Periods of modular forms of weight one
The author develops a theory of periods for modular forms of weight one, in analogy with the classic works of M. Eichler and G. Shimura. To cite one of his results, let \(G:=\text{SL}_2(\mathbb{Z})\), let \(\Gamma\) be a subgroup of finite index of \(G\), and let \(M_1(\Gamma)\) stand for the complex vector space of \(\Gamma\)-modular forms of weight one. Further, let \(V:=\left(\begin{smallmatrix} 1 & 0\\ 0 & -1\end{smallmatrix}\right)\), let \(\Gamma':=V\Gamma V\), and let \({\mathfrak M}(\Gamma): =M_1 (\Gamma)\oplus M_1(\Gamma')\). The author defines an injective map \(r: {\mathfrak M}(\Gamma)\hookrightarrow{\mathfrak F}(\Gamma)\) of the vector space \({\mathfrak M})\) into \({\mathfrak F}(\Gamma)\), the space of functions \(f:(\mathbb{R}\setminus\{0\})\times (\Gamma\setminus G)\to\mathbb{C}\) analytic in the first variable, and characterizes the image \(r({\mathfrak M}(\Gamma) )\) of that map as the subspace of those functions \(f\) in \({\mathfrak F}( \Gamma)\) which satisfy the so-called ``Manin relations'' and certain growth conditions at infinity.
scientific article; zbMATH DE number 5015059
scientific article