Über die Darstellung singulärer Modulformen halbzahligen Gewichts durch Thetareihen. (On the representation of singular modular forms of half-integral weight by theta series)
type: Item
instance of: scholarly article
title: Über die Darstellung singulärer Modulformen halbzahligen Gewichts durch Thetareihen. (On the representation of singular modular forms of half-integral weight by theta series)
zbMATH Open document ID: 0604.10012
DOI: 10.1007/BF01163352
author: Rüdiger Endres
published in: Mathematische Zeitschrift
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173766
review text: Let \(\Gamma_ n=Sp(n, {\mathbb{Z}})\) be the Siegel modular group of degree n and \([\Gamma_ n,m]\) the vector space of modular forms of weight m for \(\Gamma_ n\). A modular form \(f\in [\Gamma_ n,m]\) is called singular if its Fourier coefficients a(T) are different from 0 only when T is singular. The theta series of quadratic forms and characteristics give examples of singular modular forms. Is the vector space of singular modular forms of a given weight, for a congruence subgroup of \(\Gamma_ n\), spanned by theta series ? This question was settled by \textit{E. Freitag} [Siegel modular functions (German) (1983; Zbl 0498.10016)] for the case of the full modular group \([\Gamma_ n,m]\), \(m\in {\mathbb{Z}}\), \(m\equiv 0 mod 4\), and [Math. Z. 156, 141-155 (1977; Zbl 0359.10027)] for the case of integral weight for the group \[ \Gamma_ 0(q)=\{\left( \begin{matrix} A\quad B\\ C\quad D\end{matrix} \right)\in \Gamma_ n | \quad C\equiv 0 mod q\},\quad q\in {\mathbb{N}}. \] Under some invariance property for unimodular transformations the author proves that the vector space of singular modular forms for \(\Gamma_ n(q,2q)\rtimes Sl(n, {\mathbb{Z}})\) \((n>1)\) of half-integral weight r/2 and theta multiplier system \(\vartheta^ r_{\theta}\) is spanned by theta series. Here \(\Gamma_ n(q,2q)\) is the subgroup of theta-group \[ \Gamma_{\theta}=\{\left( \begin{matrix} A\quad B\\ C\quad D\end{matrix} \right)\in \Gamma_ n | \quad A^ tC, B^ tC\quad even\}, \] \[ \Gamma_ n(q,2q)=\{\left( \begin{matrix} A\quad B\\ C\quad D\end{matrix} \right)\in \Gamma_ n| \quad A^ tC| q, B^ tD| q\quad even,\quad \left( \begin{matrix} A\quad B\\ C\quad D\end{matrix} \right)\equiv E mod q\}. \] In the special case of singular modular forms of weight 1/2 and automorphic factor \(\vartheta_{\theta}\) of the theta function \(\vartheta (z)=\sum_{g\in {\mathbb{Z}}^ n}\exp (\pi i Z[g])\), formulas for the dimension of \([\Gamma_ n(q,2q),1/2,\quad \vartheta_{\theta}]\) and \([\Gamma_ n(2q),1/2,\quad \…
Mathematics Subject Classification ID: 11F11
zbMATH DE Number: 3975019
zbMATH Keywords: dimension formula
zbMATH Keywords: singular modular forms
zbMATH Keywords: vector space
zbMATH Keywords: spanned by theta series
zbMATH Keywords: half-integral weight
zbMATH Keywords: theta multiplier system
zbMATH Keywords: subgroup of theta- group
zbMATH Keywords: automorphic factor
reviewed by: Jannis A. Antoniadis
MaRDI profile type: Publication
cites work: BEHAVIOR OF THETA SERIES OF DEGREE $ N$ UNDER MODULAR SUBSTITUTIONS
cites work: Q5531597
cites work: Multiplikatorsysteme der symplektischen Thetagruppe
cites work: Stabile Modulformen
cites work: Die Invarianz gewisser von Thetareihen erzeugter Vektorräume unter Heckeoperatoren
cites work: Q3963086
cites work: Q5344314
cites work: Zur Theorie der Siegelschen Modulgruppe
cites work: Q5668937
cites work: On a Class of Linear Differential Equations for Automorphic Forms in Several Complex Variables
cites work: Automorphic forms of singular weight are singular forms
cites work: Q4146791
cites work: Q3960771
cites work: Prime Determinant Matrices and the Sympletic Theta Function
cites work: Evaluating symplectic Gauss sums and Jacobi Symbols
description: scientific article; zbMATH DE number 3975019