{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:7BREYV3XP5SK2QN5HIEGJPNHAS","short_pith_number":"pith:7BREYV3X","canonical_record":{"source":{"id":"1210.1559","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2012-10-04T19:41:32Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"09d4246cdd6c2a005b79195c9117b5379d06f7a2e3663ca1139db425bf8e5486","abstract_canon_sha256":"d38361717de351981a17895810b21a7c0377db3d85a9f8683b88d544869ecaea"},"schema_version":"1.0"},"canonical_sha256":"f8624c57777f64ad41bd3a0864bda70480b605b513ac14dbaa65ffbb9ae12454","source":{"kind":"arxiv","id":"1210.1559","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.1559","created_at":"2026-05-18T03:44:03Z"},{"alias_kind":"arxiv_version","alias_value":"1210.1559v1","created_at":"2026-05-18T03:44:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.1559","created_at":"2026-05-18T03:44:03Z"},{"alias_kind":"pith_short_12","alias_value":"7BREYV3XP5SK","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_16","alias_value":"7BREYV3XP5SK2QN5","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_8","alias_value":"7BREYV3X","created_at":"2026-05-18T12:26:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:7BREYV3XP5SK2QN5HIEGJPNHAS","target":"record","payload":{"canonical_record":{"source":{"id":"1210.1559","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2012-10-04T19:41:32Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"09d4246cdd6c2a005b79195c9117b5379d06f7a2e3663ca1139db425bf8e5486","abstract_canon_sha256":"d38361717de351981a17895810b21a7c0377db3d85a9f8683b88d544869ecaea"},"schema_version":"1.0"},"canonical_sha256":"f8624c57777f64ad41bd3a0864bda70480b605b513ac14dbaa65ffbb9ae12454","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:44:03.148670Z","signature_b64":"tJehfTro8Phly/RTIYH5j/vlOeSCAneUDdKRPqPckZSyl5axRCivQHYlhfexTXEzweZ/TLtEOef5hqQBXsw7Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f8624c57777f64ad41bd3a0864bda70480b605b513ac14dbaa65ffbb9ae12454","last_reissued_at":"2026-05-18T03:44:03.147922Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:44:03.147922Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1210.1559","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:44:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Y8p+QFRbSnyd6TqvxH8wuXy8sHqTZpvhdXQIX/n+0P6fHdBmuvGrRGSUZxMybbpc9NP8HGU9ahBfKtbkLFN2Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T15:04:02.715177Z"},"content_sha256":"5527a9b568ad50e8ffd0adc113d93f2868e5ce74f7b70b3cc3707ba05aec16c9","schema_version":"1.0","event_id":"sha256:5527a9b568ad50e8ffd0adc113d93f2868e5ce74f7b70b3cc3707ba05aec16c9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:7BREYV3XP5SK2QN5HIEGJPNHAS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Kottwitz-Rapoport and p-rank strata in the reduction of Shimura varieties of PEL type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.AG","authors_text":"Philipp Hartwig","submitted_at":"2012-10-04T19:41:32Z","abstract_excerpt":"We study the reduction of certain integral models of Shimura varieties of PEL type with Iwahori level structure. On these spaces we have the Kottwitz-Rapoport and the $p$-rank stratification. We show that the $p$-rank is constant on a KR stratum, generalizing a result of Ng\\^o and Genestier. We prove an abstract, uniform formula for the $p$-rank on a KR stratum. In the symplectic and in the unitary case we derive explicit formulas for its value. We apply these formulas to the question of the density of the ordinary locus and to the question of the dimension of the $p$-rank 0 locus."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.1559","kind":"arxiv","version":1},"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":"2026-05-18T03:44:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Nik/d5Mt8IvGZ0MuSkvKIqiO6L/3CnOuHtbWWckkiGbWsVct1mPR9V6uOJtBqqqQjNFEiqQKExJxdZibmACqAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T15:04:02.715611Z"},"content_sha256":"767fc916fdd8fcc89e2ab4caf63de877fe98b60f2623ac0291a0c5b5cb44fc93","schema_version":"1.0","event_id":"sha256:767fc916fdd8fcc89e2ab4caf63de877fe98b60f2623ac0291a0c5b5cb44fc93"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7BREYV3XP5SK2QN5HIEGJPNHAS/bundle.json","state_url":"https://pith