{"paper":{"title":"Character Sums, Gaussian Hypergeometric Series, and a Family of Hyperelliptic Curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Mohammad Sadek","submitted_at":"2015-07-03T13:54:14Z","abstract_excerpt":"We study the character sums \\[\\phi_{(m,n)}(a,b)=\\sum_{x\\in\\mathbb{F}_q}\\phi\\left(x(x^{m}+a)(x^{n}+b)\\right),\\textrm{ and, } \\psi_{(m,n)}(a,b)=\\sum_{x\\in\\mathbb{F}_q}\\phi\\left((x^{m}+a)(x^{n}+b)\\right)\\] where $\\phi$ is the quadratic character defined over $\\mathbb{F}_q$. These sums are expressed in terms of Gaussian hypergeometric series over $\\mathbb{F}_q$. Then we use these expressions to exhibit the number of $\\mathbb{F}_q$-rational points on families of hyperelliptic curves and their Jacobian varieties."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.00914","kind":"arxiv","version":3},"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"}