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In this paper, we show that when a non-zero modular function of arbitrary real weight for any finite index subgroup of the modular group ${\\SL}_2(\\Z)$ is lacunary, it is necessarily holomorphic on the upper-half plane, finite at the cusps and has non-negative weight."},"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":"1210.5608","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-10-20T11:51:29Z","cross_cats_sorted":[],"title_canon_sha256":"d674fb5f005c365e9dbf200596a66519dea61e54420a2a34640ad317f2ab98b3","abstract_canon_sha256":"9246206687c3eafdcc1218013b2e04c95725ede774a093622935e569270aa624"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:42:43.554779Z","signature_b64":"CieNyXz8sPKZ0toHepDoTp0JITA1KSJLDmtZ3sCNGI/N2XecyGwK6vQD7bZ3dUz+toKkM/PoUmqgvuYZ4gGmCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9422e4dd49f26c29d5725d1334f97d5e055b0d0138a465f4b0e37ad221565924","last_reissued_at":"2026-05-18T03:42:43.554300Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:42:43.554300Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The circle method and non lacunarity of Modular Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Joseph Oesterl\\'e, Sanoli Gun","submitted_at":"2012-10-20T11:51:29Z","abstract_excerpt":"Serre proved that any holomorphic cusp form of weight one for $\\Gamma_1(N)$ is lacunary while a holomorphic modular form for $\\Gamma_1(N)$ of higher integer weight is lacunary if and only if it is a linear combination of cusp forms of CM-type (see Serre, subsections 7.6 and 7.7). 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