{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:PGFENEBRU2TPMMC6VMQLCKZSR2","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":"f16d12ee130e478bd1af0a6b3aca2b0a9cfcb1c2901a37030632c973efddaffd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2013-12-30T16:35:57Z","title_canon_sha256":"b4d1d1c1d393628b05f543a2a9717f53f121c3cadae23e92d8dde70f96acdead"},"schema_version":"1.0","source":{"id":"1312.7771","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.7771","created_at":"2026-05-18T01:18:23Z"},{"alias_kind":"arxiv_version","alias_value":"1312.7771v3","created_at":"2026-05-18T01:18:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.7771","created_at":"2026-05-18T01:18:23Z"},{"alias_kind":"pith_short_12","alias_value":"PGFENEBRU2TP","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_16","alias_value":"PGFENEBRU2TPMMC6","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_8","alias_value":"PGFENEBR","created_at":"2026-05-18T12:27:54Z"}],"graph_snapshots":[{"event_id":"sha256:b2b8ec75b6ca4173e0dff55dbcfa90a30999ab19e2bea1fa99482062a71dd0a0","target":"graph","created_at":"2026-05-18T01:18:23Z","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":"We show that the modular group has an infinite family of finite index subgroups, each of which has the same trace set as the modular group itself. Various congruence subgroups of the modular group, and the Bianchi groups, are also shown to have this property. In the case of the modular group, we construct examples of such finite index subgroups.","authors_text":"Grant S. Lakeland","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2013-12-30T16:35:57Z","title":"Equivalent trace sets for arithmetic Fuchsian groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.7771","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":"sha256:0e6e3dcf5bf029b702a07360f6091f843e5e9de55874723ffac6994a41cd6c13","target":"record","created_at":"2026-05-18T01:18:23Z","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":"f16d12ee130e478bd1af0a6b3aca2b0a9cfcb1c2901a37030632c973efddaffd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2013-12-30T16:35:57Z","title_canon_sha256":"b4d1d1c1d393628b05f543a2a9717f53f121c3cadae23e92d8dde70f96acdead"},"schema_version":"1.0","source":{"id":"1312.7771","kind":"arxiv","version":3}},"canonical_sha256":"798a469031a6a6f6305eab20b12b328ebce18706aa00a1c9cc0f1c28b8d22e2a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"798a469031a6a6f6305eab20b12b328ebce18706aa00a1c9cc0f1c28b8d22e2a","first_computed_at":"2026-05-18T01:18:23.634295Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:18:23.634295Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mI3KDf+isETEnwT2AMScs6XYys6cCQAlOekVcwUQeQ25dK1YVEy4N5BNcA0Nlzkj0XRIVMWX0wic96wsncF/CA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:18:23.634895Z","signed_message":"canonical_sha256_bytes"},"source_id":"1312.7771","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0e6e3dcf5bf029b702a07360f6091f843e5e9de55874723ffac6994a41cd6c13","sha256:b2b8ec75b6ca4173e0dff55dbcfa90a30999ab19e2bea1fa99482062a71dd0a0"],"state_sha256":"bcd414110a99858e9c70d2a41e6b3ea6cd10ad9b701eafa3e1af95ea3c01553e"}