{"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). 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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.5608","kind":"arxiv","version":1},"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"}