{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2004:KBMPD2TWBW7OKSWR5MQCJ3CQ5G","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":"ba5005ca41ef1c54f2faa6c5d6a65d61763967ea25050aee7ac8c5c1efb0fb7c","cross_cats_sorted":[],"license":"","primary_cat":"math.NT","submitted_at":"2004-06-07T22:24:34Z","title_canon_sha256":"554f0ac16398c778686ab5708bdb49cff7bbd76a418caca0f7ef9b196339e6ae"},"schema_version":"1.0","source":{"id":"math/0406131","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0406131","created_at":"2026-07-04T14:38:41Z"},{"alias_kind":"arxiv_version","alias_value":"math/0406131v1","created_at":"2026-07-04T14:38:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0406131","created_at":"2026-07-04T14:38:41Z"},{"alias_kind":"pith_short_12","alias_value":"KBMPD2TWBW7O","created_at":"2026-07-04T14:38:41Z"},{"alias_kind":"pith_short_16","alias_value":"KBMPD2TWBW7OKSWR","created_at":"2026-07-04T14:38:41Z"},{"alias_kind":"pith_short_8","alias_value":"KBMPD2TW","created_at":"2026-07-04T14:38:41Z"}],"graph_snapshots":[{"event_id":"sha256:2560b9655be62c4f335b38b3c75049f868edded3780c76be447a9a10086d2f32","target":"graph","created_at":"2026-07-04T14:38:41Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/math/0406131/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let E/K be an elliptic curve defined over a number field, and let p be a prime number such that E(K) has full p-torsion. We show that the order of the p-part of the Shafarevich-Tate group of E/L is unbounded as L varies over degree p extensions of K. The proof uses O'Neil's period-index obstruction. We deduce the result from the fact that, under the same hypotheses, there exist infinitely many elements of the Weil-Chatelet group of E/K of period p and index p^2.","authors_text":"Pete L. Clark","cross_cats":[],"headline":"","license":"","primary_cat":"math.NT","submitted_at":"2004-06-07T22:24:34Z","title":"The period-index problem in WC-groups I: elliptic curves"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0406131","kind":"arxiv","version":1},"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:e9f9d2ccf42c812b28fd67f4e01fc20250f6f21c26aedfc04e73a96e800ec0ca","target":"record","created_at":"2026-07-04T14:38:41Z","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":"ba5005ca41ef1c54f2faa6c5d6a65d61763967ea25050aee7ac8c5c1efb0fb7c","cross_cats_sorted":[],"license":"","primary_cat":"math.NT","submitted_at":"2004-06-07T22:24:34Z","title_canon_sha256":"554f0ac16398c778686ab5708bdb49cff7bbd76a418caca0f7ef9b196339e6ae"},"schema_version":"1.0","source":{"id":"math/0406131","kind":"arxiv","version":1}},"canonical_sha256":"5058f1ea760dbee54ad1eb2024ec50e9a0fa12495eafa4bf8dc761bfafd0bd34","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5058f1ea760dbee54ad1eb2024ec50e9a0fa12495eafa4bf8dc761bfafd0bd34","first_computed_at":"2026-07-04T14:38:41.567110Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T14:38:41.567110Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LgMdb8D3WxVicSsnNhoQbHpVvqPuBjE5tpfB/PCm8VWcE6XhzzcYCUtUvVoSPZeW/y1Cm75Cdq0+RPzsa/NrCw==","signature_status":"signed_v1","signed_at":"2026-07-04T14:38:41.567513Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0406131","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e9f9d2ccf42c812b28fd67f4e01fc20250f6f21c26aedfc04e73a96e800ec0ca","sha256:2560b9655be62c4f335b38b3c75049f868edded3780c76be447a9a10086d2f32"],"state_sha256":"77c203b8d23284234c1b3f1185ebac181a50c6faea5045a358d8725f6e028271"}