{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:KALKHK7PRU23OPBENAUBZHS5KI","short_pith_number":"pith:KALKHK7P","canonical_record":{"source":{"id":"1409.1385","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-09-04T09:54:43Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"25d7d5b750e5a9f3f2abe2db62c6907d54f1a45480a979b4bdc1d7abb45ba69d","abstract_canon_sha256":"6822128f9f060eec40928afd111c8a1ccd1a50773b40c453ab4715c685761163"},"schema_version":"1.0"},"canonical_sha256":"5016a3abef8d35b73c2468281c9e5d5227bfa91866f03d7384b463bc30c78c72","source":{"kind":"arxiv","id":"1409.1385","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.1385","created_at":"2026-05-18T01:35:57Z"},{"alias_kind":"arxiv_version","alias_value":"1409.1385v3","created_at":"2026-05-18T01:35:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.1385","created_at":"2026-05-18T01:35:57Z"},{"alias_kind":"pith_short_12","alias_value":"KALKHK7PRU23","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_16","alias_value":"KALKHK7PRU23OPBE","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_8","alias_value":"KALKHK7P","created_at":"2026-05-18T12:28:35Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:KALKHK7PRU23OPBENAUBZHS5KI","target":"record","payload":{"canonical_record":{"source":{"id":"1409.1385","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-09-04T09:54:43Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"25d7d5b750e5a9f3f2abe2db62c6907d54f1a45480a979b4bdc1d7abb45ba69d","abstract_canon_sha256":"6822128f9f060eec40928afd111c8a1ccd1a50773b40c453ab4715c685761163"},"schema_version":"1.0"},"canonical_sha256":"5016a3abef8d35b73c2468281c9e5d5227bfa91866f03d7384b463bc30c78c72","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:35:57.233101Z","signature_b64":"rGx4LKLoBR8lZWOtDjNYpkt9S81gdmA/X61AGlEYFv+JcIeO1ZEgG3Rbr6L2ef1pMPrkCw+oBruZYF8WvkUqDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5016a3abef8d35b73c2468281c9e5d5227bfa91866f03d7384b463bc30c78c72","last_reissued_at":"2026-05-18T01:35:57.232294Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:35:57.232294Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1409.1385","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:35:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XKeEsZNtM2ftTZbwm260jNL1EF0Xb4zfcFP5ZXM01a3qud/Fjs0OL5b9YPf4qe05Zl2/FWTpZ+xvpTf/iyZwDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T05:41:32.060077Z"},"content_sha256":"3f2c2b0c9d15755caedac85d22050d87242f3a18e60bb6ad82932137090e7d83","schema_version":"1.0","event_id":"sha256:3f2c2b0c9d15755caedac85d22050d87242f3a18e60bb6ad82932137090e7d83"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:KALKHK7PRU23OPBENAUBZHS5KI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Hecke algebra isomorphisms and adelic points on algebraic groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Gunther Cornelissen, Valentijn Karemaker","submitted_at":"2014-09-04T09:54:43Z","abstract_excerpt":"Let $G$ denote a linear algebraic group over $\\mathbf{Q}$ and $K$ and $L$ two number fields. Assume that there is a group isomorphism of points on $G$ over the finite adeles of $K$ and $L$, respectively. We establish conditions on the group $G$, related to the structure of its Borel groups, under which $K$ and $L$ have isomorphic adele rings. Under these conditions, if $K$ or $L$ is a Galois extension of $\\mathbf{Q}$ and $G(\\mathbf{A}_{K,f})$ and $G(\\mathbf{A}_{L,f})$ are isomorphic, then $K$ and $L$ are isomorphic as fields. We use this result to show that if for two number fields $K$ and $L$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.1385","kind":"arxiv","version":3},"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":"2026-05-18T01:35:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"C1Sxo5rhb+Sq5D8V1o3mCfWbEeJXyL+bQ2wjdww55AIrTQBL8sHX+0vlyCcIlQ3n/PL7Et9I2YMtDPS6EbioAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T05:41:32.060441Z"},"content_sha256":"729380b89e99733162e65d3c186f926788b3c9c3309c5150df62154e94984960","schema_version":"1.0","event_id":"sha256:729380b89e99733162e65d3c186f926788b3c9c3309c5150df62154e94984960"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KALKHK7PRU23OPBENAUBZHS5KI/bundle.json","state_url":"https://pith.science/pith/KALKHK7PRU23OPBENAUBZHS5KI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KALKHK7PRU23OPBENAUBZHS5KI/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":"2026-06-24T05:41:32Z","links":{"resolver":"https://pith.science/pith/KALKHK7PRU23OPBENAUBZHS5KI","bundle":"https://pith.science/pith/KALKHK7PRU23OPBENAUBZHS5KI/bundle.json","state":"https://pith.science/pith/KALKHK7PRU23OPBENAUBZHS5KI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KALKHK7PRU23OPBENAUBZHS5KI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:KALKHK7PRU23OPBENAUBZHS5KI","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":"6822128f9f060eec40928afd111c8a1ccd1a50773b40c453ab4715c685761163","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-09-04T09:54:43Z","title_canon_sha256":"25d7d5b750e5a9f3f2abe2db62c6907d54f1a45480a979b4bdc1d7abb45ba69d"},"schema_version":"1.0","source":{"id":"1409.1385","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.1385","created_at":"2026-05-18T01:35:57Z"},{"alias_kind":"arxiv_version","alias_value":"1409.1385v3","created_at":"2026-05-18T01:35:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.1385","created_at":"2026-05-18T01:35:57Z"},{"alias_kind":"pith_short_12","alias_value":"KALKHK7PRU23","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_16","alias_value":"KALKHK7PRU23OPBE","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_8","alias_value":"KALKHK7P","created_at":"2026-05-18T12:28:35Z"}],"graph_snapshots":[{"event_id":"sha256:729380b89e99733162e65d3c186f926788b3c9c3309c5150df62154e94984960","target":"graph","created_at":"2026-05-18T01:35:57Z","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 $G$ denote a linear algebraic group over $\\mathbf{Q}$ and $K$ and $L$ two number fields. Assume that there is a group isomorphism of points on $G$ over the finite adeles of $K$ and $L$, respectively. We establish conditions on the group $G$, related to the structure of its Borel groups, under which $K$ and $L$ have isomorphic adele rings. Under these conditions, if $K$ or $L$ is a Galois extension of $\\mathbf{Q}$ and $G(\\mathbf{A}_{K,f})$ and $G(\\mathbf{A}_{L,f})$ are isomorphic, then $K$ and $L$ are isomorphic as fields. We use this result to show that if for two number fields $K$ and $L$","authors_text":"Gunther Cornelissen, Valentijn Karemaker","cross_cats":["math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-09-04T09:54:43Z","title":"Hecke algebra isomorphisms and adelic points on algebraic groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.1385","kind":"arxiv","version":3},"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":"sha256:3f2c2b0c9d15755caedac85d22050d87242f3a18e60bb6ad82932137090e7d83","target":"record","created_at":"2026-05-18T01:35:57Z","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":"6822128f9f060eec40928afd111c8a1ccd1a50773b40c453ab4715c685761163","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-09-04T09:54:43Z","title_canon_sha256":"25d7d5b750e5a9f3f2abe2db62c6907d54f1a45480a979b4bdc1d7abb45ba69d"},"schema_version":"1.0","source":{"id":"1409.1385","kind":"arxiv","version":3}},"canonical_sha256":"5016a3abef8d35b73c2468281c9e5d5227bfa91866f03d7384b463bc30c78c72","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5016a3abef8d35b73c2468281c9e5d5227bfa91866f03d7384b463bc30c78c72","first_computed_at":"2026-05-18T01:35:57.232294Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:35:57.232294Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rGx4LKLoBR8lZWOtDjNYpkt9S81gdmA/X61AGlEYFv+JcIeO1ZEgG3Rbr6L2ef1pMPrkCw+oBruZYF8WvkUqDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:35:57.233101Z","signed_message":"canonical_sha256_bytes"},"source_id":"1409.1385","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3f2c2b0c9d15755caedac85d22050d87242f3a18e60bb6ad82932137090e7d83","sha256:729380b89e99733162e65d3c186f926788b3c9c3309c5150df62154e94984960"],"state_sha256":"6263e52c99556872956380f70dbcf59fd938ed7055c9f7bed4eec48b918add47"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VzBiezWaVOueHjPYTJrsHQZZ9FcnYQLGcD4koYwhM5JnFLePHkDZV6BEMJLY1EuqIhFR3LCy+ScmbFpXFFGYAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T05:41:32.062381Z","bundle_sha256":"0f046a88024998a4c2eb32793e972963739650b2f6ba510b86a21c3da22c3ad9"}}