Thetanullwerte and stable modular forms for Hecke groups
type: Item
instance of: scholarly article
title: Thetanullwerte and stable modular forms for Hecke groups
zbMATH Open document ID: 0814.11029
DOI: 10.1007/BF02572338
author: Riccardo Salvati Manni
published in: Mathematische Zeitschrift
publication date: 1995-01-25T00:00:00Z
full work available at URL: https://eudml.org/doc/174667
review text: It is well known that the space \([\Gamma_{g,0}(q),k,\chi^ g]\) of modular forms of weight \(k\) relative to the Hecke subgroup \(\Gamma_{g,0}(q)\) of the Siegel modular group of \(\Gamma_ g = \text{Sp}_ g(\mathbb{Z})\) with character \(\chi^ g\) equals in case \(g > 2k\) the space \(B^ g_ k(q,\chi^ g)\) of stable modular forms, which itself is spanned by those theta series \(\theta_ Q\) such that the associated Dirichlet character \(\chi_ Q\) satisfies \(\chi^ g = \chi^ g_ Q\). Following the ideas and methods of previous studies where the analogous problem for the full modular group \(\Gamma_ g\) was tackled, in the present paper the author shows that the subspace \([\Gamma_{g,0}(q), \Sigma_ s, k, \chi^ g]\) spanned by homogeneous polynomials of degree \(2k\) in the Thetanullwerte equals the stable space \(B^ k_ g(q,\chi^ g)\) provided \(q\) is a power of 2, the Igusa subgroup \(\Gamma_ g(s^ 2, 2s^ 2)\) is contained in the Hecke subgroup in question and \(s\) is large enough. This equality is in fact due to the surjectivity of the restriction of the Siegel operator \(\Phi\) to the above space \([\Gamma_{g,0}(q), \Sigma_ s, k, \chi^ g]\). Finally the author gives an application to coding theory showing that certain spaces of modular forms are spanned by theta-series associated to linear codes.
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11F46
Mathematics Subject Classification ID: 94B05
zbMATH DE Number: 703845
zbMATH Keywords: Hecke subgroup
zbMATH Keywords: Siegel modular group
zbMATH Keywords: stable modular forms
zbMATH Keywords: Thetanullwerte
zbMATH Keywords: theta-series
zbMATH Keywords: linear codes
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 703845