{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:V6VKBHXFN7YCTHIYUAANCTQMXB","short_pith_number":"pith:V6VKBHXF","schema_version":"1.0","canonical_sha256":"afaaa09ee56ff0299d18a000d14e0cb87e4bded82208b3fc567bc163e134ca5a","source":{"kind":"arxiv","id":"2411.15663","version":1},"attestation_state":"computed","paper":{"title":"A Note on the Growth of Sha in Dihedral Extensions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jamie Bell","submitted_at":"2024-11-23T22:19:06Z","abstract_excerpt":"We provide a formula for the order of the Tate--Shafarevich group of elliptic curves over dihedral extensions of number fields of order $2n$, up to $4^{th}$ powers and primes dividing $n$. Specifically, for odd $n$ it is equal to the order of the Tate--Shafarevich group over the quadratic subextension. A similar formula holds for even $n$."},"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":"2411.15663","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2024-11-23T22:19:06Z","cross_cats_sorted":[],"title_canon_sha256":"6d590aaf29015f09940fd85fb90500b84732d502ca4966ebd738cc79b6b77c78","abstract_canon_sha256":"add720215a0d99a42ae51fa81a062229e42d28ccd68ab017988b0685f6bcf052"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:39:38.812421Z","signature_b64":"HgP28q7rN8hYxpdeugA7Chb/VfrQSsL32My7V56OTM+YYCy48mj3hs71HMgZUfVEgpw8hJpTsdVSjR/AYRj3CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"afaaa09ee56ff0299d18a000d14e0cb87e4bded82208b3fc567bc163e134ca5a","last_reissued_at":"2026-07-05T09:39:38.811888Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:39:38.811888Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Note on the Growth of Sha in Dihedral Extensions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jamie Bell","submitted_at":"2024-11-23T22:19:06Z","abstract_excerpt":"We provide a formula for the order of the Tate--Shafarevich group of elliptic curves over dihedral extensions of number fields of order $2n$, up to $4^{th}$ powers and primes dividing $n$. Specifically, for odd $n$ it is equal to the order of the Tate--Shafarevich group over the quadratic subextension. A similar formula holds for even $n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.15663","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.15663/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2411.15663","created_at":"2026-07-05T09:39:38.811958+00:00"},{"alias_kind":"arxiv_version","alias_value":"2411.15663v1","created_at":"2026-07-05T09:39:38.811958+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.15663","created_at":"2026-07-05T09:39:38.811958+00:00"},{"alias_kind":"pith_short_12","alias_value":"V6VKBHXFN7YC","created_at":"2026-07-05T09:39:38.811958+00:00"},{"alias_kind":"pith_short_16","alias_value":"V6VKBHXFN7YCTHIY","created_at":"2026-07-05T09:39:38.811958+00:00"},{"alias_kind":"pith_short_8","alias_value":"V6VKBHXF","created_at":"2026-07-05T09:39:38.811958+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/V6VKBHXFN7YCTHIYUAANCTQMXB","json":"https://pith.science/pith/V6VKBHXFN7YCTHIYUAANCTQMXB.json","graph_json":"https://pith.science/api/pith-number/V6VKBHXFN7YCTHIYUAANCTQMXB/graph.json","events_json":"https://pith.science/api/pith-number/V6VKBHXFN7YCTHIYUAANCTQMXB/events.json","paper":"https://pith.science/paper/V6VKBHXF"},"agent_actions":{"view_html":"https://pith.science/pith/V6VKBHXFN7YCTHIYUAANCTQMXB","download_json":"https://pith.science/pith/V6VKBHXFN7YCTHIYUAANCTQMXB.json","view_paper":"https://pith.science/paper/V6VKBHXF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2411.15663&json=true","fetch_graph":"https://pith.science/api/pith-number/V6VKBHXFN7YCTHIYUAANCTQMXB/graph.json","fetch_events":"https://pith.science/api/pith-number/V6VKBHXFN7YCTHIYUAANCTQMXB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/V6VKBHXFN7YCTHIYUAANCTQMXB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/V6VKBHXFN7YCTHIYUAANCTQMXB/action/storage_attestation","attest_author":"https://pith.science/pith/V6VKBHXFN7YCTHIYUAANCTQMXB/action/author_attestation","sign_citation":"https://pith.science/pith/V6VKBHXFN7YCTHIYUAANCTQMXB/action/citation_signature","submit_replication":"https://pith.science/pith/V6VKBHXFN7YCTHIYUAANCTQMXB/action/replication_record"}},"created_at":"2026-07-05T09:39:38.811958+00:00","updated_at":"2026-07-05T09:39:38.811958+00:00"}