{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:4NMLDUZSPRS7C7GXSIN7TONOCH","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":"ecb3b09f584c78b27a07c3f9ed7e2efe0da4b2476c3c5e25526cdb82b74f61db","cross_cats_sorted":["hep-th"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"hep-ph","submitted_at":"2021-12-29T18:57:42Z","title_canon_sha256":"fa82473be9f54354c5a71d827e10bf9c7006cc989fb765030dac172008e978fd"},"schema_version":"1.0","source":{"id":"2112.14761","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2112.14761","created_at":"2026-07-05T04:13:49Z"},{"alias_kind":"arxiv_version","alias_value":"2112.14761v1","created_at":"2026-07-05T04:13:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2112.14761","created_at":"2026-07-05T04:13:49Z"},{"alias_kind":"pith_short_12","alias_value":"4NMLDUZSPRS7","created_at":"2026-07-05T04:13:49Z"},{"alias_kind":"pith_short_16","alias_value":"4NMLDUZSPRS7C7GX","created_at":"2026-07-05T04:13:49Z"},{"alias_kind":"pith_short_8","alias_value":"4NMLDUZS","created_at":"2026-07-05T04:13:49Z"}],"graph_snapshots":[{"event_id":"sha256:9e75c60e3c9aa45a9578a5afa4980249ee5f1e643cbc0f638e0f4ae566b47806","target":"graph","created_at":"2026-07-05T04:13:49Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2112.14761/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We revisit the modular flavor symmetry from a more general perspective. The scalar modular forms of principal congruence subgroups are extended to the vector-valued modular forms, then we have more possible finite modular groups including $\\Gamma_N$ and $\\Gamma'_N$ as the flavor symmetry. The theory of vector-valued modular forms provide a method of differential equation to construct the modular multiplets, and it also reveals the simple structure of the modular invariant mass models. We review the theory of vector-valued modular forms and give general results for the lower dimensional vector-","authors_text":"Gui-Jun Ding, Xiang-Gan Liu","cross_cats":["hep-th"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"hep-ph","submitted_at":"2021-12-29T18:57:42Z","title":"Modular flavor symmetry and vector-valued modular forms"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2112.14761","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":"sha256:53ecf65ef8a4d55b26e441b19ae3ca0c5cae1f6c7063b5647a59a61c9d448b16","target":"record","created_at":"2026-07-05T04:13:49Z","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":"ecb3b09f584c78b27a07c3f9ed7e2efe0da4b2476c3c5e25526cdb82b74f61db","cross_cats_sorted":["hep-th"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"hep-ph","submitted_at":"2021-12-29T18:57:42Z","title_canon_sha256":"fa82473be9f54354c5a71d827e10bf9c7006cc989fb765030dac172008e978fd"},"schema_version":"1.0","source":{"id":"2112.14761","kind":"arxiv","version":1}},"canonical_sha256":"e358b1d3327c65f17cd7921bf9b9ae11d4a15767410e2ca5b28cd271cf490e2a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e358b1d3327c65f17cd7921bf9b9ae11d4a15767410e2ca5b28cd271cf490e2a","first_computed_at":"2026-07-05T04:13:49.052770Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:13:49.052770Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4GFdJdjiQfLstgqVSQoDf5omZsyJpz9IptQ33At6MZwh8hOTuVKSbhLQIAireMaHo/Afm7JnNjd+wraLcHnHAw==","signature_status":"signed_v1","signed_at":"2026-07-05T04:13:49.053239Z","signed_message":"canonical_sha256_bytes"},"source_id":"2112.14761","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:53ecf65ef8a4d55b26e441b19ae3ca0c5cae1f6c7063b5647a59a61c9d448b16","sha256:9e75c60e3c9aa45a9578a5afa4980249ee5f1e643cbc0f638e0f4ae566b47806"],"state_sha256":"8fdb977eea85169a259a9ef07aaf5da6a37a063b65ab75b645ac89e798be29ef"}