{"paper":{"title":"Group structures of elliptic curves over finite fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Chantal David, Dimitris Koukoulopoulos, Ethan Smith, Vorrapan Chandee","submitted_at":"2012-10-15T02:52:07Z","abstract_excerpt":"It is well-known that if $E$ is an elliptic curve over the finite field $\\mathbb{F}_p$, then $E(\\mathbb{F}_p)\\simeq\\mathbb{Z}/m\\mathbb{Z}\\times\\mathbb{Z}/mk\\mathbb{Z}$ for some positive integers $m, k$. Let $S(M,K)$ denote the set of pairs $(m,k)$ with $m\\le M$ and $k\\le K$ such that there exists an elliptic curve over some prime finite field whose group of points is isomorphic to $\\mathbb{Z}/m\\mathbb{Z}\\times\\mathbb{Z}/mk\\mathbb{Z}$. Banks, Pappalardi and Shparlinski recently conjectured that if $K\\le (\\log M)^{2-\\epsilon}$, then a density zero proportion of the groups in question actually ar"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.3880","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}