{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:ED7DYMW2UAEODN7A4XV4EEWHIZ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"f149cf1bad0ab81387cc307b348f9771555da1590ca8b0414f1a55dffe020f84","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-10-15T02:52:07Z","title_canon_sha256":"7d61315095cc84b4bd19dcad84081891919f586b34f965b953a8743671ebf73b"},"schema_version":"1.0","source":{"id":"1210.3880","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.3880","created_at":"2026-05-18T00:42:44Z"},{"alias_kind":"arxiv_version","alias_value":"1210.3880v2","created_at":"2026-05-18T00:42:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.3880","created_at":"2026-05-18T00:42:44Z"},{"alias_kind":"pith_short_12","alias_value":"ED7DYMW2UAEO","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"ED7DYMW2UAEODN7A","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"ED7DYMW2","created_at":"2026-05-18T12:27:04Z"}],"graph_snapshots":[{"event_id":"sha256:3e33dda100f105a73eaa1d1f6b2dfb99ec03d52a4989b8b8ee0217e01ebe37ad","target":"graph","created_at":"2026-05-18T00:42:44Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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","authors_text":"Chantal David, Dimitris Koukoulopoulos, Ethan Smith, Vorrapan Chandee","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-10-15T02:52:07Z","title":"Group structures of elliptic curves over finite fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.3880","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:58238114583206fb3b66ed6a96e69fd07cd611af0ee0d17aefc36357780fddeb","target":"record","created_at":"2026-05-18T00:42:44Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"f149cf1bad0ab81387cc307b348f9771555da1590ca8b0414f1a55dffe020f84","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-10-15T02:52:07Z","title_canon_sha256":"7d61315095cc84b4bd19dcad84081891919f586b34f965b953a8743671ebf73b"},"schema_version":"1.0","source":{"id":"1210.3880","kind":"arxiv","version":2}},"canonical_sha256":"20fe3c32daa008e1b7e0e5ebc212c7465c8d63df1247b0a0a0b864dcd641ddfe","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"20fe3c32daa008e1b7e0e5ebc212c7465c8d63df1247b0a0a0b864dcd641ddfe","first_computed_at":"2026-05-18T00:42:44.747355Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:42:44.747355Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bP9lDblShuG67Nr+CgI3eCzoJNRv6HrjtBHlnbZMa4rYI7Z1NUnamR+soTyrC+jStz1Y+Pdkm5a0+QUpvrPFCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:42:44.747894Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.3880","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:58238114583206fb3b66ed6a96e69fd07cd611af0ee0d17aefc36357780fddeb","sha256:3e33dda100f105a73eaa1d1f6b2dfb99ec03d52a4989b8b8ee0217e01ebe37ad"],"state_sha256":"9d011d04aed822fc204e71be248f76a10eb54aabdbc32e8a0db36b22e2584e14"}