{"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/possible-indices-for-the-galois-image-of","title":"Possible indices for the Galois image of elliptic curves over Q","arxiv_id":"1508.07663","date":"2015-08-31","proceeding":null,"authors":["David Zywina"],"abstract":"For a non-CM elliptic curve $E$ over the rationals, the Galois action on its torsion points can be expressed in terms of a Galois representation $\\rho_E : G \\to GL_2(\\hat{\\mathbb{Z}})$, where $G$ is the absolute Galois group of the rationals. A well-known theorem of Serre says that the image of $\\rho_E$ is open and hence has finite index in $GL_2(\\hat{\\mathbb{Z}})$. We will study what indices are possible assuming that we are willing to exclude a finite number of possible $j$-invariants from consideration. For example, we will show that there is a finite set $J$ of rational numbers such that if $E/\\mathbb{Q}$ is a non-CM elliptic curve with $j$-invariant not in $ J$ and with surjective mod $\\ell$ representations for all $\\ell >37$ (which conjecturally always holds), then the index $[GL_2(\\hat{\\mathbb{Z}}) : \\rho_E(G)]$ lies in the set \\[ I:= \\left\\{\\begin{array}{c}2, 4, 6, 8, 10, 12, 16, 20, 24, 30, 32, 36, 40, 48, 54, 60, 72, 84, 96, 108, 112,120, 144, \\\\192, 220, 240, 288, 336, 360, 384, 504, 576, 768, 864, 1152, 1200, 1296, 1536 \\end{array}\\right\\}. \\] Moreover, $I$ is the minimal set with this property.","url_abs":"https://arxiv.org/abs/1508.07663v2","url_pdf":"https://arxiv.org/pdf/1508.07663v2.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":"possible-indices-for-the-galois-image-of","repo_url":"https://github.com/davidzywina/possibleindices","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"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}