{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:JGWF5OO4FCUTGPNXQ5TVIBGWNY","short_pith_number":"pith:JGWF5OO4","canonical_record":{"source":{"id":"1207.4224","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-07-17T22:01:45Z","cross_cats_sorted":[],"title_canon_sha256":"6d25adec19167e66ca5a51560a381895264d2230c84a972693b8c63288e97b84","abstract_canon_sha256":"3e7035bcd5b3084f5eb3041d7db2211f38f307b739c2cce5c6422578b6c50b4b"},"schema_version":"1.0"},"canonical_sha256":"49ac5eb9dc28a9333db787675404d66e2b379d0ae7a958d31ac8cb4a39a6d68b","source":{"kind":"arxiv","id":"1207.4224","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.4224","created_at":"2026-05-18T00:40:16Z"},{"alias_kind":"arxiv_version","alias_value":"1207.4224v2","created_at":"2026-05-18T00:40:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.4224","created_at":"2026-05-18T00:40:16Z"},{"alias_kind":"pith_short_12","alias_value":"JGWF5OO4FCUT","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"JGWF5OO4FCUTGPNX","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"JGWF5OO4","created_at":"2026-05-18T12:27:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:JGWF5OO4FCUTGPNXQ5TVIBGWNY","target":"record","payload":{"canonical_record":{"source":{"id":"1207.4224","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-07-17T22:01:45Z","cross_cats_sorted":[],"title_canon_sha256":"6d25adec19167e66ca5a51560a381895264d2230c84a972693b8c63288e97b84","abstract_canon_sha256":"3e7035bcd5b3084f5eb3041d7db2211f38f307b739c2cce5c6422578b6c50b4b"},"schema_version":"1.0"},"canonical_sha256":"49ac5eb9dc28a9333db787675404d66e2b379d0ae7a958d31ac8cb4a39a6d68b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:40:16.700710Z","signature_b64":"N3HEUaZeuhWvjMtjz4JR0KH3RUMORb2T9dD6q08XScedKUC0uPzF13ULzrnkJCUc+c+sT3Hb/+ims26U6YTdAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"49ac5eb9dc28a9333db787675404d66e2b379d0ae7a958d31ac8cb4a39a6d68b","last_reissued_at":"2026-05-18T00:40:16.700012Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:40:16.700012Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1207.4224","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:40:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"B8J5lIVUFI0ZjFv9BpNjhTSzkJg0jlM1N9KTL2bN9y5tQJthiAY+KC2K5sdrbMHMQevV+uiLw7bwzY/J39/2Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T21:16:02.779290Z"},"content_sha256":"dbb428941bdc66f0f91312af8e4cade155ac459e9826dc8b9807bd8f609ae04c","schema_version":"1.0","event_id":"sha256:dbb428941bdc66f0f91312af8e4cade155ac459e9826dc8b9807bd8f609ae04c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:JGWF5OO4FCUTGPNXQ5TVIBGWNY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Modularity Lifting beyond the Taylor-Wiles Method","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"David Geraghty, Frank Calegari","submitted_at":"2012-07-17T22:01:45Z","abstract_excerpt":"We prove new modularity lifting theorems for p-adic Galois representations in situations where the methods of Wiles and Taylor--Wiles do not apply. Previous generalizations of these methods have been restricted to situations where the automorphic forms in question contribute to a single degree of cohomology. In practice, this imposes several restrictions -- one must be in a Shimura variety setting and the automorphic forms must be of regular weight at infinity. In this paper, we essentially show how to remove these restrictions.\n  Our most general result is a modularity lifting theorem which, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.4224","kind":"arxiv","version":2},"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":"2026-05-18T00:40:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GxjsagohrGe2SneHPnEXN+auImXz9eqQHx3neezyr1y100MMaVP8Q9NyG+rTdT0rMn45OGIhqRc0iKHpH/QvAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T21:16:02.779976Z"},"content_sha256":"ccf180c2b377e9a02285e52977b754b77c4dbe1ce5f92d2550b1af587ecee5ae","schema_version":"1.0","event_id":"sha256:ccf180c2b377e9a02285e52977b754b77c4dbe1ce5f92d2550b1af587ecee5ae"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JGWF5OO4FCUTGPNXQ5TVIBGWNY/bundle.json","state_url":"https://pith