{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:J5KUNFG3VLEXIAFMFKLHADLFMN","short_pith_number":"pith:J5KUNFG3","canonical_record":{"source":{"id":"1409.0370","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-09-01T11:34:41Z","cross_cats_sorted":[],"title_canon_sha256":"40a2d9cb5136698a94d7c16a665fb375ccec14ac58103d26d7b3b533ec691c8e","abstract_canon_sha256":"e6e6636be6fdad13fe5ba32be4b62b869e329307d91f5510b96178875f2b1218"},"schema_version":"1.0"},"canonical_sha256":"4f554694dbaac97400ac2a96700d65637c8dee304423fe813049e1f919fc6baa","source":{"kind":"arxiv","id":"1409.0370","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.0370","created_at":"2026-05-18T01:20:30Z"},{"alias_kind":"arxiv_version","alias_value":"1409.0370v3","created_at":"2026-05-18T01:20:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.0370","created_at":"2026-05-18T01:20:30Z"},{"alias_kind":"pith_short_12","alias_value":"J5KUNFG3VLEX","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_16","alias_value":"J5KUNFG3VLEXIAFM","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_8","alias_value":"J5KUNFG3","created_at":"2026-05-18T12:28:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:J5KUNFG3VLEXIAFMFKLHADLFMN","target":"record","payload":{"canonical_record":{"source":{"id":"1409.0370","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-09-01T11:34:41Z","cross_cats_sorted":[],"title_canon_sha256":"40a2d9cb5136698a94d7c16a665fb375ccec14ac58103d26d7b3b533ec691c8e","abstract_canon_sha256":"e6e6636be6fdad13fe5ba32be4b62b869e329307d91f5510b96178875f2b1218"},"schema_version":"1.0"},"canonical_sha256":"4f554694dbaac97400ac2a96700d65637c8dee304423fe813049e1f919fc6baa","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:20:30.052929Z","signature_b64":"uTiRlYBLMrb+vwOHKBfB4cA9db3X/8kGbyKPXNT4gXYPIYs3cKpfscuir0+YSVIrV/3pouZTqJSYQmRPtbzvAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4f554694dbaac97400ac2a96700d65637c8dee304423fe813049e1f919fc6baa","last_reissued_at":"2026-05-18T01:20:30.052295Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:20:30.052295Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1409.0370","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":"2026-05-18T01:20:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vIDhEjv7aiO5Rni98hyQ/Ee1OwfnvvOizO4tqFowcY1WZP30AOSo4RCznerShhugKUef5qiRhgtzdbZGpIgoAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T23:33:56.572186Z"},"content_sha256":"f9d3f6ffe10c91ecd95a4e6d3facc0e3ed423448926949de4b3fa9485c3a53ed","schema_version":"1.0","event_id":"sha256:f9d3f6ffe10c91ecd95a4e6d3facc0e3ed423448926949de4b3fa9485c3a53ed"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:J5KUNFG3VLEXIAFMFKLHADLFMN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Eichler cohomology in general weights using spectral theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Michael Neururer","submitted_at":"2014-09-01T11:34:41Z","abstract_excerpt":"In this paper, we construct a pairing between modular forms of positive real weight and elements of certain Eichler cohomology groups that were introduced by Knopp in 1974. We use spectral theory of automorphic forms to show that this pairing is perfect for all positive weights except 1. The approach in this paper gives a new proof of a theorem by Knopp and Mawi from 2010 for all real weights excluding 1 and also a version of this theorem for vector-valued modular forms."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.0370","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":"2026-05-18T01:20:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5qMZGIFMvGG+x+hhsjeAOIeTJWedHwO8qvPICWv56JrGwmcTf+Eb0RzSlyG9G1cLnwj8NZj8tm5K0IbDMP2DDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T23:33:56.572956Z"},"content_sha256":"e8e452fdc3d718002fb5b0f4e83b9adaf063d7d6b3d81103af406d18509de208","schema_version":"1.0","event_id":"sha256:e8e452fdc3d718002fb5b0f4e83b9adaf063d7d6b3d81103af406d18509de208"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/J5KUNFG3VLEXIAFMFKLHADLFMN/bundle.json","state_url":"https://pith