{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:4UZOYOLD74QWLMPKALMGGZQWCC","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":"9c554f3a5131bf0e5b48556587d43562eac488f9e0c95d5335599c5d831a42cb","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-06-23T08:26:31Z","title_canon_sha256":"ba13896b70efaa26513954aa7b023976ede95f83d64e2d108fbbcc144c0df960"},"schema_version":"1.0","source":{"id":"0906.4189","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0906.4189","created_at":"2026-05-17T23:53:20Z"},{"alias_kind":"arxiv_version","alias_value":"0906.4189v3","created_at":"2026-05-17T23:53:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0906.4189","created_at":"2026-05-17T23:53:20Z"},{"alias_kind":"pith_short_12","alias_value":"4UZOYOLD74QW","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"4UZOYOLD74QWLMPK","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"4UZOYOLD","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:a8a891f86df7343d72927e9815fcfd4672259dcbd75d6c26805f67e2aacbfa88","target":"graph","created_at":"2026-05-17T23:53:20Z","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 prove modularity lifting theorems for l-adic Galois representations of any dimension satisfying a unitary type condition and a Fontaine-Laffaille type condition at l. This extends the results of Clozel, Harris and Taylor, and the subsequent work by Taylor. The proof uses the Taylor-Wiles method, as improved by Diamond, Fujiwara, Kisin and Taylor, applied to Hecke algebras of unitary groups, and results of Labesse on stable base change and descent from unitary groups to GL_n.","authors_text":"Lucio Guerberoff","cross_cats":["math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-06-23T08:26:31Z","title":"Modularity lifting theorems for Galois representations of unitary type"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.4189","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:db397f96f7bac961a338459734e0d4a8273b788cce8b1f2425481e0b283b9e2d","target":"record","created_at":"2026-05-17T23:53:20Z","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":"9c554f3a5131bf0e5b48556587d43562eac488f9e0c95d5335599c5d831a42cb","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-06-23T08:26:31Z","title_canon_sha256":"ba13896b70efaa26513954aa7b023976ede95f83d64e2d108fbbcc144c0df960"},"schema_version":"1.0","source":{"id":"0906.4189","kind":"arxiv","version":3}},"canonical_sha256":"e532ec3963ff2165b1ea02d863661610af78f1dca9be8c6080413c19e929e406","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e532ec3963ff2165b1ea02d863661610af78f1dca9be8c6080413c19e929e406","first_computed_at":"2026-05-17T23:53:20.004366Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:53:20.004366Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"d5QB7yY2DGovZTgAjpcSawUgZ6Uio8uQlrU8zOJPOdK3x5UHjKfBcAqjTI+1EtXCLgAPguti4S0n9e96b5cjDQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:53:20.004994Z","signed_message":"canonical_sha256_bytes"},"source_id":"0906.4189","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:db397f96f7bac961a338459734e0d4a8273b788cce8b1f2425481e0b283b9e2d","sha256:a8a891f86df7343d72927e9815fcfd4672259dcbd75d6c26805f67e2aacbfa88"],"state_sha256":"e7bfab645a2cdf74de81b1f6af6e00a382247c138582f7f44267c2299c26c373"}