Stabile Modulformen und Eisensteinreihen. (Stable modular forms and Eisenstein series)
type: Item
instance of: scholarly article
title: Stabile Modulformen und Eisensteinreihen. (Stable modular forms and Eisenstein series)
zbMATH Open document ID: 0596.10023
author: Rainer Weissauer
published in: Lecture Notes in Mathematics
publication date: 1986-01-01T00:00:00Z
review text: Let \({\mathcal H}_ n:=\{Z=Z^{(n)}=Z'\); Im Z\(>0\}\) denote the Siegel half- space of degree n. Given \(0\leq j\leq n\), an even integer k and a cusp form g on \({\mathcal H}_ j\) of weight k, one defines the Klingen-Eisenstein series by \[ G(Z,s):=\sum_{M}g(\pi (M<Z>))\quad \det (CZ+D)^{-k}\quad (\frac{\det Im(\pi (M<Z>))}{\det Im(M<Z>)})^{-s}, \] where M runs through a set of representatives of the Siegel modular group over a certain subgroup and \(\pi\) denotes the projection of an \(n\times n\) matrix onto its upper left \(j\times j\) corner. G(Z,s) converges for complex s, whenever Re s is sufficiently large. It was proved by \textit{R. P. Langlands} [On the functional equations satisfied by Eisenstein series (Lect. Notes Math. 544) (1976; Zbl 0332.10018)] that G(Z,s) admits a meromorphic continuation to the whole complex s-plane. The author makes strong use of Langlands theory of Eisenstein series and spectral decomposition in order to derive properties of the functions, which are obtained by analytic continuation, and their behavior under the Siegel \(\Phi\)-operator. If \(k>(n+j+3)/2\) the Klingen-Eisenstein series G(Z,s) is regular at \(s=0\). The function \(G(Z):=G(Z,0)\) becomes a Siegel modular form satisfying \(\Phi^{n-j}G=g\). If \(k=(n+j+3)/2\) then G(Z,s) turns out to be regular at \(s=0\) again, but not always holomorphic. In the case \(k=(n+j+1)/2\) the function G(Z,0) proves to be holomorphic, but might identically vanish. If \(k<(n+j+1)/2\) the procedure of Hecke summation has to be modified. In this case \(G(Z):=_{s=s_ 0}G(Z,s)\), where \(s_ 0=(n+j+1)/2-k\), yields a holomorphic modular form, which might identically vanish. The results are used in order to characterize the subspace of stable modular forms. Especially the vector space of all holomorphic cusp forms on \({\mathcal H}_ n\) with respect to an irreducible representation \((V_{\rho},\rho)\) with weight k(\(\rho)\) consists of finite linear combinations of theta series \(\vartheta_{S,P…
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11E45
Mathematics Subject Classification ID: 11-02
zbMATH DE Number: 3959538
zbMATH Keywords: Siegel modular form
zbMATH Keywords: stable modular forms
zbMATH Keywords: theta series
zbMATH Keywords: Eisenstein series
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0072235
OpenAlex ID: W619236162
DOI: 10.1007/BFB0072235
Recommended article: Thetanullwerte and Stable Modular Forms II
Recommended article: Q3726202
Recommended article: Thetanullwerte and Stable Modular Forms
Recommended article: Thetanullwerte and stable modular forms for Hecke groups
Recommended article: On differential modular forms and some analytic relations between Eisenstein series
Recommended article: Eisenstein series in the Kohnen plus space for Hilbert modular forms
Recommended article: Q3824483
Recommended article: Q3722573
Recommended article: Periods of modular forms and identities between Eisenstein series
Recommended article: Generalized Eisenstein series and several modular transformation formulae
description: scientific article; zbMATH DE number 3959538