Abstract
The aim of this paper is to give another short proof of the Saito–Kurokawa lift based on a converse theorem of Imai as was already done by Duke and Imamoglu. In contrast to their proof we avoid spectral analysis but use a real analytic Eisenstein series in a suitable Rankin–Selberg integral involving Siegel’s theta series.
Highlights
The Saito–Kurokawa lift gives a correspondence between classical modular forms contained in the Kohnen space and the Maass Spezialschar of Siegel modular forms
By the work of Maass, Andrianov and Zagier this lift was originally produced as a composition of two liftings involving Jacobi forms, see [3]
Krieg [9] generalized this lifting to the case of Hermitian modular forms
Summary
The Saito–Kurokawa lift gives a correspondence between classical modular forms contained in the Kohnen space and the Maass Spezialschar of Siegel modular forms. Krieg [9] generalized this lifting to the case of Hermitian modular forms. A different strategy of proof in the case of Siegel modular forms was given by Duke and Imamoglu in [2] who used a converse theorem of Imai [5]. Imai’s converse theorem makes use of spectral theory of the Laplacian and in order to establish the lift, Duke and Imamoglu prove functional equations for certain spectral Koecher– Maass series. Instead a real analytic Eisenstein series and. This is an Open Access article published by World Scientific Publishing Company. In a forthcoming paper, we shall apply our method to the higher dimensional case
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