{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:BFUF6D4AVCHTN2X5PQ4FMSMKRG","short_pith_number":"pith:BFUF6D4A","canonical_record":{"source":{"id":"0705.1899","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2007-05-14T09:46:35Z","cross_cats_sorted":[],"title_canon_sha256":"c2421b5cdda277f2fa71c0aaa82d77d90fc7a9f95492486edfa6c22a834d9aec","abstract_canon_sha256":"ee01319d8b085ec5afedb2acb408685becc87e28cb40315cb988457c013df091"},"schema_version":"1.0"},"canonical_sha256":"09685f0f80a88f36eafd7c3856498a89a936128821d8a78ab55e3a00e26ca63f","source":{"kind":"arxiv","id":"0705.1899","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0705.1899","created_at":"2026-05-18T03:12:48Z"},{"alias_kind":"arxiv_version","alias_value":"0705.1899v2","created_at":"2026-05-18T03:12:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0705.1899","created_at":"2026-05-18T03:12:48Z"},{"alias_kind":"pith_short_12","alias_value":"BFUF6D4AVCHT","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"BFUF6D4AVCHTN2X5","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"BFUF6D4A","created_at":"2026-05-18T12:25:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:BFUF6D4AVCHTN2X5PQ4FMSMKRG","target":"record","payload":{"canonical_record":{"source":{"id":"0705.1899","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2007-05-14T09:46:35Z","cross_cats_sorted":[],"title_canon_sha256":"c2421b5cdda277f2fa71c0aaa82d77d90fc7a9f95492486edfa6c22a834d9aec","abstract_canon_sha256":"ee01319d8b085ec5afedb2acb408685becc87e28cb40315cb988457c013df091"},"schema_version":"1.0"},"canonical_sha256":"09685f0f80a88f36eafd7c3856498a89a936128821d8a78ab55e3a00e26ca63f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:12:48.867072Z","signature_b64":"8/WFpKoutELIu8015ScX1PduDYxTpfrmeEXHAM+AcOuOJfsYcgLpdBWbLZLCAjVFYzelneAqJ0Hw1/khYReBAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"09685f0f80a88f36eafd7c3856498a89a936128821d8a78ab55e3a00e26ca63f","last_reissued_at":"2026-05-18T03:12:48.866322Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:12:48.866322Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0705.1899","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:12:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CnhKOYUYmBkm2cpRjNUCKFHLP/6LvTOJk/jHOeSmpIV0sZYfRoPfUBzVo5pBvkzgvI3viXSRHNVg/cyL3I5kDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T08:48:44.020045Z"},"content_sha256":"d329181c0797924729ce27aac778bb657fb82be2c82f6a1cd8266470c39f31fd","schema_version":"1.0","event_id":"sha256:d329181c0797924729ce27aac778bb657fb82be2c82f6a1cd8266470c39f31fd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:BFUF6D4AVCHTN2X5PQ4FMSMKRG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Self-duality of Selmer groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Tim Dokchitser, Vladimir Dokchitser","submitted_at":"2007-05-14T09:46:35Z","abstract_excerpt":"The first part of the paper gives a new proof of self-duality for Selmer groups: if A is an abelian variety over a number field K, and F/K is a Galois extension with Galois group G, then the Q_pG-representation naturally associated to the p-infinity Selmer group of A/F is self-dual. The second part describes a method for obtaining information about parities of Selmer ranks from the local Tamagawa numbers of A in intermediate extensions of F/K."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0705.1899","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"2026-05-18T03:12:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/q48Mp+NQy6BWW4hC3wd3uzwe22ibcTk0WNb9xO5eIZc89XxyX2k+S77tSbbFTDqnzU8xw+Hze4ILbMfkWfIBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T08:48:44.020390Z"},"content_sha256":"1a7c9920bf141710ad701ac16c49e13018216c4ea52462b4d68a264343b3b274","schema_version":"1.0","event_id":"sha256:1a7c9920bf141710ad701ac16c49e13018216c4ea52462b4d68a264343b3b274"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BFUF6D4AVCHTN2X5PQ4FMSMKRG/bundle.json","state_url":"https://pith.science/pith/BFUF6D4AVCHTN2X5PQ4FMSMKRG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BFUF6D4AVCHTN2X5PQ4FMSMKRG/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":"2026-06-28T08:48:44Z","links":{"resolver":"https://pith.science/pith/BFUF6D4AVCHTN2X5PQ4FMSMKRG","bundle":"https://pith.science/pith/BFUF6D4AVCHTN2X5PQ4FMSMKRG/bundle.json","state":"https://pith.science/pith/BFUF6D4AVCHTN2X5PQ4FMSMKRG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BFUF6D4AVCHTN2X5PQ4FMSMKRG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:BFUF6D4AVCHTN2X5PQ4FMSMKRG","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":"ee01319d8b085ec5afedb2acb408685becc87e28cb40315cb988457c013df091","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2007-05-14T09:46:35Z","title_canon_sha256":"c2421b5cdda277f2fa71c0aaa82d77d90fc7a9f95492486edfa6c22a834d9aec"},"schema_version":"1.0","source":{"id":"0705.1899","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0705.1899","created_at":"2026-05-18T03:12:48Z"},{"alias_kind":"arxiv_version","alias_value":"0705.1899v2","created_at":"2026-05-18T03:12:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0705.1899","created_at":"2026-05-18T03:12:48Z"},{"alias_kind":"pith_short_12","alias_value":"BFUF6D4AVCHT","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"BFUF6D4AVCHTN2X5","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"BFUF6D4A","created_at":"2026-05-18T12:25:55Z"}],"graph_snapshots":[{"event_id":"sha256:1a7c9920bf141710ad701ac16c49e13018216c4ea52462b4d68a264343b3b274","target":"graph","created_at":"2026-05-18T03:12:48Z","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":"The first part of the paper gives a new proof of self-duality for Selmer groups: if A is an abelian variety over a number field K, and F/K is a Galois extension with Galois group G, then the Q_pG-representation naturally associated to the p-infinity Selmer group of A/F is self-dual. The second part describes a method for obtaining information about parities of Selmer ranks from the local Tamagawa numbers of A in intermediate extensions of F/K.","authors_text":"Tim Dokchitser, Vladimir Dokchitser","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2007-05-14T09:46:35Z","title":"Self-duality of Selmer groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0705.1899","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:d329181c0797924729ce27aac778bb657fb82be2c82f6a1cd8266470c39f31fd","target":"record","created_at":"2026-05-18T03:12:48Z","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":"ee01319d8b085ec5afedb2acb408685becc87e28cb40315cb988457c013df091","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2007-05-14T09:46:35Z","title_canon_sha256":"c2421b5cdda277f2fa71c0aaa82d77d90fc7a9f95492486edfa6c22a834d9aec"},"schema_version":"1.0","source":{"id":"0705.1899","kind":"arxiv","version":2}},"canonical_sha256":"09685f0f80a88f36eafd7c3856498a89a936128821d8a78ab55e3a00e26ca63f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"09685f0f80a88f36eafd7c3856498a89a936128821d8a78ab55e3a00e26ca63f","first_computed_at":"2026-05-18T03:12:48.866322Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:12:48.866322Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8/WFpKoutELIu8015ScX1PduDYxTpfrmeEXHAM+AcOuOJfsYcgLpdBWbLZLCAjVFYzelneAqJ0Hw1/khYReBAA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:12:48.867072Z","signed_message":"canonical_sha256_bytes"},"source_id":"0705.1899","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d329181c0797924729ce27aac778bb657fb82be2c82f6a1cd8266470c39f31fd","sha256:1a7c9920bf141710ad701ac16c49e13018216c4ea52462b4d68a264343b3b274"],"state_sha256":"6651c12c0880a4f9cd0a07615f4b978f1d4d6000a00f0c3b941aa0fa6cca0b97"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WLWNgstxrIeEPlj5NAy6rz2Vfrq98XRmK0yqgcMIJPZmrGzCjdI9534e0ErPjhqQvQHmYcjAoy9d4nif8xaOCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T08:48:44.022257Z","bundle_sha256":"f56ec453ab15931a6835bcafbd250c9e82f57a28992ec533baa2138bd0042e58"}}