The first sentence of @ccorn's answer to a previous question of mine was:
“Because of the modular symmetries of $j(\tau)$, the zeros of $j(\tau)$ are precisely the $\operatorname{SL}(2,\mathbb{Z})$-transforms of the fundamental-domain zero $\zeta_3=\mathrm{e}^{2\pi\mathrm{i}/3}$.”
To what extent can $j(\tau)$ be written as an infinite product over those zeros?, something like this:
$$j(\tau)\stackrel{?}{=}u(\tau)\cdot\prod_{A\in\mathbf{SL}(2,\mathbf{Z})}\left[1-\frac{\tau}{A(e^{2\pi i/3})}\right]^{3}$$
And, if such an expression were valid, what would $u(\tau)$ be?