{"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/on-the-acyclicity-of-reductions-of-elliptic","title":"On the acyclicity of reductions of elliptic curves modulo primes in arithmetic progressions","arxiv_id":"2206.00872","date":"2022-06-02","proceeding":null,"authors":["Nathan Jones","Sung Min Lee"],"abstract":"Let $E$ be an elliptic curve defined over $\\mathbb{Q}$ and, for a prime $p$ of good reduction for $E$ let $\\tilde{E}_p$ denote the reduction of $E$ modulo $p$. Inspired by an elliptic curve analogue of Artin's primitive root conjecture posed by S. Lang and H. Trotter in 1977, J-P. Serre adapted methods of C. Hooley to prove a GRH-conditional asymptotic formula for the number of primes $p \\leq x$ for which the group $\\tilde{E}_p(\\mathbb{F}_p)$ is cyclic. More recently, Akbal and G\\\"{u}lo$\\breve{\\text{g}}$lu considered the question of cyclicity of $\\tilde{E}_p(\\mathbb{F}_p)$ under the additional restriction that $p$ lie in an arithmetic progression. In this note, we study the issue of which arithmetic progressions $a \\bmod n$ have the property that, for all but finitely many primes $p \\equiv a \\bmod n$, the group $\\tilde{E}_p(\\mathbb{F}_p)$ is not cyclic, answering a question of Akbal and G\\\"{u}lo$\\breve{\\text{g}}$lu on this issue.","url_abs":"https://arxiv.org/abs/2206.00872v3","url_pdf":"https://arxiv.org/pdf/2206.00872v3.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":"on-the-acyclicity-of-reductions-of-elliptic","repo_url":"https://github.com/ncjones-uic/acyclicreductions","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}