{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:ED7DYMW2UAEODN7A4XV4EEWHIZ","short_pith_number":"pith:ED7DYMW2","schema_version":"1.0","canonical_sha256":"20fe3c32daa008e1b7e0e5ebc212c7465c8d63df1247b0a0a0b864dcd641ddfe","source":{"kind":"arxiv","id":"1210.3880","version":2},"attestation_state":"computed","paper":{"title":"Group structures of elliptic curves over finite fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Chantal David, Dimitris Koukoulopoulos, Ethan Smith, Vorrapan Chandee","submitted_at":"2012-10-15T02:52:07Z","abstract_excerpt":"It is well-known that if $E$ is an elliptic curve over the finite field $\\mathbb{F}_p$, then $E(\\mathbb{F}_p)\\simeq\\mathbb{Z}/m\\mathbb{Z}\\times\\mathbb{Z}/mk\\mathbb{Z}$ for some positive integers $m, k$. Let $S(M,K)$ denote the set of pairs $(m,k)$ with $m\\le M$ and $k\\le K$ such that there exists an elliptic curve over some prime finite field whose group of points is isomorphic to $\\mathbb{Z}/m\\mathbb{Z}\\times\\mathbb{Z}/mk\\mathbb{Z}$. Banks, Pappalardi and Shparlinski recently conjectured that if $K\\le (\\log M)^{2-\\epsilon}$, then a density zero proportion of the groups in question actually ar"},"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":"1210.3880","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-10-15T02:52:07Z","cross_cats_sorted":[],"title_canon_sha256":"7d61315095cc84b4bd19dcad84081891919f586b34f965b953a8743671ebf73b","abstract_canon_sha256":"f149cf1bad0ab81387cc307b348f9771555da1590ca8b0414f1a55dffe020f84"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:44.747894Z","signature_b64":"bP9lDblShuG67Nr+CgI3eCzoJNRv6HrjtBHlnbZMa4rYI7Z1NUnamR+soTyrC+jStz1Y+Pdkm5a0+QUpvrPFCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"20fe3c32daa008e1b7e0e5ebc212c7465c8d63df1247b0a0a0b864dcd641ddfe","last_reissued_at":"2026-05-18T00:42:44.747355Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:44.747355Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Group structures of elliptic curves over finite fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Chantal David, Dimitris Koukoulopoulos, Ethan Smith, Vorrapan Chandee","submitted_at":"2012-10-15T02:52:07Z","abstract_excerpt":"It is well-known that if $E$ is an elliptic curve over the finite field $\\mathbb{F}_p$, then $E(\\mathbb{F}_p)\\simeq\\mathbb{Z}/m\\mathbb{Z}\\times\\mathbb{Z}/mk\\mathbb{Z}$ for some positive integers $m, k$. Let $S(M,K)$ denote the set of pairs $(m,k)$ with $m\\le M$ and $k\\le K$ such that there exists an elliptic curve over some prime finite field whose group of points is isomorphic to $\\mathbb{Z}/m\\mathbb{Z}\\times\\mathbb{Z}/mk\\mathbb{Z}$. Banks, Pappalardi and Shparlinski recently conjectured that if $K\\le (\\log M)^{2-\\epsilon}$, then a density zero proportion of the groups in question actually ar"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.3880","kind":"arxiv","version":2},"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":"1210.3880","created_at":"2026-05-18T00:42:44.747435+00:00"},{"alias_kind":"arxiv_version","alias_value":"1210.3880v2","created_at":"2026-05-18T00:42:44.747435+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.3880","created_at":"2026-05-18T00:42:44.747435+00:00"},{"alias_kind":"pith_short_12","alias_value":"ED7DYMW2UAEO","created_at":"2026-05-18T12:27:04.183437+00:00"},{"alias_kind":"pith_short_16","alias_value":"ED7DYMW2UAEODN7A","created_at":"2026-05-18T12:27:04.183437+00:00"},{"alias_kind":"pith_short_8","alias_value":"ED7DYMW2","created_at":"2026-05-18T12:27:04.183437+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/ED7DYMW2UAEODN7A4XV4EEWHIZ","json":"https://pith.science/pith/ED7DYMW2UAEODN7A4XV4EEWHIZ.json","graph_json":"https://pith.science/api/pith-number/ED7DYMW2UAEODN7A4XV4EEWHIZ/graph.json","events_json":"https://pith.science/api/pith-number/ED7DYMW2UAEODN7A4XV4EEWHIZ/events.json","paper":"https://pith.science/paper/ED7DYMW2"},"agent_actions":{"view_html":"https://pith.science/pith/ED7DYMW2UAEODN7A4XV4EEWHIZ","download_json":"https://pith.science/pith/ED7DYMW2UAEODN7A4XV4EEWHIZ.json","view_paper":"https://pith.science/paper/ED7DYMW2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1210.3880&json=true","fetch_graph":"https://pith.science/api/pith-number/ED7DYMW2UAEODN7A4XV4EEWHIZ/graph.json","fetch_events":"https://pith.science/api/pith-number/ED7DYMW2UAEODN7A4XV4EEWHIZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ED7DYMW2UAEODN7A4XV4EEWHIZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ED7DYMW2UAEODN7A4XV4EEWHIZ/action/storage_attestation","attest_author":"https://pith.science/pith/ED7DYMW2UAEODN7A4XV4EEWHIZ/action/author_attestation","sign_citation":"https://pith.science/pith/ED7DYMW2UAEODN7A4XV4EEWHIZ/action/citation_signature","submit_replication":"https://pith.science/pith/ED7DYMW2UAEODN7A4XV4EEWHIZ/action/replication_record"}},"created_at":"2026-05-18T00:42:44.747435+00:00","updated_at":"2026-05-18T00:42:44.747435+00:00"}