Spaces of holomorphic functions equivalent to the even Maass cusp forms
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.
scientific article; zbMATH DE number 1026486