{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:2Y3TFC3XZFKK6ZNORBN7O4U3FM","short_pith_number":"pith:2Y3TFC3X","canonical_record":{"source":{"id":"2410.06346","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2024-10-08T20:31:47Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"0ff3fbeb6dfdb2f4b895523ac617c9be18cb67158ee2568c88bc3cf368d04845","abstract_canon_sha256":"5abdd56cb0dc375545f36ffba5047971203392de593acc27b32ce5f275da6462"},"schema_version":"1.0"},"canonical_sha256":"d637328b77c954af65ae885bf7729b2b1dd8131369f1b7160f3aabc0dfed582c","source":{"kind":"arxiv","id":"2410.06346","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2410.06346","created_at":"2026-07-05T12:06:02Z"},{"alias_kind":"arxiv_version","alias_value":"2410.06346v4","created_at":"2026-07-05T12:06:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2410.06346","created_at":"2026-07-05T12:06:02Z"},{"alias_kind":"pith_short_12","alias_value":"2Y3TFC3XZFKK","created_at":"2026-07-05T12:06:02Z"},{"alias_kind":"pith_short_16","alias_value":"2Y3TFC3XZFKK6ZNO","created_at":"2026-07-05T12:06:02Z"},{"alias_kind":"pith_short_8","alias_value":"2Y3TFC3X","created_at":"2026-07-05T12:06:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:2Y3TFC3XZFKK6ZNORBN7O4U3FM","target":"record","payload":{"canonical_record":{"source":{"id":"2410.06346","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2024-10-08T20:31:47Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"0ff3fbeb6dfdb2f4b895523ac617c9be18cb67158ee2568c88bc3cf368d04845","abstract_canon_sha256":"5abdd56cb0dc375545f36ffba5047971203392de593acc27b32ce5f275da6462"},"schema_version":"1.0"},"canonical_sha256":"d637328b77c954af65ae885bf7729b2b1dd8131369f1b7160f3aabc0dfed582c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:06:02.275240Z","signature_b64":"pSHO0u/OipWmwePE5qDdrSScxCMf/FUOsd3E7zkiWcO6Z5qL7RyzTHs87pvClcqPJz5HtjXcjNeO/gVpD2AYDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d637328b77c954af65ae885bf7729b2b1dd8131369f1b7160f3aabc0dfed582c","last_reissued_at":"2026-07-05T12:06:02.274692Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:06:02.274692Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2410.06346","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T12:06:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YyEFMHz3WW3Szl1z330gewk41YM1PXfg2pDc8yGkDJzP9JAVhDZDGn32+bDtF0dU2KMk8uL+6aRcPMLAmQR7DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T02:50:18.989615Z"},"content_sha256":"bfe3fedf2b4441979bf07f1001cc78bb3308683d785b6b108776b88dc3ffc972","schema_version":"1.0","event_id":"sha256:bfe3fedf2b4441979bf07f1001cc78bb3308683d785b6b108776b88dc3ffc972"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:2Y3TFC3XZFKK6ZNORBN7O4U3FM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A remark on the Langlands correspondence for tori","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.RT","authors_text":"Eric Opdam, Marcelo De Martino","submitted_at":"2024-10-08T20:31:47Z","abstract_excerpt":"For an algebraic torus defined over a local (or global) field $F$, a celebrated result of R.P. Langlands establishes a natural homomorphism from the group of continuous cohomology classes of the Weil group, valued in the dual torus, onto the space of complex characters of the rational points of the torus (or automorphic characters in the global case). We expand on this result by detailing its topological aspects. We show that if we topologize the relevant spaces of continuous homomorphisms and continuous cochains using the compact-open topology, Langlands's map becomes a (surjective, finite-to"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.06346","kind":"arxiv","version":4},"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/2410.06346/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":"2026-07-05T12:06:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KF/zoMQVVTDgTBAdXuknVZUJoi+bNXcAVwkLoJZ0m882hX0Q+Kjhq6ViXgFzWJ3dz0NzeoGWjF0enkRZbA19DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T02:50:18.990321Z"},"content_sha256":"769ad38f649bb7e49a1f5a2fda76eabc7bd4b18ae5099b374d2319dec5cde529","schema_version":"1.0","event_id":"sha256:769ad38f649bb7e49a1f5a2fda76eabc7bd4b18ae5099b374d2319dec5cde529"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2Y3TFC3XZFKK6ZNORBN7O4U3FM/bundle.json","state_url":"https://pith.science/pith/2Y3TFC3XZFKK6ZNORBN7O