Jacobi forms (awaiting review)

A Jacobi form (of full level) of weight $k$ and index $N$ is a holomorphic function $f:\mathcal{H}\times\C\to \C$ where $\mathcal{H}$ is the upper half plane that satisfies the following conditions.

• For all $\left(\begin{matrix}a&b\\c&d\end{matrix} \right) \in SL(2,{\Bbb Z})$, we have $$f\left(\frac{a\tau+b}{c\tau+d}, \frac{z}{c\tau+d}\right)= (c\tau+d)^k e^{2\pi i\, mcz^2 /(c\tau+d)} f(\tau,z)$$
• $f$ has a Fourier expansion $$f(\tau,z)=\sum_{n,r:4nN-r^2\ge0} c(n,r) e^{2\pi i (n\tau+r z)}.$$

The space of Jacobi forms is denoted by $J_{k,N}$. If furthermore, the Fourier expansion is over only those $n,r$ for which $4Nn-r^2>0$, we say it is a Jacobi cusp form, and denote the space of Jacobi cusp forms by $J_{k,N}^{\text{cusp}}$.