{"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/triples-of-singular-moduli-with-rational","title":"Triples of singular moduli with rational product","arxiv_id":"2005.04164","date":"2020-05-08","proceeding":null,"authors":["Guy Fowler"],"abstract":"We show that all triples $(x_1,x_2,x_3)$ of singular moduli satisfying $x_1 x_2 x_3 \\in \\mathbb{Q}^{\\times}$ are \"trivial\". That is, either $x_1, x_2, x_3 \\in \\mathbb{Q}$; some $x_i \\in \\mathbb{Q}$ and the remaining $x_j, x_k$ are distinct, of degree $2$, and conjugate over $\\mathbb{Q}$; or $x_1, x_2, x_3$ are pairwise distinct, of degree $3$, and conjugate over $\\mathbb{Q}$. This theorem is best possible and is the natural three dimensional analogue of a result of Bilu, Luca, and Pizarro-Madariaga in two dimensions. It establishes an explicit version of the Andr\\'e--Oort conjecture for the family of subvarieties $V_{\\alpha} \\subset \\mathbb{C}^3$ defined by an equation $x_1 x_2 x_3 = \\alpha \\in \\mathbb{Q}$.","url_abs":"http://arxiv.org/abs/2005.04164v2","url_pdf":"http://arxiv.org/pdf/2005.04164v2.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":"triples-of-singular-moduli-with-rational","repo_url":"https://github.com/guyfowler/rationaltriples","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}