Given a holomorphic Jacobi form $\varphi_{k,1} \in \mathbb{J}_{k,1}$ of weight $k$ and index 1, we know we can use the Hecke operators $V_{m}$ to lift $\varphi_{k,1}$ to a Siegel modular form

$$\mathscr{V}(\varphi_{k,1}) = \sum_{m=0}^{\infty} p^{m} \big(\varphi_{k,1}\big| V_{m}\big)(\tau, z)$$

of degree two and weight $k$. This map is an isomorphism onto the Maass 'Spezialschar' inside of the space of weight $k$ degree two Siegel modular forms $\mathbb{M}_{k}\big(Sp_{4}(\mathbb{Z})\big)$. This can all be found in Eichler and Zagier.

Borcherds extended this story to weakly holomorphic Jacobi forms $\varphi_{k,1} \in \mathbb{J}^{\text{wh}}_{k,1}$. This is Theorem 9.3 in (https://math.berkeley.edu/~reb/papers/on2/on2.pdf). So we "lift" $\varphi_{k,1}$ to a meromorphic Siegel modular form of degree two, weight $k$, and with respect to $O_{M}(\mathbb{Z})^{+}$. But there are a handful of points I'm stuck on:

  1. Does this theorem cover the case of automorphic Siegel modular forms with respect to $Sp_{4}(\mathbb{Z}),$ with $\varphi_{k,1} \in \mathbb{J}^{\text{wh}}_{k,1}$ transforming under the usual Jacobi group $SL_{2}(\mathbb{Z}) \rtimes \mathbb{Z}^{2}$?

  2. Borcherds has a special section for weight $k \leq 0$. I'm more or less fine with $k=0$, but I'm confused about the negative weight case. I'm wanting to perform this lifting on the unique weak Jacobi form of weight -2 and index 1 $$\varphi_{-2,1} = \frac{\vartheta^{2}_{1}(\tau, z)}{\eta^{6}(\tau)}.$$ Is this out there in the literature somewhere? Borcherds claims that lifting a weakly holomorphic Jacobi form of negative weight is certainly possible, but doesn't produce anything "new". Applying this to $\varphi_{-2,1}$ above I should get a meromorphic Siegel modular form associated to $\tfrac{d^{3}}{d \tau}f(\tau)$, but I'm not exactly sure what to make of this.

EDIT: Apparently the slogan for (1) is that "Siegel modular forms with respect to $Sp_{4}(\mathbb{Z})$ are simply automorphic forms on the Type IV domain with respect to $O^{+}_{M}(\mathbb{Z})$", but if someone could help me understand what this means that would be great!



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