\bar{\\mathbb{Q}}/\\mathbb{Q})$-orbits of cusps of $X_1(N)_{\\bar{\\mathbb{Q}}}$ for $N\\leq 55$, $N \\neq 54$.","url_abs":"https://arxiv.org/abs/2007.13929v2","url_pdf":"https://arxiv.org/pdf/2007.13929v2.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":"sporadic-cubic-torsion","repo_url":"https://github.com/jmorrow4692/SporadicCubicTorsion","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}