On Poincaré series of exponential type on Sp\(_ 2\)
type: Item
instance of: scholarly article
title: On Poincaré series of exponential type on Sp\(_ 2\)
zbMATH Open document ID: 0790.11038
DOI: 10.1007/BF02941348
author: Winfried Kohnen
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1994-06-26T00:00:00Z
review text: Let \(\Gamma_ 2=\text{Sp}_ 2(\mathbb{Z})\) denote the Siegel modular group of degree 2. If \(T\) is a positive definite half-integral \(2\times 2\) matrix, the Poincaré series of exponential type \[ P_{k,T}(Z)= \sum_{M={AB\choose CD}:{I^*\choose 0I}\backslash\Gamma_ 2} \text{det}(CZ+D)^{-k} e^{2\pi i\text{ tr}(TM\langle Z\rangle)}, \] \(k>4\) even, is known to be a Siegel cusp form of weight \(k\). If \(k\geq 8\) is even, the author can describe a representation of \(P_{k,T}\) as an infinite linear combination of Poincaré series on the Jacobi group \(\Gamma^ J_ 1= \text{SL}_ 2(\mathbb{Z})\ltimes \mathbb{Z}^ 2\). This yields the Fourier-Jacobi expansion of \(P_{k,T}\). As an application an elementary formula for the Fourier coefficients of \(P_{k,T}\) is derived. Moreover the adjoint with respect to the Petersson inner product of the maps sending a Siegel cusp form onto its various Fourier-Jacobi coefficients can be described.
Mathematics Subject Classification ID: 11F46
Mathematics Subject Classification ID: 11F30
zbMATH DE Number: 492599
zbMATH Keywords: Poincaré series of exponential type
zbMATH Keywords: Siegel cusp form
zbMATH Keywords: Fourier-Jacobi expansion
zbMATH Keywords: Fourier coefficients
zbMATH Keywords: Petersson inner product
reviewed by: Aloys Krieg
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 492599