Jacobi forms and differential operators: odd weights (Q2360741)

Let \(J_{k,m}(N)\) denote the space of Jacobi forms of weight \(k\) and index \(m\) for the Jacobi group \(\Gamma_0(N)\ltimes\mathbb{Z}^2\) of level \(N\). Eichler and Zagier introduced a family of differential operators \(D_{\nu}\), mapping a Jacobi form of weight \(k\) to a modular form of weight \(k+\nu\). An unpublished question of Böcherer concerns the possibility of removing one non-trivial differential operator from the direct sum \(\oplus_{\nu=0}^{2m}D_\nu\) without affecting the injectivity. When \(k\) is even, \textit{S. Das} and \textit{B. Ramakrishnan} [J. Number Theory 149, 351--367 (2015; Zbl 1387.11032)] showed the last operator \(D_{2m}\) can be removed when \(k,m\) satisfy certain conditions. This paper under review discusses the case of odd \(k\). In view of that the even-indexed differential opertors do not affect injectivity of the direct sum operator, and that the last operator \(D_{2m-1}\) is also removable (see p. 114 of this paper), the authors consider the operator \(i_{2\nu-1}(k,m,N)\) for odd \(k\) and \(1\neq\nu\leq m-1\), \(m\geq3\), which is obtained by deleting \(D_{2\nu-1}\) from the direct sum \(D_1\oplus\cdots\oplus D_{2\nu-1}\oplus\cdots\oplus D_{2m-3}\) of odd-indexed operators. The main result of this paper is the following Theorem 1.1. Let \(k\geq3\) be an odd integer. Then (i) \(i_{2m-3}(k,m,N)\) is injective for all \(N\geq1\) with \(m-k\geq4\). (ii) \(i_{2m-3}(k,m,N)\) is injective for \(N\) square-free and \(m\) odd with \(m-k\geq2\). (iii) \(i_{2m-3}(k,m,N)\) is injective for \(N=1\) and \(m\) odd with \(m-k\geq2\).