Construction of Jacobi forms
A Jacobi form \(\varphi\) with weight \(k\) and index \(m\) on \(SL(2, \mathbb{Z})\) is defined to be a holomorphic function \(\varphi:{\mathcal H}\times \mathbb{C}\to \mathbb{C}\) satisfying the two transformation equations, for given positive integers \(k\) and \(m\), \[ \begin{aligned} (c\tau+ d)^{-k} e^{2\pi im {{-cz^ 2} \over {c\tau+d}}} \varphi \Biggl( {{a\tau+b} \over {c\tau+d}}, {z \over {c\tau+d}} \Biggr) &= \varphi (\tau, z),\\ e^{2\pi im (\lambda^ 2\tau+ 2\lambda z)} \varphi (\tau,z+ \lambda\tau+ \mu) &= \varphi (\tau, z), \end{aligned} \] for any \((\lambda, \mu)\in \mathbb{Z}^ 2\) and \(\left( \begin{smallmatrix} a &b\\ c&d \end{smallmatrix} \right)\in SL(2, \mathbb{Z})\), and it has a Fourier expansion of the form \[ \varphi(\tau, z)= \sum^ \infty _{\substack{ n=1\\ r\in \mathbb{Z}, r^ 2\leq 4nm }} c(n,r) \xi^ r q^ n, \] where \(q= e^{2\pi i\tau}\) and \(\xi= e^{2\pi iz}\). The authors adapt the Rankin-Selberg convolution to give a construction of Jacobi forms on the modular group \(\Gamma(1)\). Given two Jacobi cusp forms on \(\Gamma (1)\), one of weight \(k\) and index \(m\) and the other of weight \(\ell\) and index 0, the construction yields a third Jacobi cusp form, of weight \(k-\ell\) and index \(m\). The method is subject to the restriction \(k>\ell +5>5\).
scientific article; zbMATH DE number 784100