# The Geometry of Jacobi Forms and their Asymptotic Expansions

A Jacobi form of weight $k$ and index $m$ is a meromorphic function $\varphi: \mathbb{H} \times \mathbb{C} \to \mathbb{C}$ satisfying

$$\varphi\bigg(\frac{a \tau + b}{c \tau + d}, \frac{z}{c \tau + d}\bigg) = (c \tau + d)^{k} e^{\frac{2 \pi i mc}{c \tau +d} z^{2}}\varphi(\tau, z)$$

$$\varphi(\tau, z + \lambda \tau + \mu) = e^{-2 \pi i m(\lambda^{2}\tau + 2 \lambda z)} \varphi(\tau, z).$$

Let $\pi : \mathcal{C} \to \overline{\mathcal{M}}_{1,1}$ be the universal elliptic curve. We can identify the surface $\mathcal{C} \cong \overline{\mathcal{M}}_{1,2}$, and we can realize $\overline{\mathcal{M}}_{1,1}$ as a divisor of $\mathcal{C}$. The general fiber $F$ gives another independent divisor of $\mathcal{C}$.

The way I understand it (please correct me if I'm wrong) a Jacobi form $\varphi$ is a section of a line bundle $\mathcal{E} \to \mathcal{C}$ such that the weight and index are encoded as

$$k = \text{deg}\big( \mathcal{E}|_{\overline{\mathcal{M}}_{1,1}}\big), \,\,\,\,\,\,\,\,\,\,\,\,\, m = \text{deg}\big(\mathcal{E} |_{F}\big)$$

I believe one should think of $\tau$ as a coordinate on $\overline{\mathcal{M}}_{1,1}$ and $z$ as a coordinate in the fiber direction. If $z=0$, indeed we just get a section of a line bundle over $\overline{\mathcal{M}}_{1,1}$, consistent with $\varphi(\tau, 0)$ being an ordinary modular form of weight $k$.

The Jacobi form I'm interested in is $\varphi_{-2,1}$ which is the unique weak Jacobi form of weight -2 and index 1. Its inverse is a meromorphic Jacobi form of weight 2 and index -1. It is well known that we have an asymptotic expansion around $z=0$,

$$\frac{1}{\varphi_{-2,1}}(\tau, z) = \sum_{g=0}^{\infty} z^{2g-2} \mathcal{P}_{g}(\tau),$$

where $\mathcal{P}_{g}(\tau)$ is a quasi-modular form of weight $g$. Basically, I want to understand the geometry of an expansion of this form.

First off, applying the elliptic transformation law of Jacobi forms (the second of the two transformation equations above) to this expansion gives nonsense, which makes sense because it holds only for small $z$. In addition, we lose all information about the index, since that is associated to the fiber directions (i.e. general $z$). However, I would expect it to encode the weight perfectly. And it almost does! We can try to apply a modular transformation

$$\frac{1}{\varphi_{-2,1}}\big(\frac{a \tau + b}{c \tau + d}, \frac{z}{c \tau + d}\big)= \sum_{g=0}^{\infty} \big( \frac{z}{c \tau + d}\big)^{2g-2} \mathcal{P}_{g}\big(\frac{a \tau + b}{c \tau + d}\big),$$

and notice that if $\mathcal{P}_{g}$ were modular, the cancellation of the automorphy factors $c \tau + d$ would work out perfectly to give us the actual weight. So my first question is: why does the quasi-modularity arise to spoil this determination of the weight? It comes so close to working, which would make geometric sense.

Secondly, putting aside issues of quasi-modularity, $\mathcal{P}_{g}$ should be a section of a line bundle $\mathcal{E}_{g} \to \overline{\mathcal{M}}_{1,1}$. The equation just above seems to indicate that we can interpret $z$ as a coordinate on, or section of the dual bundle $\mathcal{E}_{g}^{\vee}$, at least in the expansion. This would explain the cancellation of the automorphy factors geometrically. Is this or something like it true?

Finally, doing an expansion around $z=0$, I would expect the normal bundle of $\overline{\mathcal{M}}_{1,1}$ in $\mathcal{C}$ to play a role. What is this normal bundle, and does it play a role in this story?