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Let $\\mathcal{C}$ be the curve defined by the equation $y^{\\ell}=P(x)$, and take the points on $\\mathcal{C}$ to lie in the rectangle $[0,p-1]^2$. In this paper, we study the distribution of the number of points on $\\mathcal{C}$ inside a small rectangle among residue classes modulo $m$ when we move the rectangle around in $[0,p-1]^2$."},"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":"1110.4693","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-10-21T03:11:46Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"4389538023b1408a66ed8ec4528120b217f2cc5b3c5b64afe7dbdfd2c1d35951","abstract_canon_sha256":"df674f416cd2f395aff1be1e8f22bef942a555ec75ab4d849fd4dfa597716ae8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:10:35.294495Z","signature_b64":"67YQ0KylU4x3L2EB8mC4sh7LDT6Y7T2Dv2VH/ml57gISXB6emYHdAJ/Dq+iHLyZRCCcKf4QJu8pxTRoRSxudDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6a8ebe25ed87f01fcbbc4447744522b8d647f74f4409a900a34e30caca569c8f","last_reissued_at":"2026-05-18T04:10:35.293648Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:10:35.293648Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the distribution of the number of points on a family of curves over finite fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Alexandru Zaharescu, Kit-Ho Mak","submitted_at":"2011-10-21T03:11:46Z","abstract_excerpt":"Let $p$ be a large prime, $\\ell\\geq 2$ be a positive integer, $m\\geq 2$ be an integer relatively prime to $\\ell$ and $P(x)\\in\\mathbb{F}_p[x]$ be a polynomial which is not a complete $\\ell'$-th power for any $\\ell'$ for which $GCD(\\ell',\\ell)=1$. Let $\\mathcal{C}$ be the curve defined by the equation $y^{\\ell}=P(x)$, and take the points on $\\mathcal{C}$ to lie in the rectangle $[0,p-1]^2$. In this paper, we study the distribution of the number of points on $\\mathcal{C}$ inside a small rectangle among residue classes modulo $m$ when we move the rectangle around in $[0,p-1]^2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.4693","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1110.4693","created_at":"2026-05-18T04:10:35.293779+00:00"},{"alias_kind":"arxiv_version","alias_value":"1110.4693v1","created_at":"2026-05-18T04:10:35.293779+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.4693","created_at":"2026-05-18T04:10:35.293779+00:00"},{"alias_kind":"pith_short_12","alias_value":"NKHL4JPNQ7YB","created_at":"2026-05-18T12:26:37.096874+00:00"},{"alias_kind":"pith_short_16","alias_value":"NKHL4JPNQ7YB7S54","created_at":"2026-05-18T12:26:37.096874+00:00"},{"alias_kind":"pith_short_8","alias_value":"NKHL4JPN","created_at":"2026-05-18T12:26:37.096874+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/NKHL4JPNQ7YB7S54IRDXIRJCXD","json":"https://pith.science/pith/NKHL4JPNQ7YB7S54IRDXIRJCXD.json","graph_json":"https://pith.science/api/pith-number/NKHL4JPNQ7YB7S54IRDXIRJCXD/graph.json","events_json":"https://pith.science/api/pith-number/NKHL4JPNQ7YB7S54IRDXIRJCXD/events.json","paper":"https://pith.science/paper/NKHL4JPN"},"agent_actions":{"view_html":"https://pith.science/pith/NKHL4JPNQ7YB7S54IRDXIRJCXD","download_json":"https://pith.science/pith/NKHL4JPNQ7YB7S54IRDXIRJCXD.json","view_paper":"https://pith.science/paper/NKHL4JPN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1110.4693&json=true","fetch_graph":"https://pith.science/api/pith-number/NKHL4JPNQ7YB7S54IRDXIRJCXD/graph.json","fetch_events":"https://pith.science/api/pith-number/NKHL4JPNQ7YB7S54IRDXIRJCXD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NKHL4JPNQ7YB7S54IRDXIRJCXD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NKHL4JPNQ7YB7S54IRDXIRJCXD/action/storage_attestation","attest_author":"https://pith.science/pith/NKHL4JPNQ7YB7S54IRDXIRJCXD/action/author_attestation","sign_citation":"https://pith.science/pith/NKHL4JPNQ7YB7S54IRDXIRJCXD/action/citation_signature","submit_replication":"https://pith.science/pith/NKHL4JPNQ7YB7S54IRDXIRJCXD/action/replication_record"}},"created_at":"2026-05-18T04:10:35.293779+00:00","updated_at":"2026-05-18T04:10:35.293779+00:00"}