{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:V6VKBHXFN7YCTHIYUAANCTQMXB","short_pith_number":"pith:V6VKBHXF","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"},"canonical_sha256":"afaaa09ee56ff0299d18a000d14e0cb87e4bded82208b3fc567bc163e134ca5a","source":{"kind":"arxiv","id":"2411.15663","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.15663","created_at":"2026-07-05T09:39:38Z"},{"alias_kind":"arxiv_version","alias_value":"2411.15663v1","created_at":"2026-07-05T09:39:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.15663","created_at":"2026-07-05T09:39:38Z"},{"alias_kind":"pith_short_12","alias_value":"V6VKBHXFN7YC","created_at":"2026-07-05T09:39:38Z"},{"alias_kind":"pith_short_16","alias_value":"V6VKBHXFN7YCTHIY","created_at":"2026-07-05T09:39:38Z"},{"alias_kind":"pith_short_8","alias_value":"V6VKBHXF","created_at":"2026-07-05T09:39:38Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:V6VKBHXFN7YCTHIYUAANCTQMXB","target":"record","payload":{"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"},"canonical_sha256":"afaaa09ee56ff0299d18a000d14e0cb87e4bded82208b3fc567bc163e134ca5a","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"},"source_kind":"arxiv","source_id":"2411.15663","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T09:39:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8Rbn0pnXYcuoaCoTaIlt5wqMr3qCTaePlZvl+VF2xRPLR8pdA+VNr6eHa66Ow/23d4+VjcK4Y8vkXPGMaX1eCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T22:35:46.590786Z"},"content_sha256":"02d15fa18b6d3dc40713dc4dbedff6e74feca293848c877bc84887c3d0f9baa7","schema_version":"1.0","event_id":"sha256:02d15fa18b6d3dc40713dc4dbedff6e74feca293848c877bc84887c3d0f9baa7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:V6VKBHXFN7YCTHIYUAANCTQMXB","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T09:39:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wVp23VayjBIy9BTSAVATAa8Nm8nAVmRYegXkQuxeAoG98M94Co0QTExj+D9c/+Ie+u9wNlfMR8n6OZXcxOUCAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T22:35:46.591171Z"},"content_sha256":"6e680370558184fa45d11794c51ec48a453da452fde4e2af12300f0cfaca41a0","schema_version":"1.0","event_id":"sha256:6e680370558184fa45d11794c51ec48a453da452fde4e2af12300f0cfaca41a0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/V6VKBHXFN7YCTHIYUAANCTQMXB/bundle.json","state_url":"https://pith.science/pith/V6VKBHXFN7YCTHIYUAANCTQMXB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/V6VKBHXFN7YCTHIYUAANCTQMXB/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-12T22:35:46Z","links":{"resolver":"https://pith.science/pith/V6VKBHXFN7YCTHIYUAANCTQMXB","bundle":"https://pith.science/pith/V6VKBHXFN7YCTHIYUAANCTQMXB/bundle.json","state":"https://pith.science/pith/V6VKBHXFN7YCTHIYUAANCTQMXB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/V6VKBHXFN7YCTHIYUAANCTQMXB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:V6VKBHXFN7YCTHIYUAANCTQMXB","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":"add720215a0d99a42ae51fa81a062229e42d28ccd68ab017988b0685f6bcf052","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2024-11-23T22:19:06Z","title_canon_sha256":"6d590aaf29015f09940fd85fb90500b84732d502ca4966ebd738cc79b6b77c78"},"schema_version":"1.0","source":{"id":"2411.15663","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.15663","created_at":"2026-07-05T09:39:38Z"},{"alias_kind":"arxiv_version","alias_value":"2411.15663v1","created_at":"2026-07-05T09:39:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.15663","created_at":"2026-07-05T09:39:38Z"},{"alias_kind":"pith_short_12","alias_value":"V6VKBHXFN7YC","created_at":"2026-07-05T09:39:38Z"},{"alias_kind":"pith_short_16","alias_value":"V6VKBHXFN7YCTHIY","created_at":"2026-07-05T09:39:38Z"},{"alias_kind":"pith_short_8","alias_value":"V6VKBHXF","created_at":"2026-07-05T09:39:38Z"}],"graph_snapshots":[{"event_id":"sha256:6e680370558184fa45d11794c51ec48a453da452fde4e2af12300f0cfaca41a0","target":"graph","created_at":"2026-07-05T09:39:38Z","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/2411.15663/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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 $n$.","authors_text":"Jamie Bell","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2024-11-23T22:19:06Z","title":"A Note on the Growth of Sha in Dihedral Extensions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.15663","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":"sha256:02d15fa18b6d3dc40713dc4dbedff6e74feca293848c877bc84887c3d0f9baa7","target":"record","created_at":"2026-07-05T09:39:38Z","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":"add720215a0d99a42ae51fa81a062229e42d28ccd68ab017988b0685f6bcf052","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2024-11-23T22:19:06Z","title_canon_sha256":"6d590aaf29015f09940fd85fb90500b84732d502ca4966ebd738cc79b6b77c78"},"schema_version":"1.0","source":{"id":"2411.15663","kind":"arxiv","version":1}},"canonical_sha256":"afaaa09ee56ff0299d18a000d14e0cb87e4bded82208b3fc567bc163e134ca5a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"afaaa09ee56ff0299d18a000d14e0cb87e4bded82208b3fc567bc163e134ca5a","first_computed_at":"2026-07-05T09:39:38.811888Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:39:38.811888Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HgP28q7rN8hYxpdeugA7Chb/VfrQSsL32My7V56OTM+YYCy48mj3hs71HMgZUfVEgpw8hJpTsdVSjR/AYRj3CA==","signature_status":"signed_v1","signed_at":"2026-07-05T09:39:38.812421Z","signed_message":"canonical_sha256_bytes"},"source_id":"2411.15663","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:02d15fa18b6d3dc40713dc4dbedff6e74feca293848c877bc84887c3d0f9baa7","sha256:6e680370558184fa45d11794c51ec48a453da452fde4e2af12300f0cfaca41a0"],"state_sha256":"7c15fede90e7423284a6d05a9e6a24b0ad3abca08bd528fe23fb42b431c451c9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HvDoEVXqST1iahtFAnCQLWj+rvrAJ4nr9wNyJYUnIYzFN1SBWc5SMTL3D/EfFZzI7CuIYf+ZsdZdWSDJnsWdCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T22:35:46.593777Z","bundle_sha256":"b5fc391445b128321238e64d0b9d56d44443a9b04a4efcb900d60b4e24ed8e3b"}}