{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:NGCBQRUQEZ6S7XNDXQTCQEN6E6","short_pith_number":"pith:NGCBQRUQ","canonical_record":{"source":{"id":"1606.02477","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-06-08T09:50:44Z","cross_cats_sorted":[],"title_canon_sha256":"0b1d13b03947244e224dbdb7e7695560fb565ce2bb0033feafc526a8c3473960","abstract_canon_sha256":"67fca075aee2db1882bfd831cf6723d15e34980e9891a4b697b5f8e9d0a7106a"},"schema_version":"1.0"},"canonical_sha256":"6984184690267d2fdda3bc262811be27aac3e6ec7ccfebc53bb04a46f7452271","source":{"kind":"arxiv","id":"1606.02477","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.02477","created_at":"2026-05-18T00:22:11Z"},{"alias_kind":"arxiv_version","alias_value":"1606.02477v3","created_at":"2026-05-18T00:22:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.02477","created_at":"2026-05-18T00:22:11Z"},{"alias_kind":"pith_short_12","alias_value":"NGCBQRUQEZ6S","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"NGCBQRUQEZ6S7XND","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"NGCBQRUQ","created_at":"2026-05-18T12:30:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:NGCBQRUQEZ6S7XNDXQTCQEN6E6","target":"record","payload":{"canonical_record":{"source":{"id":"1606.02477","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-06-08T09:50:44Z","cross_cats_sorted":[],"title_canon_sha256":"0b1d13b03947244e224dbdb7e7695560fb565ce2bb0033feafc526a8c3473960","abstract_canon_sha256":"67fca075aee2db1882bfd831cf6723d15e34980e9891a4b697b5f8e9d0a7106a"},"schema_version":"1.0"},"canonical_sha256":"6984184690267d2fdda3bc262811be27aac3e6ec7ccfebc53bb04a46f7452271","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:22:11.265690Z","signature_b64":"u4nyWxtdNLRM1DHlqkLfTCKaXDEpIe8meWDukTb7gxgkxp8sUBsZxfNNip3Up0PwfeVn3SFTwRPrW7aRM068Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6984184690267d2fdda3bc262811be27aac3e6ec7ccfebc53bb04a46f7452271","last_reissued_at":"2026-05-18T00:22:11.265217Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:22:11.265217Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1606.02477","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":"2026-05-18T00:22:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GUPemMREijCCbYCRpm30+nA5veu3wuvEe4lzgeMCbHGlyj/SiHhXZSLhTNlY5mHrTLt8u6cYzUg0pxUB2Y2uCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T23:03:20.566136Z"},"content_sha256":"895e92f79fe17eb6c6e73593ba37c437546599dfd58fcb8219170c42fb1c6ee5","schema_version":"1.0","event_id":"sha256:895e92f79fe17eb6c6e73593ba37c437546599dfd58fcb8219170c42fb1c6ee5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:NGCBQRUQEZ6S7XNDXQTCQEN6E6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Kuznetsov Trace Formula for $\\mathrm{PGL}_2(\\mathbb{C})$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Zhi Qi","submitted_at":"2016-06-08T09:50:44Z","abstract_excerpt":"In this note, using a representation theoretic method of Cogdell and Piatetski-Shapiro, we prove the Kuznetsov trace formula for an arbitrary discrete group $\\Gamma$ in $\\mathrm{PGL}_2(\\mathbb{C})$ that is cofinite but not cocompact. An essential ingredient is a kernel formula, recently proved by the author, on Bessel functions for $\\mathrm{PGL}_2(\\mathbb{C})$. This approach avoids the difficult analysis in the existing method due to Bruggeman and Motohashi."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.02477","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":"2026-05-18T00:22:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WA/WfBgWAXuTTfRQEM8dk28yk0fJFncyp7CTS3nfS5c4GBU4qxghC4S3O8S3x5A6sS5l2LI2pzZV7LbPoJdbAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T23:03:20.566536Z"},"content_sha256":"f9b79b622926eb44e41dae8f93d2651c60499aa458b8a6a89a179f67e782671a","schema_version":"1.0","event_id":"sha256:f9b79b622926eb44e41dae8f93d2651c60499aa458b8a6a89a179f67e782671a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NGCBQRUQEZ6S7XNDXQTCQEN6E6/bundle.json","state_url":"https://pith.science/pith/NGCBQRUQEZ6S7XNDXQTCQEN6E6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NGCBQRUQEZ6S7XNDXQTCQEN6E6/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":"2026-07-01T23:03:20Z","links":{"resolver":"https://pith.science/pith/NGCBQRUQEZ6S7XNDXQTCQEN6E6","bundle":"https://pith.science/pith/NGCBQRUQEZ6S7XNDXQTCQEN6E6/bundle.json","state":"https://pith.science/pith/NGCBQRUQEZ6S7XNDXQTCQEN6E6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NGCBQRUQEZ6S7XNDXQTCQEN6E6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:NGCBQRUQEZ6S7XNDXQTCQEN6E6","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":"67fca075aee2db1882bfd831cf6723d15e34980e9891a4b697b5f8e9d0a7106a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-06-08T09:50:44Z","title_canon_sha256":"0b1d13b03947244e224dbdb7e7695560fb565ce2bb0033feafc526a8c3473960"},"schema_version":"1.0","source":{"id":"1606.02477","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.02477","created_at":"2026-05-18T00:22:11Z"},{"alias_kind":"arxiv_version","alias_value":"1606.02477v3","created_at":"2026-05-18T00:22:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.02477","created_at":"2026-05-18T00:22:11Z"},{"alias_kind":"pith_short_12","alias_value":"NGCBQRUQEZ6S","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"NGCBQRUQEZ6S7XND","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"NGCBQRUQ","created_at":"2026-05-18T12:30:32Z"}],"graph_snapshots":[{"event_id":"sha256:f9b79b622926eb44e41dae8f93d2651c60499aa458b8a6a89a179f67e782671a","target":"graph","created_at":"2026-05-18T00:22:11Z","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":"In this note, using a representation theoretic method of Cogdell and Piatetski-Shapiro, we prove the Kuznetsov trace formula for an arbitrary discrete group $\\Gamma$ in $\\mathrm{PGL}_2(\\mathbb{C})$ that is cofinite but not cocompact. An essential ingredient is a kernel formula, recently proved by the author, on Bessel functions for $\\mathrm{PGL}_2(\\mathbb{C})$. This approach avoids the difficult analysis in the existing method due to Bruggeman and Motohashi.","authors_text":"Zhi Qi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-06-08T09:50:44Z","title":"On the Kuznetsov Trace Formula for $\\mathrm{PGL}_2(\\mathbb{C})$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.02477","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":"sha256:895e92f79fe17eb6c6e73593ba37c437546599dfd58fcb8219170c42fb1c6ee5","target":"record","created_at":"2026-05-18T00:22:11Z","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":"67fca075aee2db1882bfd831cf6723d15e34980e9891a4b697b5f8e9d0a7106a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-06-08T09:50:44Z","title_canon_sha256":"0b1d13b03947244e224dbdb7e7695560fb565ce2bb0033feafc526a8c3473960"},"schema_version":"1.0","source":{"id":"1606.02477","kind":"arxiv","version":3}},"canonical_sha256":"6984184690267d2fdda3bc262811be27aac3e6ec7ccfebc53bb04a46f7452271","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6984184690267d2fdda3bc262811be27aac3e6ec7ccfebc53bb04a46f7452271","first_computed_at":"2026-05-18T00:22:11.265217Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:22:11.265217Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"u4nyWxtdNLRM1DHlqkLfTCKaXDEpIe8meWDukTb7gxgkxp8sUBsZxfNNip3Up0PwfeVn3SFTwRPrW7aRM068Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:22:11.265690Z","signed_message":"canonical_sha256_bytes"},"source_id":"1606.02477","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:895e92f79fe17eb6c6e73593ba37c437546599dfd58fcb8219170c42fb1c6ee5","sha256:f9b79b622926eb44e41dae8f93d2651c60499aa458b8a6a89a179f67e782671a"],"state_sha256":"16e7134e84651eb3f9c5bd22ea56d6956f56fefea18c658a77a9e4186da0bff5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"S6pM96XZdbAkw1r3ans5LvoBM5C+e0BkliKwYD+TLDDgcsK4FKC8THnjC4PBgT4Mae1V1Z2qJMhN5rY4RJx9Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-01T23:03:20.568666Z","bundle_sha256":"f6276d26042b3215e4e39dc0553b87cf40c731b765984180161cc2a9a42edc4f"}}