A note on the restriction map for Jacobi forms (Q1969651)

Let \(J_{k,m}(\Gamma_0(N),\chi)\) be the space of holomorphic Jacobi forms of weight \(k\), index \(m\), level \(N\) and character \(\chi\). Jacobi forms \(\phi(\tau,z)\in J_{k,m}(\Gamma_0(N),\chi)\) give rise to modular forms of weight \(k\) by restriction to \(z=0\), and more generally modular forms of weight \(k+\nu\) by certain differential operators \(D_\nu\), which are polynomials in \(\partial/\partial\tau\) and \(\partial/\partial z\), evaluated at \(z=0\). In this paper the authors give two explicit descriptions of the kernel \(J_{k,m}(\Gamma_0(N),\chi)^o\) of \(D_0\) in the special case of index \(1\): One description is in terms of modular forms of weight \(k-1\), the second one is a precise description of the image of that kernel under \(D_2\) (i.e. in terms of modular forms of weight \(k+2\)). Indeed, the authors obtain the commuting diagram of isomorphisms between \(J_{k,1}(\Gamma_0(N),\chi)^o\) and two spaces of modular forms \(M^{k-1}(\Gamma_0(N),\chi\bar{\omega})\) (weight \(k-1\)) and \(S^{k+2}(\Gamma_0(N),\chi)^o\) (weight \(k+2\)). Also, they prove that the last two spaces are related to each other by multiplication by a fixed modular form of weight \(3\).