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_{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.