Type: Article
Publication Date: 1979-12-01
Citations: 7
DOI: https://doi.org/10.21099/tkbjm/1496158689
Let $F$ be a positive integral symmetric matrix of degree $m$ , and $Z$ a variable on the Siegel space $H_{n}$ of degree $n$ .Let $\Phi$ be a spherical function of order $\nu$ with respect to $F$ which is of the formwith an $m\times n$ matrix $\eta$ such that ${}^{t}\eta\eta=0$ if $\nu>1$ .We define a theta series associated with $F$ by settingwhere $U,$ $V$ are $m\times n$ real matrices, tr denotes the trace of a corresponding square matrix and $G$ runs through all $m\times n$ integral matrices.We write simply $\theta_{F,U.V}(Z)$ for the theta series $\theta_{F,U,V}(Z;\Phi)$ when $\Phi$ is of order $0$ .For congruence subgroups of $SL_{2}(Z)$ the transformation formulas for theta series of degree 1 associated with $F$ are well known.For example, we can find transformation formulas for theta series of degree 1 in [7], [8], in which multi- pliers are explicitly determined.Transformation formulas for the theta series $\theta_{F,U.V}(Z;\Phi)$ of degree $n\geq 1$ are also established in [1] in the case where $F$ is even and $U,$ $V$ are zero (the condition on $U,$ $V$ is not necessary if $\Phi$ is of order $0[9]$ ).Using these results we can get many examples of Siegel modular forms for congruence subgroups.In this paper we determine a transformation formula for the theta series $\theta_{F,U,V}(Z;\Phi)$ associated with a positive integral symmetric matrix $F$ and any real matrices $U,$ $V$ and using this, we get some examples of cusp forms for some congruence subgroups $\Gamma^{\prime}$ of $sp_{n}(Z)$ .Cusp forms of weight $n+1$ for $\Gamma^{\prime}$ induce differential forms of the first kind on the nonsingular model of the modular function field with respect to $\Gamma^{\prime}$ .Our result shows that the geometric genus of the nonsingular model of the modular function field with respect to $\Gamma^{\prime}$ is positive.