On weak Jacobi forms of rank two (Q6052930)

Classical Jacobi forms and weak Jacobi forms as introduced by \textit{M. Eichler} and \textit{D. Zagier} [The theory of Jacobi forms. Boston-Basel-Stuttgart: Birkhäuser (1985; Zbl 0554.10018)] are modular forms for the group \begin{gather*} \mathrm{SL}_2(\mathbb{R}) \ltimes \big( \mathbb{R}^2 \mathop{\widetilde\times} \mathbb{R}) \text{,} \end{gather*} where the second factor is a central extension isomorphic to the usual Heisenberg group. Such Jacobi forms come with, besides the weight~\(k \in \mathbb{Z}\), a Jacobi index~\(m \in \mathbb{Z}\), and hence yield a bigraded ring. Generators for the ring of linear combinations of weak Jacobi forms (of even weight) \begin{gather*} \mathrm{J}^{\mathrm{w}}_{2 \bullet, \bullet} = \mathbb{C}\big[ \phi_{-2,1}, \phi_{0,1}, E_4, E_6 \big] \quad\text{and}\quad \mathrm{J}^{\mathrm{w}}_{\bullet, \bullet} = \mathrm{J}^{\mathrm{w}}_{2 \bullet, \bullet}\big[ \phi_{-1,2} \big] \end{gather*} were determined by Eichler-Zagier. In particular, these rings are finitely generated, while after some confusion in the literature [\textit{B. Runge}, Acta Math. 175, No. 2, 165--196 (1995; Zbl 0882.11028)] it now seems by yet unpublished work of \textit{A. M. Botero} et al. [``Rings of Siegel-Jacobi forms of bounded relative index are not finitely generated'', Preprint, \url{arXiv:2203.14583}] that the analogous rings of linear combinations of Jacobi forms are not finitely generated even when bounding the ratio of~\(m\) and~\(k\). The foundations of natural generalizations to symplectic groups~\(\mathrm{Sp}_g(\mathbb{R})\) and to other central extensions~\((\mathbb{R}^{2} \otimes V) \mathop{\widetilde\times} \mathbb{R}\) were published by Ziegler. One reason to study them is their appearance in Fourier-Jacobi expansions of Siegel, orthogonal, and Hermitian modular forms. The authors study the ring of linear combinations of weak Jacobi forms for the group \begin{gather*} \mathrm{SL}_2(\mathbb{R}) \ltimes \big( (\mathbb{R}^2 \otimes V) \mathop{\widetilde\times} \mathbb{R}) \text{,}\quad V = \mathbb{R}^2 \text{,} \end{gather*} for which the Jacobi index~\(m\) is naturally an integral binary quadratic form (represented in the paper as is customary by integral symmetric matrices). Specifically, they determine generators for \begin{gather*} \mathrm{J}^{\mathrm{w}}_{2 \bullet, 2 \left(\begin{smallmatrix} \bullet & \bullet \\ \bullet & \bullet \end{smallmatrix}\right)} \quad\text{and}\quad \mathrm{J}^{\mathrm{w}}_{\bullet, \left(\begin{smallmatrix} \bullet & \bullet \\ \bullet & \bullet \end{smallmatrix}\right)} \text{.} \end{gather*} The proof largely follows ideas of classical work by Igusa. After finding preimages of Eichler-Zagier's generators under suitable pullback maps (which correspond to~\(\mathbb{R} \rightarrow V = \mathbb{R}^2\), \(v \mapsto (v,0)\), \(v \mapsto (0,v)\) and~\(v \mapsto (v,v)\)), they use a further weak Jacobi form whose divisor corresponds to the images of these maps. An induction argument on the weight then yields the result.