The circle method and non-lacunarity of meromorphic modular forms

Abstract

Abstract Serre proved that any holomorphic cusp form of weight one for Γ1(N) is lacunary while a holomorphic modular form for Γ1(N) of higher integer weight is lacunary if and only if it is a linear combination of cusp forms of CM-type (see [Publ. Math. I.H.E.S. 54 (1981), 323–401, Sections 7.6 and 7.7]). In this paper, we show that when a non-zero meromorphic modular form of arbitrary real weight for any finite index subgroup of the modular group SL 2 ( ℤ ) ${\operatorname{SL}}_2(\mathbb {Z})$ is lacunary, it is necessarily holomorphic on the upper-half plane, finite at the cusps and has non-negative weight.