{"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/explicit-open-images-for-elliptic-curves-over","title":"Explicit open images for elliptic curves over $\\mathbb{Q}$","arxiv_id":"2206.14959","date":"2022-06-30","proceeding":null,"authors":["David Zywina"],"abstract":"For a non-CM elliptic curve $E$ defined over $\\mathbb{Q}$, the Galois action on its torsion points gives rise to a Galois representation $\\rho_E: Gal(\\overline{\\mathbb{Q}}/\\mathbb{Q})\\to GL_2(\\widehat{\\mathbb{Z}})$ 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 $GL_2(\\widehat{\\mathbb{Z}})$. We describe an algorithm that computes the image of $\\rho_E$ up to conjugacy in $GL_2(\\widehat{\\mathbb{Z}})$; this algorithm is practical and has been implemented. Up to a positive answer to a uniformity question of Serre and finding all the rational points on a finite number of explicit modular curves of genus at least $2$, we give a complete classification of the groups $\\rho_E(Gal(\\overline{\\mathbb{Q}}/\\mathbb{Q}))\\cap SL_2(\\widehat{\\mathbb{Z}})$ and the indices $[GL_2(\\widehat{\\mathbb{Z}}):\\rho_E(Gal(\\overline{\\mathbb{Q}}/\\mathbb{Q}))]$ for non-CM elliptic curves $E/\\mathbb{Q}$. Much of the paper is dedicated to the efficient computation of modular curves via modular forms expressed in terms of Eisenstein series.","url_abs":"https://arxiv.org/abs/2206.14959v2","url_pdf":"https://arxiv.org/pdf/2206.14959v2.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":"explicit-open-images-for-elliptic-curves-over","repo_url":"https://github.com/davidzywina/openimage","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"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}