Note on Siegel-Eisenstein series of low weight
type: Item
instance of: scholarly article
title: Note on Siegel-Eisenstein series of low weight
zbMATH Open document ID: 0867.11035
DOI: 10.1007/BF02940802
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1997-08-10T00:00:00Z
review text: The author mainly proves the results announced in [Proc. Japan. Acad., Ser. A 70, 115-117 (1994; Zbl 0815.11026)]. Let \(\Gamma_n(K)\) denote the Hermitian modular group of degree \(n\) over the imaginary quadratic number field \(K\) of class number 1. The author considers the Siegel-Eisenstein series \[ E^{(n)}_k(Z,s):= \sum_{(\begin{smallmatrix} A\\ C\end{smallmatrix}\begin{smallmatrix} B\\ D\end{smallmatrix}):(\begin{smallmatrix} *\\ 0\end{smallmatrix} \begin{smallmatrix} *\\ *\end{smallmatrix})\backslash\Gamma_n(K)}\text{det}(CZ+ D)^{-k}|\text{det}(CZ+ D)|^{-s}, \] which converges absolutely for \(\text{Re}(s)+ k>2n\). At first he calculates the Fourier expansion expressing the coefficients in terms of Koecher-Maaß zeta-functions. It is shown that \(E^{(n)}_k(Z,s)\) possesses a meromorphic continuation to the whole \(s\)-plane and satisfies a functional equation under \(s\mapsto 2n-s\). Now Shimura's results are applied in order to derive conditions when Eisenstein series of low weight are holomorphic in \(Z\) as well as identities with theta series.
Mathematics Subject Classification ID: 11F55
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 11F66
zbMATH DE Number: 977864
zbMATH Keywords: Hermitian modular group
zbMATH Keywords: Siegel-Eisenstein series
zbMATH Keywords: Fourier expansion
zbMATH Keywords: Koecher-Maaß zeta-functions
zbMATH Keywords: meromorphic continuation
zbMATH Keywords: functional equation
zbMATH Keywords: theta series
author: Shoyu Nagaoka
reviewed by: Aloys Krieg
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 977864