.science/pith/JGWF5OO4FCUTGPNXQ5TVIBGWNY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JGWF5OO4FCUTGPNXQ5TVIBGWNY/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":"2026-06-05T21:16:02Z","links":{"resolver":"https://pith.science/pith/JGWF5OO4FCUTGPNXQ5TVIBGWNY","bundle":"https://pith.science/pith/JGWF5OO4FCUTGPNXQ5TVIBGWNY/bundle.json","state":"https://pith.science/pith/JGWF5OO4FCUTGPNXQ5TVIBGWNY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JGWF5OO4FCUTGPNXQ5TVIBGWNY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:JGWF5OO4FCUTGPNXQ5TVIBGWNY","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":"3e7035bcd5b3084f5eb3041d7db2211f38f307b739c2cce5c6422578b6c50b4b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-07-17T22:01:45Z","title_canon_sha256":"6d25adec19167e66ca5a51560a381895264d2230c84a972693b8c63288e97b84"},"schema_version":"1.0","source":{"id":"1207.4224","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.4224","created_at":"2026-05-18T00:40:16Z"},{"alias_kind":"arxiv_version","alias_value":"1207.4224v2","created_at":"2026-05-18T00:40:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.4224","created_at":"2026-05-18T00:40:16Z"},{"alias_kind":"pith_short_12","alias_value":"JGWF5OO4FCUT","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"JGWF5OO4FCUTGPNX","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"JGWF5OO4","created_at":"2026-05-18T12:27:11Z"}],"graph_snapshots":[{"event_id":"sha256:ccf180c2b377e9a02285e52977b754b77c4dbe1ce5f92d2550b1af587ecee5ae","target":"graph","created_at":"2026-05-18T00:40:16Z","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 new modularity lifting theorems for p-adic Galois representations in situations where the methods of Wiles and Taylor--Wiles do not apply. Previous generalizations of these methods have been restricted to situations where the automorphic forms in question contribute to a single degree of cohomology. In practice, this imposes several restrictions -- one must be in a Shimura variety setting and the automorphic forms must be of regular weight at infinity. In this paper, we essentially show how to remove these restrictions.\n  Our most general result is a modularity lifting theorem which, ","authors_text":"David Geraghty, Frank Calegari","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-07-17T22:01:45Z","title":"Modularity Lifting beyond the Taylor-Wiles Method"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.4224","kind":"arxiv","version":2},"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:dbb428941bdc66f0f91312af8e4cade155ac459e9826dc8b9807bd8f609ae04c","target":"record","created_at":"2026-05-18T00:40:16Z","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":"3e7035bcd5b3084f5eb3041d7db2211f38f307b739c2cce5c6422578b6c50b4b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-07-17T22:01:45Z","title_canon_sha256":"6d25adec19167e66ca5a51560a381895264d2230c84a972693b8c63288e97b84"},"schema_version":"1.0","source":{"id":"1207.4224","kind":"arxiv","version":2}},"canonical_sha256":"49ac5eb9dc28a9333db787675404d66e2b379d0ae7a958d31ac8cb4a39a6d68b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"49ac5eb9dc28a9333db787675404d66e2b379d0ae7a958d31ac8cb4a39a6d68b","first_computed_at":"2026-05-18T00:40:16.700012Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:40:16.700012Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"N3HEUaZeuhWvjMtjz4JR0KH3RUMORb2T9dD6q08XScedKUC0uPzF13ULzrnkJCUc+c+sT3Hb/+ims26U6YTdAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:40:16.700710Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.4224","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dbb428941bdc66f0f91312af8e4cade155ac459e9826dc8b9807bd8f609ae04c","sha256:ccf180c2b377e9a02285e52977b754b77c4dbe1ce5f92d2550b1af587ecee5ae"],"state_sha256":"44716c71f86d8ccd95ebf414031af3ab7219ba283e82973e2f6470c28bd7c7ab"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7GV/SNr49SqC5zdJFDgcBNmuJhotg6FWxm5PxewQ/hycRlYZJHTi6/BtGnxBNIZi2LFdCy7ymHVgGDlcnBt8Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T21:16:02.784160Z","bundle_sha256":"64b8e77580d19835ad5d4cc13f5cccd92e59f7bd2345d5256d43ed70cd71e793"}}