The circle method and non-lacunarity of meromorphic modular forms (Q2515812)

The authors consider the lacunaricity of meromorphic modular forms \(f\) of arbitrary real weight for finite index subgroups of the modular group \(\mathrm{SL}_2(\mathbb{Z})\). They show the following three results: {\parindent=0.7cm\begin{itemize}\item[--] If \(f\) is lacunary, then \(f\) is holomorphic on the complex upper half plane \(\mathbb{H}\). \item[--] If the weight is strictly negative, then \(f\) is always strongly non-lacunary. \item[--] The same holds if \(k\geq 0\) and \(f\) is not finite at the cusps. \end{itemize}} To obtain their results, the authors review the Hardy-Rademacher circle method and extend this theory to the case \(k\geq 0\). Moreover, they present a variant of this method using Poincaré series. These results complement work of \textit{J.-P. Serre} [Publ. Math., Inst. Hautes Étud. Sci. 54, 123--201 (1981; Zbl 0496.12011)] on holomorphic cusp forms.