A Hecke correspondence theorem for automorphic integrals with rational period functions
type: Item
instance of: scholarly article
title: A Hecke correspondence theorem for automorphic integrals with rational period functions
zbMATH Open document ID: 0742.11028
author: John H. Hawkins
author: Marvin Knopp
published in: Illinois Journal of Mathematics
publication date: 1992-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11F12
zbMATH DE Number: 4216378
zbMATH Keywords: modular integral
zbMATH Keywords: Riemann-Hecke correspondence
zbMATH Keywords: entire automorphic forms
zbMATH Keywords: Dirichlet series with functional equations
zbMATH Keywords: Mellin transform
zbMATH Keywords: rational periodic functions
zbMATH Keywords: meromorphic continuation
zbMATH Keywords: automorphic integrals
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4216378