{"paper":{"title":"Edwards Curves and Gaussian Hypergeometric Series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Mohammad Sadek, Nermine El-Sissi","submitted_at":"2015-01-14T21:54:40Z","abstract_excerpt":"Let $E$ be an elliptic curve described by either an Edwards model or a twisted Edwards model over $\\mathbb{F}_p$, namely, $E$ is defined by one of the following equations $x^2+y^2=a^2(1+x^2y^2),\\, a^5-a\\not\\equiv 0$ mod $p$, or, $ax^2+y^2=1+dx^2y^2,\\,ad(a-d)\\not\\equiv0$ mod $p$, respectively. We express the number of rational points of $E$ over $\\mathbb{F}_p$ using the Gaussian hypergeometric series $\\displaystyle {_2F_1}\\left(\\begin{matrix}\n  \\phi&\\phi\n  {} & \\epsilon\n  \\end{matrix}\\Big| x\\right)$ where $\\epsilon$ and $\\phi$ are the trivial and quadratic characters over $\\mathbb{F}_p$ respect"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.03526","kind":"arxiv","version":1},"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"}