{"paper":{"title":"Some properties of the distribution of the numbers of points on elliptic curves over a finite prime field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"James Mc Laughlin, Saiying He","submitted_at":"2019-01-03T03:59:30Z","abstract_excerpt":"Let $p \\geq 5$ be a prime and for $a, b \\in \\mathbb{F}_{p}$, let $E_{a,b}$ denote the elliptic curve over $\\mathbb{F}_{p}$ with equation $y^2=x^3+a\\,x + b$. As usual define the trace of Frobenius $a_{p,\\,a,\\,b}$ by \\begin{equation*}\n  \\#E_{a,b}(\\mathbb{F}_{p}) = p+1 -a_{p,\\,a,\\,b}. \\end{equation*} We use elementary facts about exponential sums and known results about binary quadratic forms over finite fields to evaluate the sums $\\sum_{t\\in\\mathbb{F}_{p}} a_{p,\\, t,\\, b}$, $\\sum _{t \\in \\mathbb{F}_{p}} a_{p,\\,a,\\, t}$,\n  $ \\sum_{t=0}^{p-1}a_{p,\\,t,\\,b}^{2}$, $ \\sum_{t=0}^{p-1}a_{p,\\,a,\\,t}^{2}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.00604","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"}