On Hermitian Eisenstein series
type: Item
instance of: scholarly article
title: On Hermitian Eisenstein series
zbMATH Open document ID: 0815.11026
author: Shoyu Nagaoka
published in: Proceedings of the Japan Academy. Series A
publication date: 1994-12-18T00:00:00Z
review text: Let \(\Gamma_ n(K)\) denote the Hermitian modular group of degree \(n\) in the sense of \textit{H. Braun} [Ann. Math., II. Ser. 50, 827-855 (1949; Zbl 0038.238)] associated with an imaginary quadratic number field \(K\) of class number 1. The author considers the Eisenstein series \[ E_ k^{(n)} (Z,s) : =\sum_{{AB \choose CD} : {** \choose 0*} \backslash \Gamma_ n (K)} \text{det} (CZ + D)^{-k} |\;\text{det} (CZ + D) |^{-s}, \] which converges absolutely if and only if \(\text{Re} (s) + k>2n\). At first the author shows that \(E_ k^{(n)}(Z,s)\) can be meromorphically continued to the whole \(s\)-plane and satisfies a functional equation under \(s \mapsto 2n - s\). Now the author makes use of \textit{G. Shimura}'s work [Duke Math. J. 50, 417-476 (1983; Zbl 0519.10019)] and the theory of singular modular forms. As an application he obtains conditions when Eisenstein series of low weight are holomorphic in \(Z\) as well as certain identities with theta series e.g. \[ E_ 4^{(4)} (Z,0) = 2 \theta^{(4)} (Z;I), \] where \(I\) denotes the Iyanaga form.
Mathematics Subject Classification ID: 11F55
Mathematics Subject Classification ID: 11F66
zbMATH DE Number: 701463
zbMATH Keywords: Hermitian modular forms
zbMATH Keywords: differential operators
zbMATH Keywords: Eisenstein series
zbMATH Keywords: functional equation
zbMATH Keywords: singular modular forms
zbMATH Keywords: theta series
zbMATH Keywords: Iyanaga form
reviewed by: Aloys Krieg
MaRDI profile type: Publication
full work available at URL: pjaa.70.115
OpenAlex ID: W2061518261
cites work: ON THE ANALYTIC PROPERTIES OF STANDARD ZETA FUNCTIONS OF SIEGEL MODULAR FORMS
cites work: Hermitian modular functions
cites work: Q3963086
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cites work: On Eisenstein series
cites work: Differential operators and the singular values of Eisenstein series
DOI: 10.3792/PJAA.70.115
description: scientific article; zbMATH DE number 701463