Spaces of holomorphic functions equivalent to the even Maass cusp forms
type: Item
instance of: scholarly article
title: Spaces of holomorphic functions equivalent to the even Maass cusp forms
zbMATH Open document ID: 0922.11043
author: J. Lewis
published in: Inventiones Mathematicae
publication date: 1999-03-16T00:00:00Z
review text: The aim of the paper under review is to establish a most remarkable bijection between the space \({\mathcal E}^0_{s,e}\) of even Maaß cusp forms \(u\) on the modular group such that \[ -y^2\Biggl({\partial^2\over\partial x^2}+ {\partial^2\over\partial y^2}\Biggr) u= s(1- s)u \] (where \(s= {1\over 2}+ ir\) with \(r\in\mathbb{R}\)) and a certain space \({\mathcal I}{\mathcal E}^0_s\) of entire functions on \(\mathbb{C}\). The functions \(g\in{\mathcal I}{\mathcal E}^0_s\) satisfy (among other conditions) the integral equation \[ g(z)= \int^\infty_0 J_{2s- 1}(2\sqrt{zx}) \Biggl({x\over z}\Biggr)^{s-{1\over 2}} {g(x)\over e^x- 1} dx; \] the bijection from \({\mathcal E}^0_{s,e}\) to \({\mathcal I}{\mathcal E}^0_s\) is given by an integral transform of Hankel type. In the course of the proof a second type of functions \(\psi\) comes up which are also equivalent to the even Maaß cusp forms under consideration: These functions are holomorphic on the cut plane, satisfy a growth condition and the functional equation \[ \psi(z)- \psi(z+ 1)= z^{-2s}\psi\Biggl(1+ {1\over z}\Biggr) \] with \(s\) as above. The map \(u\mapsto \psi\) is effected by a simple integral transform. The functions satisfying the last functional equation (plus some further conditions) may be considered as the period functions corresponding to the even Maaß forms. These functions are also connected with D. Mayer's dynamical systems approach to the Selberg zeta function.
Mathematics Subject Classification ID: 11F37
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11F72
zbMATH DE Number: 1026486
zbMATH Keywords: Laplace transform
zbMATH Keywords: Hankel transform
zbMATH Keywords: even Maaß cusp forms
zbMATH Keywords: entire functions
zbMATH Keywords: period functions
reviewed by: Jürgen Elstrodt
MaRDI profile type: Publication
full work available at URL: s002220050120
OpenAlex ID: W2026303491
DOI: 10.1007/S002220050120
description: scientific article; zbMATH DE number 1026486