{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:CFGBZOLS5XI3HGORKKHRAVRQAT","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":"bce2f30803c04eeb99cc01d76b388e35bd221a078ce0d35d769d4125015dcb01","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-09T12:52:46Z","title_canon_sha256":"4691c7b04825ca23a32528144424fb15547f7712bd9314cf5a5636357fdcab68"},"schema_version":"1.0","source":{"id":"1407.2465","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.2465","created_at":"2026-05-18T02:48:01Z"},{"alias_kind":"arxiv_version","alias_value":"1407.2465v1","created_at":"2026-05-18T02:48:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.2465","created_at":"2026-05-18T02:48:01Z"},{"alias_kind":"pith_short_12","alias_value":"CFGBZOLS5XI3","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_16","alias_value":"CFGBZOLS5XI3HGOR","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_8","alias_value":"CFGBZOLS","created_at":"2026-05-18T12:28:22Z"}],"graph_snapshots":[{"event_id":"sha256:6cc72157c7c30c444d00c7c7c2ada546d7cfdf89cbe3f1052e0da14977d57de7","target":"graph","created_at":"2026-05-18T02:48:01Z","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":"For an elliptic curve over the rational number field and a prime number $p$, we study the structure of the classical Selmer group of $p$-power torsion points. In our previous paper \\cite{Ku6}, assuming the main conjecture and the non-degeneracy of the $p$-adic height pairing, we proved that the structure of the Selmer group with respect to $p$-power torsion points is determined by some analytic elements $\\tilde{\\delta}_{m}$ defined from modular symbols. In this paper, we do not assume the main conjecture nor the non-degeneracy of the $p$-adic height pairing, and study the structure of Selmer g","authors_text":"Masato Kurihara","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-09T12:52:46Z","title":"The structure of Selmer groups of elliptic curves and modular symbols"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.2465","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":"sha256:160a4d77a149d63d1af96d458685d33012c59977b744e1b6c47fbd6f6fca39f4","target":"record","created_at":"2026-05-18T02:48:01Z","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":"bce2f30803c04eeb99cc01d76b388e35bd221a078ce0d35d769d4125015dcb01","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-09T12:52:46Z","title_canon_sha256":"4691c7b04825ca23a32528144424fb15547f7712bd9314cf5a5636357fdcab68"},"schema_version":"1.0","source":{"id":"1407.2465","kind":"arxiv","version":1}},"canonical_sha256":"114c1cb972edd1b399d1528f10563004ee5f7c99d03c8c962fc9ebe309df080d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"114c1cb972edd1b399d1528f10563004ee5f7c99d03c8c962fc9ebe309df080d","first_computed_at":"2026-05-18T02:48:01.500861Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:48:01.500861Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mjy1vWiSaYVyV8MrUbWLBPsX4dsxNXvZWEO4FEPnrSekCzr7Y9DHy8MddB8Yn37miQtae+brCGukd7tRxeJvAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:48:01.501291Z","signed_message":"canonical_sha256_bytes"},"source_id":"1407.2465","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:160a4d77a149d63d1af96d458685d33012c59977b744e1b6c47fbd6f6fca39f4","sha256:6cc72157c7c30c444d00c7c7c2ada546d7cfdf89cbe3f1052e0da14977d57de7"],"state_sha256":"bb7e4cab768436bd0b6917d247998ad6bbfb77cbc34f0e9fad125b0ab4e8a578"}