The following construction of something very nearly a Siegel modular form of degree 2 arose in my research. I'm outside the worlds of automorphic forms and number theory, so I'm wondering if it resembles anything well-known or interesting.
Let $F(\tau, z, \sigma)$ be a (meromorphic) degree 2 Siegel modular form of weight $k>0$, and let $r \in \mathbb{Z}_{>0}$. I was compelled to consider the following meromorphic function on the Siegel half-space $\mathbb{H}_{2}$:
$$\Phi_{r}(\tau, z, \sigma) := \sum_{d|r}F\big(d\tau, z, \tfrac{6}{d}\sigma\big).$$
Has a construction like this been studied before, or arisen anywhere? Maybe there's some context or interpretation someone can offer? As I'll briefly sketch below, it's very nearly a Siegel form on the congruence subgroup $\Gamma_{0}^{(2)}(r) \subset Sp_{4}(\mathbb{Z})$. In particular, notice that we retain the symmetry between $\tau$ and $\sigma$...
From the Fourier-Jacobi expansion of $F$, one can easily write down the corresponding expansion of $\Phi_{r}$, where $Q=e^{2 \pi i \sigma}$:
$$\Phi_{r}(\tau, z, \sigma) = \sum_{m=0}^{\infty} Q^{m} \varphi_{k,m/r}(\tau, z).$$
One can show that $\varphi_{k, m/r}$ behave like Jacobi forms of weight $k$ and rational index $m/r\in \mathbb{Q}$ on the group $\Gamma_{0}(r) \rtimes (r\mathbb{Z} \times \mathbb{Z})$. So this is very nearly the Fourier-Jacobi expansion of a Siegel form, but the index, and the exponent of $Q$ differ by the factor of $1/r$. So $\Phi_{r}$ transforms properly under the following:
$$(\tau, z, \sigma) \mapsto \bigg(\frac{a \tau +b}{c\tau + d}, \frac{z}{c \tau +d}, \sigma - \frac{cz^{2}}{\textbf{r}(c \tau +d)}\bigg)$$
for $(a,b,c,d) \in \Gamma_{0}(r),$ and for $\lambda \in \textbf{r}\mathbb{Z}$ and $\mu \in \mathbb{Z}$:
$$ (\tau, z, \sigma) \mapsto \bigg(\tau, z+ \lambda\tau + \mu, \sigma +\frac{1}{\textbf{r}}(\lambda^{2}\tau + 2 \lambda z)\bigg).$$
The factors of $1/r$ which I've bolded above seem to spoil this action truly coming from the action of $\Gamma_{0}^{(2)}(r)$ on $\mathbb{H}_{2}$. Is there some group closely related to $Sp_{4}(\mathbb{Z})$ that I should be thinking about instead, like similitudes or something?
