Eisenstein series on quaternion half-space of degree 2
type: Item
instance of: scholarly article
title: Eisenstein series on quaternion half-space of degree 2
zbMATH Open document ID: 0759.11019
author: Shoyu Nagaoka
published in: Proceedings of the Japan Academy. Series A
publication date: 1993-01-17T00:00:00Z
review text: The author considers the Eisenstein series of Siegel type on the quaternion half-space of degree 2 \[ E_ k(Z,s):=(\text{det} Y)^{s/2}\sum_{\textstyle {{*\;*\choose C D}:{*\;*\choose 0\;*}\backslash\Gamma_ 2}}|\text{det}(CZ+D)|^{- s}\text{det}(CZ+D)^{-k}, \] where \(\Gamma_ 2\) denotes the modular group over the Hurwitz order [for details cf. the reviewer, Modular forms on half-spaces of quaternions (Lect. Notes Math. 1143) (Springer, Berlin (1985; Zbl 0564.10032)]). By calculating the Siegel series explicitly the meromorphic continuation and functional equation of \(E_ 0(Z,s)\) under \(s\mapsto 6-s\) and \(Z\mapsto{^ tZ}\) are proved. Moreover, the Fourier expansion of \(E_ k(Z,0)\), \(k\geq 4\) even, is determined, which involves divisor sums only. A part of these results were independently communicated by \textit{H. H. Kim} [Manuscr. Math. 76, 85-104 (1992); see the review above].
Mathematics Subject Classification ID: 11F55
zbMATH DE Number: 97939
zbMATH Keywords: Siegel-Eisenstein series
zbMATH Keywords: Eisenstein series
zbMATH Keywords: quaternion half-space
zbMATH Keywords: meromorphic continuation
zbMATH Keywords: functional equation
zbMATH Keywords: Fourier expansion
reviewed by: Aloys Krieg
MaRDI profile type: Publication
full work available at URL: pjaa.68.101
OpenAlex ID: W2026236982
cites work: Dirichletsche Reihe mit Funktionalgleichung in der Theorie der Modulfunktion 2. Grades
cites work: Modular forms on half-spaces of quaternions
cites work: The elementary divisor theory over the Hurwitz order of integral quaternions
cites work: The Maaß space and Hecke operators
cites work: Siegel's modular forms and Dirichlet series. Course given at the University of Maryland, 1969-1970
cites work: An explicit formula for Siegel series
cites work: Confluent hypergeometric functions on tube domains
cites work: On Eisenstein series
DOI: 10.3792/PJAA.68.101
description: scientific article; zbMATH DE number 97939