Assume a holomorphic function from a product of two upper half planes $Z: \mathbb{H}_+\times \mathbb{H}_+\rightarrow \mathbb{C}$ with an expansion of the form $$ Z(\tau,\tau') = \sum_{(h,h')\in S} a_{h,h'} \ q^{h} \ (q')^{h'} \ , \qquad \quad (q\equiv e^{2\pi i \tau}, \ q'\equiv e^{2\pi i \tau'}) $$ where
- $S$ is a discrete subset in $\mathbb{R}_{[h_0,+\infty)}\times \mathbb{R}_{[h_0,+\infty)}$, with $h_0<0$ a strictly negative constant. We also assume that $(h_0,h_0) \in S$.
- $a_{h,h'}\ge 0$. We normalize the function by setting $a_{h_0,h_0}=1$.
- $Z$ is invariant under the diagonal modular group, meaning $$ Z \left(\frac{a \tau+b}{c \tau+d},\frac{a \tau'-b}{-c \tau'+d}\right)=Z(\tau,\tau') $$ for $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \text{SL}(2,\mathbb{Z})$.
We conjecture that such a function is uniformly bounded on two arbitrary Ford circles (denoted by $C_{\frac{\alpha}{\gamma}}$ for $\frac{\alpha}{\gamma} \in \mathbb{Q} \cup \{\infty\}$), i.e. $|Z(\tau,\tau')|\leq \mathsf{C}$ when $\tau \in C_{\frac{a}{c}},\tau' \in C_{\frac{a'}{c'}}$. The constant $\mathsf{C}$ should not depend on any of the parameters $(a,c,a',c',\tau,\tau')$.
It is simple to verify this for some examples, such as $Z(\tau,\tau')=j(\tau)j (\tau')$ with $j$ the j-function, or more generally by using Cauchy-Schwartz $Z(\tau,\tau')=f(\tau)^T f(\tau')$ with $f$ a vector-valued modular function with positive Fourier coefficients. In these cases, the optimal bound is in fact $Z(i,i)$. However, we could not find a simple argument showing this in general, nor could we find a counter example.
