Mixed automorphic forms and differential equations
To define mixed automorphic forms one needs a holomorphic map \(\omega\) from the upper half plane \({\mathcal H}\) to itself, two cofinite discrete groups, \(G\) and \(\Gamma\), acting in \({\mathcal H}\), and a surjective group homomorphism \(\chi: G\to \Gamma\), such that \(\omega (g\cdot z)=\chi (g)\cdot \omega (z)\) for all \(g\in G\). It is also desirable that \(\omega\) has an extension to the cusps of \(G\) in such a way that it induces a holomorphic map \(X_ G\to X_{\Gamma}\) between the compactifications of the quotients. A mixed automorphic form of type \((k,m)\) is a meromorphic function \(f\) on \({\mathcal H}\) that satisfies the transformation behavior \[ f(\frac{az+b}{cz+d})=(cz+d)^ k(c_{\gamma} \omega(z)+d_{\gamma})^ m f(z) \text{ for all } \left( \begin{matrix} a & b\\ c & d\end{matrix} \right)\in G \text{, with } \left( \begin{matrix} a_{\gamma} & b_{\gamma}\\ c_{\gamma} & d_{\gamma}\end{matrix} \right)=\chi \left( \begin{matrix} a & b\\ c & d\end{matrix} \right), \] and a meromorphy condition at the cusps of \(G\). Take \(m=0\) to get classical meromorphic automorphic forms for \(G\). The derivative \(z\mapsto \omega'(z)\) is a mixed automorphic form of type (2,- 2). \textit{P. Stiller} [Special values of Dirichlet series, monodromy, and the periods of automorphic forms, Mem. Am. Math. Soc. 299 (1984; Zbl 0536.10023) section 3 bis], associates `generalized automorphic forms' to certain inhomogeneous differential equations, and studies this relation. These generalized automorphic forms are mixed automorphic forms of type \((0,k+2)\). The paper under review reformulates Stiller's results in terms of mixed automorphic forms of type \((2,k)\). This is easily done: multiply Stiller's generalized automorphic forms by \(\omega '\).
scientific article; zbMATH DE number 4212143