A Hecke correspondence theorem for automorphic integrals with rational period functions
This article carries over to a broader context the Riemann-Hecke correspondence between entire automorphic forms and Dirichlet series with functional equations, related by means of the Mellin transform. The broader context is that of entire automorphic (or modular) integrals, with rational period functions, on \(\Gamma_ \vartheta\) (the subgroup of \(SL(2,\mathbb{Z})\) connected with the modular form \(\vartheta(z))\). These are functions \(f\), holomorphic in the upper half-plane \(H\), such that \[ f(z)=\sum^ \infty_{n=0}a_ ne^{\pi inz},\quad z^{-k}f(- 1/z)=f(z)+q(z),\quad z\hbox { in } H,\leqno (1) \] where \(k\in\mathbb{Z}\), \(a_ n=O(n^ r)\), \(n\to+\infty\), and \(q(z)\) is a rational function of \(z\). The integer \(k\) is called the weight of \(f\), and \(q(z)\) the rational period function associated to \(f\). In the case when \(f\) is an automorphic form of weight \(k\) on \(\Gamma_ \vartheta\) (i.e. \(q=0\) in (1)), the Mellin transform \[ \Phi_ f(s)=\pi^{-s}\Gamma(s)\sum^ \infty_{n=1}a_ nn^{-s} \] has a meromorphic continuation to the entire \(s\)-plane and satisfies the classical (Hecke) functional equation, \(\Phi_ s(k-s)=(- 1)^{k/2}\Phi_ f(s)\). It is proved here that in the more general case of automorphic integrals, the meromorphic continuation still obtains, the Hecke functional equation now replaced by \[ \Phi_ f(k-s)=(- 1)^{k/2}\Phi(s)+R_ k(s),\leqno (2) \] where \(R_ k(s)\) is a finite linear combination of products of beta functions with exponential functions. It is of interest to note that the ordinary hypergeometric functions occur prominently in the derivation of (2). As in Hecke's theory there is a convex theorem as well.
Dirichlet series with functional equations
scientific article; zbMATH DE number 4216378