On Poincaré series and restriction mappings
Let \(J_{n-1,1}^{k,\text{cusp}}(m)\) denote the space of Jacobi cusp forms of degree \((n-1,1)\), weight \(k\) and index \(m\). Let \(S_n^k\) denote the space of Siegel cusp forms of degree \(n\) and weight \(k\) for the Siegel modular group \(\Gamma_n\). Fix \(n,m,k\in \mathbb{N}\), \(k\) even, \(k> 2n> 2\), and consider the linear map \(K_m^{(n)}: J_{n-1,1}^{k, \text{cusp}}(m)\to S_n^k\), \(\Phi\mapsto K_m^{(n)} (Z,\Phi)\) (described on p. 82). The first result of this paper is the explicit formula for \(K_m^{(n)} \left(\left( \begin{smallmatrix} Z&0\\ 0&W \end{smallmatrix} \right), \Phi\right)\), with \((Z,W)\in \mathbb{H}_{n-1} \times \mathbb{H}_1\) (Satz 2.1). The second result (Satz 3.1) is the calculation of the scalar product \[ \bigl\langle \bigl\langle K_m^{(2)} \left(\left( \begin{smallmatrix} \tau &0\\ 0 &\tau' \end{smallmatrix} \right), \Phi_m\right);\;\varphi_1(\tau) \varphi_2(\tau') \bigr\rangle \bigr\rangle, \] where \(\Phi_m\in J_{1,1}^{k, \text{cusp}}(m)\), and \(\varphi_1, \varphi_2\in S_1^k\), are elliptic cusp forms of weight \(k\). The results are part of a series of papers (by the author) on pullbacks of Eisenstein series, Jacobi forms and \(L\)-functions.
scientific article; zbMATH DE number 1416674