{"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/equations-in-three-singular-moduli-the-equal","title":"Equations in three singular moduli: the equal exponent case","arxiv_id":"2105.12696","date":"2021-05-26","proceeding":null,"authors":["Guy Fowler"],"abstract":"Let $a \\in \\mathbb{Z}_{>0}$ and $\\epsilon_1, \\epsilon_2, \\epsilon_3 \\in \\{\\pm 1\\}$. We classify explicitly all singular moduli $x_1, x_2, x_3$ satisfying either $\\epsilon_1 x_1^a + \\epsilon_2 x_2^a + \\epsilon_3 x_3^a \\in \\mathbb{Q}$ or $(x_1^{\\epsilon_1} x_2^{\\epsilon_2} x_3^{\\epsilon_3})^{a} \\in \\mathbb{Q}^{\\times}$. In particular, we show that all the solutions in singular moduli $x_1, x_2, x_3$ to the Fermat equations $x_1^a + x_2^a + x_3^a= 0$ and $x_1^a + x_2^a - x_3^a= 0$ satisfy $x_1 x_2 x_3 = 0$. Our proofs use a generalisation of a result of Faye and Riffaut on the fields generated by sums and products of two singular moduli, which we also establish.","url_abs":"https://arxiv.org/abs/2105.12696v2","url_pdf":"https://arxiv.org/pdf/2105.12696v2.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":"equations-in-three-singular-moduli-the-equal","repo_url":"https://github.com/guyfowler/three_singular_moduli","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}