Über die Darstellung singulärer Modulformen halbzahligen Gewichts durch Thetareihen. (On the representation of singular modular forms of half-integral weight by theta series)
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 \([\Gamma_ n(2q),1/2,\quad \vartheta_{\theta}]\) are given.
scientific article; zbMATH DE number 3975019
scientific article