Dirichlet series corresponding to Siegel's modular forms of degree n with level N
type: Item
instance of: scholarly article
title: Dirichlet series corresponding to Siegel's modular forms of degree n with level N
zbMATH Open document ID: 0714.11031
author: T. Arakawa
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1990-01-01T00:00:00Z
review text: Let \(F(Z)=\sum_{T}a(T)e^{2\pi i trace(TZ)}\) be a Siegel modular form of degree n and weight k. A natural type of Dirichlet series attached to F is \[ L(F,s):=\sum_{\{T\}}a(T)/\epsilon (T) \det (T)^ s, \] where \(\{\) \(T\}\) runs through all positive definite semi-integral symmetric n by n matrices modulo the action of Gl(n,\({\mathbb{Z}})\) and \(\epsilon\) (T) denotes the number of units of T. Such series were formerly studied by Koecher and Maaß [see e.g. \textit{H. Maaß}'s book ``Siegel's modular forms and Dirichlet series'', Lect. Notes Math. 216 (1971; Zbl 0224.10028)], their residues however remained unknown, until the author (using Klingen type Eisenstein series) proved a residue formula for them [Math. Ann. 238, 157-173 (1978; Zbl 0373.10017)]. In the paper under review the author gives another proof of the residue formula, which is more complicated than the previous one, but it has two advantages: It avoids the use of Klingen-Eisenstein series and the restriction on the weight is weaker \((k>n-1\) instead of \(k>2n)\). The author's result covers the case of Siegel modular forms for congruence subgroups of type \(\Gamma^ n_ 0(N).\) By essentially the same method the author also computes the residues of Koecher-Epstein zeta-functions attached to positive definite quadratic forms. That residue formula was claimed in \textit{M. Koecher}'s dissertation [J. Reine Angew. Math. 192, 1-23 (1953; Zbl 0053.055)], but - as is well known - there is a gap in Koecher's proof.
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11F46
Mathematics Subject Classification ID: 11E45
zbMATH DE Number: 4175033
zbMATH Keywords: Siegel modular form
zbMATH Keywords: Dirichlet series
zbMATH Keywords: residue formula
zbMATH Keywords: residues of Koecher-Epstein zeta-functions
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178227657
OpenAlex ID: W2030312722
cites work: Q3754056
cites work: Dirichlet series corresponding to Siegel's modular forms
cites work: Zum Darstellungssatz für Siegelsche Modulformen
cites work: Q3325811
cites work: Über Dirichlet-Reihen mit Funktionalgleichung.
cites work: Siegel's modular forms and Dirichlet series. Course given at the University of Maryland, 1969-1970
cites work: Vektorwertige Siegelsche Modulformen kleinen Gewichtes.
DOI: 10.2748/TMJ/1178227657
description: scientific article; zbMATH DE number 4175033