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We discuss refinements and applications of this result when $V$ is holomorphic, in particular we prove that if $g = e^{h(0)}$ is a finite order automorphism then Tr$_V q^{L(0)-c/24}g$ is a modular function of weight 0 on a congruence subgroup of $SL_2(Z)$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1103.0994","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2011-03-04T23:21:29Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"5d230accc5240e027181a11d5b0832435ed16669ae8244e6a97ccceb618de868","abstract_canon_sha256":"ffbf42c64990cfb7603b6587195285295c45e7d18d03a6c655d8cf108a9970ff"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:34:45.984943Z","signature_b64":"6/Od/CD2WyG/DP++EQTEE7oKKvma+NuXA/EOk1Cj+hQn1WHWpvDIrseP0aQD+UaWWRlb7yzVH0TDwv6+8zOkBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e1083bd0da184249fbcab34fa2a9f993cfdfe675811bdffb04f0064d2f71871f","last_reissued_at":"2026-05-18T01:34:45.984242Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:34:45.984242Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Vertex operator algebras and weak Jacobi forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.QA","authors_text":"Geoffrey Mason, Matthew Krauel","submitted_at":"2011-03-04T23:21:29Z","abstract_excerpt":"Let $V$ be a strongly regular vertex operator algebra. 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