{"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/abelian-varieties-of-prescribed-order-over","title":"Abelian varieties of prescribed order over finite fields","arxiv_id":"2106.13651","date":"2021-06-25","proceeding":null,"authors":["Raymond van Bommel","Edgar Costa","Wanlin Li","Bjorn Poonen","Alexander Smith"],"abstract":"Given a prime power $q$ and $n \\gg 1$, we prove that every integer in a large subinterval of the Hasse--Weil interval $[(\\sqrt{q}-1)^{2n},(\\sqrt{q}+1)^{2n}]$ is $#A(\\mathbb{F}_q)$ for some geometrically simple ordinary principally polarized abelian variety $A$ of dimension $n$ over $\\mathbb{F}_q$. As a consequence, we generalize a result of Howe and Kedlaya for $\\mathbb{F}_2$ to show that for each prime power $q$, every sufficiently large positive integer is realizable, i.e., $#A(\\mathbb{F}_q)$ for some abelian variety $A$ over $\\mathbb{F}_q$. Our result also improves upon the best known constructions of sequences of simple abelian varieties with point counts towards the extremes of the Hasse--Weil interval. A separate argument determines, for fixed $n$, the largest subinterval of the Hasse--Weil interval consisting of realizable integers, asymptotically as $q \\to \\infty$; this gives an asymptotically optimal improvement of a 1998 theorem of DiPippo and Howe. Our methods are effective: We prove that if $q \\le 5$, then every positive integer is realizable, and for arbitrary $q$, every positive integer $\\ge q^{3 \\sqrt{q} \\log q}$ is realizable.","url_abs":"https://arxiv.org/abs/2106.13651v1","url_pdf":"https://arxiv.org/pdf/2106.13651v1.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":"abelian-varieties-of-prescribed-order-over","repo_url":"https://github.com/edgarcosta/abvar-fq-orders","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}