Let $J\_{k,m}(N)$ be the space of Jacobi forms of weight $k$, index $m$, and congruence subgroup $\Gamma\_{0}(N) \rtimes \mathbb{Z}^{2}$. I do not believe it is relevant here to specify what type of Jacobi form (weak, holomorphic, etc). I can't find hardly any references on how the index, and the congruence subgroup change when scaling the variables. So my question is simply: 



> 
> Given $\varphi(\tau, z) \in J\_{k,m}(1)$, what is the index and congruence subgroup of the Jacobi form
>  $$\psi\_{d\_{1}, d\_{2}}(\tau, z) = \varphi(d\_{1} \tau, d\_{2} z)$$
>  for general $d\_{1}, d\_{2} \in \mathbb{Z}\_{>0}$? 
> 
> 
> 


There are a few relevant things I know. For an ordinary modular form $f(\tau)$ of weight $k$ on $SL\_{2}(\mathbb{Z})$, it's straightforward to use the "slash operator" to show that $f(d \tau)$ is a weight $k$ form on $\Gamma\_{0}(d)$. (See Exercise 1.2.11 of Diamond and Shurman)


I also know that in the case of $d\_{1}=1$, $\varphi(\tau, dz) \in J\_{k, d^{2}m}(1)$. This is easy to show from the transformations, and moreover it coincides with a well-known Hecke-like operator


$$U\_{d} : J\_{k,m}(1) \to J\_{k, d^{2}m}(1).$$


I'm unsure what the right statement is for general $d\_{1}, d\_{2}$ specifically perhaps $\varphi(d \tau, z)$ and $\varphi(d \tau, d z)$. 


Part of what's confusing me is that if $F(\tau, z, \sigma)$ is a weight $k$, degree 2 Siegel modular form, then it has a Fourier-Jacobi expansion in $Q = e^{2 \pi i \sigma}$:


$$F(\tau, z, \sigma) = \sum\_{m=0}^{\infty} Q^{m} \varphi\_{k,m}(\tau, z), \,\,\,\,\,\,\,\,\,\,\,\,\,\, \varphi\_{k,m} \in J\_{k,m}(1)$$


I believe it's true that $F(d\tau, dz, d\sigma)$ is a weight $k$ form for the degree 2 congruence subgroup $\Gamma^{(2)}\_{0}(d) \subset Sp\_{4}(\mathbb{Z})$. So we have a Fourier-Jacobi expansion:


$$F(d\tau, dz, d\sigma) = \sum\_{m=0}^{\infty} Q^{dm} \varphi\_{k,m}(d\tau, dz).$$


But I think we need the index of the Jacobi forms to match the exponent of $Q$. So somehow, $\varphi\_{k,m}(d\tau, dz) \in J\_{k, dm}(d)$? So the index is multiplied by $d$, not $d^{2}$ in this case. 



Index and congruence subgroup from scaling variables of Jacobi form
$\varphi(d\_1\tau,d\_2 z)$ will generally be a Jacobi form for a congruence subgroup of this kind only when $d\_1 | d\_2$. Otherwise it will not transform correctly under $(\tau,z) \mapsto (\tau,z+\tau)$.


In this case its congruence group is $\Gamma\_0(d\_1) \rtimes \mathbb{Z}^2$ and its index is $m\frac{d\_2^2}{d\_1}$, as you can see using the Fourier-Jacobi expansions (the index indeed matches the exponent) or by checking the transformations directly.


