{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:7YYXIBGXTRVPBQT4AOVXAIHIHE","short_pith_number":"pith:7YYXIBGX","canonical_record":{"source":{"id":"1210.5727","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-10-21T13:24:04Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"9d98f641f97be0ba5d892bc15c83c711f0b226e619439c868154146a56797c1f","abstract_canon_sha256":"59c7a7951cf203516a6aacad774215f9d8a09023ec271d6420d03d1c2cf82c38"},"schema_version":"1.0"},"canonical_sha256":"fe317404d79c6af0c27c03ab7020e83935637457b0dcc265745a0df4a5020489","source":{"kind":"arxiv","id":"1210.5727","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.5727","created_at":"2026-05-18T00:44:30Z"},{"alias_kind":"arxiv_version","alias_value":"1210.5727v2","created_at":"2026-05-18T00:44:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.5727","created_at":"2026-05-18T00:44:30Z"},{"alias_kind":"pith_short_12","alias_value":"7YYXIBGXTRVP","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"7YYXIBGXTRVPBQT4","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"7YYXIBGX","created_at":"2026-05-18T12:26:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:7YYXIBGXTRVPBQT4AOVXAIHIHE","target":"record","payload":{"canonical_record":{"source":{"id":"1210.5727","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-10-21T13:24:04Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"9d98f641f97be0ba5d892bc15c83c711f0b226e619439c868154146a56797c1f","abstract_canon_sha256":"59c7a7951cf203516a6aacad774215f9d8a09023ec271d6420d03d1c2cf82c38"},"schema_version":"1.0"},"canonical_sha256":"fe317404d79c6af0c27c03ab7020e83935637457b0dcc265745a0df4a5020489","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:30.217244Z","signature_b64":"yAQdICTcr2hawiCUcF4O5X68gyRRubBwQfXU4cWUaJlluvMiFeqWw3Wvi+m9ca96GHZ1gO0Y5gKWIluDgJtQDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fe317404d79c6af0c27c03ab7020e83935637457b0dcc265745a0df4a5020489","last_reissued_at":"2026-05-18T00:44:30.216692Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:30.216692Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1210.5727","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":"2026-05-18T00:44:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tyQqTsBlpk1eNT//4RGeg0cfyygYomjQkEX06Zho4HFZNTV57QiblNZm+QolUdTdhXDaPXHhzEgbU7pxnfdACg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T22:52:13.936810Z"},"content_sha256":"dea634c086057d8a275966a6aa46937988866db61cb5a7c77e6f225272346d00","schema_version":"1.0","event_id":"sha256:dea634c086057d8a275966a6aa46937988866db61cb5a7c77e6f225272346d00"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:7YYXIBGXTRVPBQT4AOVXAIHIHE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Norms as products of linear polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Alexei Skorobogatov, Damaris Schindler","submitted_at":"2012-10-21T13:24:04Z","abstract_excerpt":"Let F be a number field, and let F\\subset K be a field extension of degree n. Suppose that we are given 2r sufficiently general linear polynomials in r variables over F. Let X be the variety over F such that the F-points of X bijectively correspond to the representations of the product of these polynomials by a norm from K to F. Combining the circle method with descent we prove that the Brauer-Manin obstruction is the only obstruction to the Hasse principle and weak approximation on any smooth and projective model of X."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.5727","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":"2026-05-18T00:44:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VvNT0Aq96T6WVGNYP0Llw98aBe1CM0TauKzgrOM4z46Jo2iwGI10LouQ7e7pQT2v44PvH7zsb4ouy2h+lApzDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T22:52:13.937520Z"},"content_sha256":"94cdcd788e443167b43d4fa5fbc348634b6c6fd24766f25ffe892441811379c1","schema_version":"1.0","event_id":"sha256:94cdcd788e443167b43d4fa5fbc348634b6c6fd24766f25ffe892441811379c1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7YYXIBGXTRVPBQT4AOVXAIHIHE/bundle.json","state_url":"https://pith.science