{"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/open-image-computations-for-elliptic-curves","title":"Open image computations for elliptic curves over number fields","arxiv_id":"2403.16147","date":"2024-03-24","proceeding":null,"authors":["David Zywina"],"abstract":"For a non-CM elliptic curve $E$ defined over a number field $K$, the Galois action on its torsion points gives rise to a Galois representation $\\rho_E: Gal(\\overline{K}/K)\\to GL_2(\\widehat{\\mathbb{Z}})$ that is unique up to isomorphism. A renowned theorem of Serre says that the image of $\\rho_E$ is an open, and hence finite index, subgroup of $GL_2(\\widehat{\\mathbb{Z}})$. In an earlier work of the author, an algorithm was given, and implemented, that computed the image of $\\rho_E$ up to conjugacy in $GL_2(\\widehat{\\mathbb{Z}})$ in the special case $K=\\mathbb{Q}$. A fundamental ingredient of this earlier work was the Kronecker-Weber theorem whose conclusion fails for number fields $K\\neq \\mathbb{Q}$. We shall give an overview of an analogous algorithm for a general number field and work out the required group theory. We also give some bounds on the index in Serre's theorem for a typical elliptic curve over a fixed number field.","url_abs":"https://arxiv.org/abs/2403.16147v2","url_pdf":"https://arxiv.org/pdf/2403.16147v2.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":"open-image-computations-for-elliptic-curves","repo_url":"https://github.com/davidzywina/agreeablegroups","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}