# Index and congruence subgroup from scaling variables of Jacobi form

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.

$$\varphi(d_1\tau,d_2 z)$$ 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.
• This is great, thanks a lot! Maybe I'm just not looking in the right places, but it would be nice if this was remarked in the literature somewhere. By the way, you say a "congruence subgroup of this kind." Is something like, say, $\varphi(d\tau, z)$ possibly a Jacobi form for some more exotic subgroup? – Benighted Sep 25 '19 at 16:18
• @Benighted Sure, you'll have to replace the $\mathbb{Z}^2$ by something else, like $d \mathbb{Z}^2$, and the index will no longer be an integer. – user146384 Sep 30 '19 at 7:54