.science/pith/7BREYV3XP5SK2QN5HIEGJPNHAS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7BREYV3XP5SK2QN5HIEGJPNHAS/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":"2026-08-04T15:04:02Z","links":{"resolver":"https://pith.science/pith/7BREYV3XP5SK2QN5HIEGJPNHAS","bundle":"https://pith.science/pith/7BREYV3XP5SK2QN5HIEGJPNHAS/bundle.json","state":"https://pith.science/pith/7BREYV3XP5SK2QN5HIEGJPNHAS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7BREYV3XP5SK2QN5HIEGJPNHAS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:7BREYV3XP5SK2QN5HIEGJPNHAS","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":"d38361717de351981a17895810b21a7c0377db3d85a9f8683b88d544869ecaea","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2012-10-04T19:41:32Z","title_canon_sha256":"09d4246cdd6c2a005b79195c9117b5379d06f7a2e3663ca1139db425bf8e5486"},"schema_version":"1.0","source":{"id":"1210.1559","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.1559","created_at":"2026-05-18T03:44:03Z"},{"alias_kind":"arxiv_version","alias_value":"1210.1559v1","created_at":"2026-05-18T03:44:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.1559","created_at":"2026-05-18T03:44:03Z"},{"alias_kind":"pith_short_12","alias_value":"7BREYV3XP5SK","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_16","alias_value":"7BREYV3XP5SK2QN5","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_8","alias_value":"7BREYV3X","created_at":"2026-05-18T12:26:56Z"}],"graph_snapshots":[{"event_id":"sha256:767fc916fdd8fcc89e2ab4caf63de877fe98b60f2623ac0291a0c5b5cb44fc93","target":"graph","created_at":"2026-05-18T03:44:03Z","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 study the reduction of certain integral models of Shimura varieties of PEL type with Iwahori level structure. On these spaces we have the Kottwitz-Rapoport and the $p$-rank stratification. We show that the $p$-rank is constant on a KR stratum, generalizing a result of Ng\\^o and Genestier. We prove an abstract, uniform formula for the $p$-rank on a KR stratum. In the symplectic and in the unitary case we derive explicit formulas for its value. We apply these formulas to the question of the density of the ordinary locus and to the question of the dimension of the $p$-rank 0 locus.","authors_text":"Philipp Hartwig","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2012-10-04T19:41:32Z","title":"Kottwitz-Rapoport and p-rank strata in the reduction of Shimura varieties of PEL type"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.1559","kind":"arxiv","version":1},"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":"sha256:5527a9b568ad50e8ffd0adc113d93f2868e5ce74f7b70b3cc3707ba05aec16c9","target":"record","created_at":"2026-05-18T03:44:03Z","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":"d38361717de351981a17895810b21a7c0377db3d85a9f8683b88d544869ecaea","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2012-10-04T19:41:32Z","title_canon_sha256":"09d4246cdd6c2a005b79195c9117b5379d06f7a2e3663ca1139db425bf8e5486"},"schema_version":"1.0","source":{"id":"1210.1559","kind":"arxiv","version":1}},"canonical_sha256":"f8624c57777f64ad41bd3a0864bda70480b605b513ac14dbaa65ffbb9ae12454","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f8624c57777f64ad41bd3a0864bda70480b605b513ac14dbaa65ffbb9ae12454","first_computed_at":"2026-05-18T03:44:03.147922Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:44:03.147922Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tJehfTro8Phly/RTIYH5j/vlOeSCAneUDdKRPqPckZSyl5axRCivQHYlhfexTXEzweZ/TLtEOef5hqQBXsw7Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:44:03.148670Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.1559","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5527a9b568ad50e8ffd0adc113d93f2868e5ce74f7b70b3cc3707ba05aec16c9","sha256:767fc916fdd8fcc89e2ab4caf63de877fe98b60f2623ac0291a0c5b5cb44fc93"],"state_sha256":"66fe90389a10e7af70b0a8b81ef05aef77cabcef44ec54badc103365c65bde59"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Z2RFA4mNnakeybnMC8M6yDJCYwaurvahQr+z0jTOwsrWoM0HmU9rHO7TjF0haxNXcNwEaJ488Q82LwyXNkf4AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-04T15:04:02.719615Z","bundle_sha256":"f6652d1aa6a29b6ae309af9c3a5046e0a1490b3fc6d4e2735cfa10cbac90cb8a"}}