Explicit formulae of Siegel Eisenstein series
type: Item
instance of: scholarly article
title: Explicit formulae of Siegel Eisenstein series
zbMATH Open document ID: 0867.11034
DOI: 10.1007/BF02678184
author: Atsushi Haruki
published in: Manuscripta Mathematica
publication date: 1997-08-10T00:00:00Z
full work available at URL: https://eudml.org/doc/156255
review text: Let \(\Gamma_m\) denote the Siegel modular group of degree \(m\). The author investigates the Eisenstein series \[ E^{(m)}_k(Z,s):= \text{det}(\text{Im }z)^s\sum_{(\begin{smallmatrix} a\\ c\end{smallmatrix} \begin{smallmatrix} b\\ d\end{smallmatrix}):(\begin{smallmatrix} *\\ 0\end{smallmatrix} \begin{smallmatrix} *\\ *\end{smallmatrix})\backslash\Gamma_m}\text{det}(cz+ d)^{-k}|\text{det}(cz+ d)|^{-2s}, \] which converges absolutely for \(\text{Re}(s)>(m+1-k)/2\). As is known, \(E^{(m)}_k(z,s)\) has a meromorphic continuation to the whole \(s\)-plane and satisfies a functional equation. The author quotes explicit formulae of the Fourier expansion of \(E^{(m)}_k(z,s)\) from \textit{S. Mizumoto} [Math. Ann. 297, 581-625 (1993; Zbl 0786.11024)]. The orders (as functions of \(s\)) of the Fourier coefficients at \(s=0\) are investigated. They turn out to be holomorphic except for \(k=(m+2)/2\). In the main part, functional equations for some pairs of Fourier coefficients are given. This leads to explicit formulae. Thus, the author is able to reprove \textit{R. Weissauer's} results [Stabile Modulformen und Eisensteinreihen, Lect. Notes Math. 1219, Springer-Verlag, Berlin (1986; Zbl 0596.10023)] on the holomorphy of \(E^{(m)}_k(Z,s)\) at \(s=0\) and the structure of the Fourier coefficients.
Mathematics Subject Classification ID: 11F46
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 11F66
zbMATH DE Number: 980720
zbMATH Keywords: orders of Fourier coefficients
zbMATH Keywords: Siegel modular group
zbMATH Keywords: Eisenstein series
zbMATH Keywords: Fourier expansion
zbMATH Keywords: functional equations
reviewed by: Aloys Krieg
MaRDI profile type: Publication
cites work: Dirichlet series corresponding to Siegel's modular forms of degree n with level N
cites work: Über die Fourierkoeffizienten der Siegelschen Eisensteinreihen
cites work: Dirichletsche Reihe mit Funktionalgleichung in der Theorie der Modulfunktion 2. Grades
cites work: Dirichlet series in the theory of Siegel modular forms
cites work: Q3995202
cites work: Class numbers, Jacobi forms and Siegel-Eisenstein series of weight 2 on \(\text{Sp}_ 2(\mathbb{Z})\)
cites work: On the functional equations satisfied by Eisenstein series
cites work: Siegel's modular forms and Dirichlet series. Course given at the University of Maryland, 1969-1970
cites work: Eisenstein series for Siegel modular groups
cites work: A note on the Siegel-Eisenstein series of weight 2 on \(Sp_ 2(\mathbb{Z})\)
cites work: Note on Siegel-Eisenstein series
cites work: Confluent hypergeometric functions on tube domains
cites work: On Eisenstein series
cites work: Nearly holomorphic functions on Hermitian symmetric spaces
cites work: Stabile Modulformen und Eisensteinreihen. (Stable modular forms and Eisenstein series)
description: scientific article; zbMATH DE number 980720