4U3FM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2Y3TFC3XZFKK6ZNORBN7O4U3FM/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":"2026-08-13T02:50:18Z","links":{"resolver":"https://pith.science/pith/2Y3TFC3XZFKK6ZNORBN7O4U3FM","bundle":"https://pith.science/pith/2Y3TFC3XZFKK6ZNORBN7O4U3FM/bundle.json","state":"https://pith.science/pith/2Y3TFC3XZFKK6ZNORBN7O4U3FM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2Y3TFC3XZFKK6ZNORBN7O4U3FM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:2Y3TFC3XZFKK6ZNORBN7O4U3FM","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":"5abdd56cb0dc375545f36ffba5047971203392de593acc27b32ce5f275da6462","cross_cats_sorted":["math.NT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2024-10-08T20:31:47Z","title_canon_sha256":"0ff3fbeb6dfdb2f4b895523ac617c9be18cb67158ee2568c88bc3cf368d04845"},"schema_version":"1.0","source":{"id":"2410.06346","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2410.06346","created_at":"2026-07-05T12:06:02Z"},{"alias_kind":"arxiv_version","alias_value":"2410.06346v4","created_at":"2026-07-05T12:06:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2410.06346","created_at":"2026-07-05T12:06:02Z"},{"alias_kind":"pith_short_12","alias_value":"2Y3TFC3XZFKK","created_at":"2026-07-05T12:06:02Z"},{"alias_kind":"pith_short_16","alias_value":"2Y3TFC3XZFKK6ZNO","created_at":"2026-07-05T12:06:02Z"},{"alias_kind":"pith_short_8","alias_value":"2Y3TFC3X","created_at":"2026-07-05T12:06:02Z"}],"graph_snapshots":[{"event_id":"sha256:769ad38f649bb7e49a1f5a2fda76eabc7bd4b18ae5099b374d2319dec5cde529","target":"graph","created_at":"2026-07-05T12:06:02Z","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/2410.06346/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For an algebraic torus defined over a local (or global) field $F$, a celebrated result of R.P. Langlands establishes a natural homomorphism from the group of continuous cohomology classes of the Weil group, valued in the dual torus, onto the space of complex characters of the rational points of the torus (or automorphic characters in the global case). We expand on this result by detailing its topological aspects. We show that if we topologize the relevant spaces of continuous homomorphisms and continuous cochains using the compact-open topology, Langlands's map becomes a (surjective, finite-to","authors_text":"Eric Opdam, Marcelo De Martino","cross_cats":["math.NT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2024-10-08T20:31:47Z","title":"A remark on the Langlands correspondence for tori"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.06346","kind":"arxiv","version":4},"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:bfe3fedf2b4441979bf07f1001cc78bb3308683d785b6b108776b88dc3ffc972","target":"record","created_at":"2026-07-05T12:06:02Z","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":"5abdd56cb0dc375545f36ffba5047971203392de593acc27b32ce5f275da6462","cross_cats_sorted":["math.NT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2024-10-08T20:31:47Z","title_canon_sha256":"0ff3fbeb6dfdb2f4b895523ac617c9be18cb67158ee2568c88bc3cf368d04845"},"schema_version":"1.0","source":{"id":"2410.06346","kind":"arxiv","version":4}},"canonical_sha256":"d637328b77c954af65ae885bf7729b2b1dd8131369f1b7160f3aabc0dfed582c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d637328b77c954af65ae885bf7729b2b1dd8131369f1b7160f3aabc0dfed582c","first_computed_at":"2026-07-05T12:06:02.274692Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T12:06:02.274692Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pSHO0u/OipWmwePE5qDdrSScxCMf/FUOsd3E7zkiWcO6Z5qL7RyzTHs87pvClcqPJz5HtjXcjNeO/gVpD2AYDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T12:06:02.275240Z","signed_message":"canonical_sha256_bytes"},"source_id":"2410.06346","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bfe3fedf2b4441979bf07f1001cc78bb3308683d785b6b108776b88dc3ffc972","sha256:769ad38f649bb7e49a1f5a2fda76eabc7bd4b18ae5099b374d2319dec5cde529"],"state_sha256":"ba4d2f519119a9a6328f201772f30b2a706020e4cae8983510519e8bd1128c54"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bRUfL9WUM1vgMxzc0SLhaLj27Vn7XTkudwkv9UJQ8RPDLu3qeFCd8kzdD5hCJhOood2ad/VM47mMvEV5mipLAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T02:50:18.995872Z","bundle_sha256":"3c19e19379bc150c16c49fd3c37d068ac3bd3424236187a08d3621f48d96df07"}}