Free modules of vector-valued modular forms and applications to Jacobi forms

Abstract

The space $M(\rho ;\mathcal{E})$ of weakly holomorphic modular forms for a dimension-$r$ complex representation $\rho$ of ${\mathrm{SL}}_2(\mathbb{Z})$ with growth prescribed by $\mathcal{E}$ at the cusp is a free module over $\mathbb{C}[E_4,E_6]$. We first prove that a $\mathbb{C}[E_4,E_6]$-linearly independent homogeneous family $F_1,\dots ,F_r$ in this module gives a basis if and only if ${\sum }_{i=1}^r\mathrm{wt}(F_i)=12\mathrm{tr}\mathcal{E}$. We then extend this criterion to Jacobi forms. As applications, we determine the generator weights of free modules of $\mathrm{O}(D_{8d}^{+})$-invariant Jacobi forms of index $D_{8d}^{+}(p)$, where $D_{8d}^{+}$ is an even unimodular lattice extending $D_{8d}$, $d\ge 2$, and $p=2,3,5$.

arXiv comment36 pages