{"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"}