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We explicitly find the number of $\\mathbb{F}_q$-points on $E_d$ and $E'_d$ in terms of special values of ${_{d}}F_{d-1}$ and ${_{d-1}}F_{d-2}$ Gaussian hypergeometric series with characters of orders $d-1$, $d$, $2(d-1)$, $2d$, and $2d(d-1)$ as parameters. This gives a solution to a problem posed by Ken Ono \\cite[p. 204]{ono2} on special values of ${_{n+1}}F_n$ Gaussian hypergeometric series for $n > 2$. We"},"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":"1311.4695","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-11-19T11:16:40Z","cross_cats_sorted":[],"title_canon_sha256":"2c081c462dacf4d6b2460b24ecf5f300cdc1114bfb04d5d111d7012beab3a09a","abstract_canon_sha256":"15d6247c1565534e4d6c18c2ab681586d4f6e4a9e05c1d08b1a434ec834c80db"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:06:43.686964Z","signature_b64":"SDwwJlZIL1/OjlzyitejxVh2QVu3DRL+736yN1zFAb70/wNzv+RWNqNIIrQTns6UVsmNDqq2vg79lkpwcsR1BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a55d19eef44508950acb466a861dc6a6324dec59c6c0d01f3edd02486ec685ce","last_reissued_at":"2026-05-18T03:06:43.686367Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:06:43.686367Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hyperelliptic curves over $\\mathbb{F}_q$ and Gaussian hypergeometric series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Gautam Kalita, Rupam Barman","submitted_at":"2013-11-19T11:16:40Z","abstract_excerpt":"Let $d\\geq2$ be an integer. Denote by $E_d$ and $E'_{d}$ the hyperelliptic curves over $\\mathbb{F}_q$ given by $$E_d: y^2=x^d+ax+b~~~ \\text{and} ~~~E'_d: y^2=x^d+ax^{d-1}+b,$$ respectively. We explicitly find the number of $\\mathbb{F}_q$-points on $E_d$ and $E'_d$ in terms of special values of ${_{d}}F_{d-1}$ and ${_{d-1}}F_{d-2}$ Gaussian hypergeometric series with characters of orders $d-1$, $d$, $2(d-1)$, $2d$, and $2d(d-1)$ as parameters. This gives a solution to a problem posed by Ken Ono \\cite[p. 204]{ono2} on special values of ${_{n+1}}F_n$ Gaussian hypergeometric series for $n > 2$. 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