/pith/7YYXIBGXTRVPBQT4AOVXAIHIHE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7YYXIBGXTRVPBQT4AOVXAIHIHE/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":"2026-06-07T22:52:13Z","links":{"resolver":"https://pith.science/pith/7YYXIBGXTRVPBQT4AOVXAIHIHE","bundle":"https://pith.science/pith/7YYXIBGXTRVPBQT4AOVXAIHIHE/bundle.json","state":"https://pith.science/pith/7YYXIBGXTRVPBQT4AOVXAIHIHE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7YYXIBGXTRVPBQT4AOVXAIHIHE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:7YYXIBGXTRVPBQT4AOVXAIHIHE","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":"59c7a7951cf203516a6aacad774215f9d8a09023ec271d6420d03d1c2cf82c38","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-10-21T13:24:04Z","title_canon_sha256":"9d98f641f97be0ba5d892bc15c83c711f0b226e619439c868154146a56797c1f"},"schema_version":"1.0","source":{"id":"1210.5727","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.5727","created_at":"2026-05-18T00:44:30Z"},{"alias_kind":"arxiv_version","alias_value":"1210.5727v2","created_at":"2026-05-18T00:44:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.5727","created_at":"2026-05-18T00:44:30Z"},{"alias_kind":"pith_short_12","alias_value":"7YYXIBGXTRVP","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"7YYXIBGXTRVPBQT4","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"7YYXIBGX","created_at":"2026-05-18T12:26:58Z"}],"graph_snapshots":[{"event_id":"sha256:94cdcd788e443167b43d4fa5fbc348634b6c6fd24766f25ffe892441811379c1","target":"graph","created_at":"2026-05-18T00:44:30Z","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":"Let F be a number field, and let F\\subset K be a field extension of degree n. Suppose that we are given 2r sufficiently general linear polynomials in r variables over F. Let X be the variety over F such that the F-points of X bijectively correspond to the representations of the product of these polynomials by a norm from K to F. Combining the circle method with descent we prove that the Brauer-Manin obstruction is the only obstruction to the Hasse principle and weak approximation on any smooth and projective model of X.","authors_text":"Alexei Skorobogatov, Damaris Schindler","cross_cats":["math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-10-21T13:24:04Z","title":"Norms as products of linear polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.5727","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:dea634c086057d8a275966a6aa46937988866db61cb5a7c77e6f225272346d00","target":"record","created_at":"2026-05-18T00:44:30Z","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":"59c7a7951cf203516a6aacad774215f9d8a09023ec271d6420d03d1c2cf82c38","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-10-21T13:24:04Z","title_canon_sha256":"9d98f641f97be0ba5d892bc15c83c711f0b226e619439c868154146a56797c1f"},"schema_version":"1.0","source":{"id":"1210.5727","kind":"arxiv","version":2}},"canonical_sha256":"fe317404d79c6af0c27c03ab7020e83935637457b0dcc265745a0df4a5020489","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fe317404d79c6af0c27c03ab7020e83935637457b0dcc265745a0df4a5020489","first_computed_at":"2026-05-18T00:44:30.216692Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:44:30.216692Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yAQdICTcr2hawiCUcF4O5X68gyRRubBwQfXU4cWUaJlluvMiFeqWw3Wvi+m9ca96GHZ1gO0Y5gKWIluDgJtQDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:44:30.217244Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.5727","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dea634c086057d8a275966a6aa46937988866db61cb5a7c77e6f225272346d00","sha256:94cdcd788e443167b43d4fa5fbc348634b6c6fd24766f25ffe892441811379c1"],"state_sha256":"a835262ce9f1f9caeb36f53c0a9fa5418a9188dd792f03923385be897ea3f9f0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+OFhJ0w4fbQjINOvRyJGhqdGdPOEfE/XU0LGIdlyhSvQ9Y5oFX4/fjAwqcG29ckwAwaneiXZggTmxfoP8xN1Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T22:52:13.941463Z","bundle_sha256":"59e01deff8b25368dcf4497532a4f8cead4e18c09b7e51cc1bd657a09640a9c9"}}