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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1507.00914","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-07-03T13:54:14Z","cross_cats_sorted":[],"title_canon_sha256":"389f2cdd2015975eadb1e37944d22f961c7e4c16703f163ab5c2195d1c800746","abstract_canon_sha256":"3c3553e4b08de58a47bd3d03653f9da0fa3eb0eb295cb2ab7ed0089aa82f2d0e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:24.638391Z","signature_b64":"MItIVL4dDiaQ7iR5f+rDCy1hdAh36xEWYQyzCbn59LSXuVhwrxM3q7L7wrd26idzjNsMhQviJE48PgXn742fDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9ad826d31f4b0487a3e0d0262a5f5ef2ea5181cddcb8c477addb1038a70a6101","last_reissued_at":"2026-05-18T00:53:24.637980Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:24.637980Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\mathbb{F}_q$. 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