.science/pith/J5KUNFG3VLEXIAFMFKLHADLFMN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/J5KUNFG3VLEXIAFMFKLHADLFMN/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":"2026-06-11T23:33:56Z","links":{"resolver":"https://pith.science/pith/J5KUNFG3VLEXIAFMFKLHADLFMN","bundle":"https://pith.science/pith/J5KUNFG3VLEXIAFMFKLHADLFMN/bundle.json","state":"https://pith.science/pith/J5KUNFG3VLEXIAFMFKLHADLFMN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/J5KUNFG3VLEXIAFMFKLHADLFMN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:J5KUNFG3VLEXIAFMFKLHADLFMN","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":"e6e6636be6fdad13fe5ba32be4b62b869e329307d91f5510b96178875f2b1218","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-09-01T11:34:41Z","title_canon_sha256":"40a2d9cb5136698a94d7c16a665fb375ccec14ac58103d26d7b3b533ec691c8e"},"schema_version":"1.0","source":{"id":"1409.0370","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.0370","created_at":"2026-05-18T01:20:30Z"},{"alias_kind":"arxiv_version","alias_value":"1409.0370v3","created_at":"2026-05-18T01:20:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.0370","created_at":"2026-05-18T01:20:30Z"},{"alias_kind":"pith_short_12","alias_value":"J5KUNFG3VLEX","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_16","alias_value":"J5KUNFG3VLEXIAFM","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_8","alias_value":"J5KUNFG3","created_at":"2026-05-18T12:28:33Z"}],"graph_snapshots":[{"event_id":"sha256:e8e452fdc3d718002fb5b0f4e83b9adaf063d7d6b3d81103af406d18509de208","target":"graph","created_at":"2026-05-18T01:20:30Z","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 paper, we construct a pairing between modular forms of positive real weight and elements of certain Eichler cohomology groups that were introduced by Knopp in 1974. We use spectral theory of automorphic forms to show that this pairing is perfect for all positive weights except 1. The approach in this paper gives a new proof of a theorem by Knopp and Mawi from 2010 for all real weights excluding 1 and also a version of this theorem for vector-valued modular forms.","authors_text":"Michael Neururer","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-09-01T11:34:41Z","title":"Eichler cohomology in general weights using spectral theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.0370","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:f9d3f6ffe10c91ecd95a4e6d3facc0e3ed423448926949de4b3fa9485c3a53ed","target":"record","created_at":"2026-05-18T01:20:30Z","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":"e6e6636be6fdad13fe5ba32be4b62b869e329307d91f5510b96178875f2b1218","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-09-01T11:34:41Z","title_canon_sha256":"40a2d9cb5136698a94d7c16a665fb375ccec14ac58103d26d7b3b533ec691c8e"},"schema_version":"1.0","source":{"id":"1409.0370","kind":"arxiv","version":3}},"canonical_sha256":"4f554694dbaac97400ac2a96700d65637c8dee304423fe813049e1f919fc6baa","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4f554694dbaac97400ac2a96700d65637c8dee304423fe813049e1f919fc6baa","first_computed_at":"2026-05-18T01:20:30.052295Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:20:30.052295Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uTiRlYBLMrb+vwOHKBfB4cA9db3X/8kGbyKPXNT4gXYPIYs3cKpfscuir0+YSVIrV/3pouZTqJSYQmRPtbzvAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:20:30.052929Z","signed_message":"canonical_sha256_bytes"},"source_id":"1409.0370","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f9d3f6ffe10c91ecd95a4e6d3facc0e3ed423448926949de4b3fa9485c3a53ed","sha256:e8e452fdc3d718002fb5b0f4e83b9adaf063d7d6b3d81103af406d18509de208"],"state_sha256":"b13f03464b136f8795b899542a56a3bbd26e592d66dd4888de3b43cc99b5f2ad"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IuPvtBlm427j/Tzw0ldlC4pYNCtjiRAUIbUBb995fPx4D/GtAmHKUGBm/BfnPvj7CJQg85JFrdalAftAoi9xDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T23:33:56.577355Z","bundle_sha256":"269d306097ebae33b76109f0ffa9d7fbcb653ecb71056e1c125d5f4a1d55cd6a"}}