Thetanullwerte and stable modular forms for Hecke groups
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.
scientific article; zbMATH DE number 703845