{"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/symbol-length-in-brauer-groups-of-elliptic","title":"Symbol Length in Brauer Groups of Elliptic Curves","arxiv_id":"2107.10886","date":"2021-07-22","proceeding":null,"authors":["Mateo Attanasio","Caroline Choi","Andrei Mandelshtam","Charlotte Ure"],"abstract":"Let $\\ell$ be an odd prime, and let $K$ be a field of characteristic not $2,3,$ or $\\ell$ containing a primitive $\\ell$-th root of unity. For an elliptic curve $E$ over $K$, we consider the standard Galois representation $$\\rho_{E,\\ell}: \\text{Gal}(\\overline{K}/K) \\rightarrow \\text{GL}_2(\\mathbb{F}_{\\ell}),$$ and denote the fixed field of its kernel by $L$. Recently, the last author gave an algorithm to compute elements in the Brauer group explicitly, deducing an upper bound of $2(\\ell+1)(\\ell-1)$ on the symbol length in $\\mathbin{_{\\ell}\\text{Br}(E)} / \\mathbin{_{\\ell}\\text{Br}(K)}$. More precisely, the symbol length is bounded above by $2[L:K]$. We improve this bound to $[L:K]-1$ if $\\ell \\nmid [L:K]$. Under the additional assumption that $\\text{Gal}(L/K)$ contains an element of order $d > 1$, we further reduce it to $(1-\\frac{1}{d})[L:K]$. In particular, these bounds hold for all CM elliptic curves, in which case we deduce a general upper bound of $\\ell + 1$. We provide an algorithm implemented in SageMath to compute these symbols explicitly over number fields.","url_abs":"https://arxiv.org/abs/2107.10886v2","url_pdf":"https://arxiv.org/pdf/2107.10886v2.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":"symbol-length-in-brauer-groups-of-elliptic","repo_url":"https://github.com/team-brauer/brauer-groups","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}