It's well-known that $SL_2(\mathbb{Z}) \backslash \mathfrak{h}$ is a coarse moduli space for complex elliptic curves. Thus, I would expect this to be related to the pullback of $\mathcal{M}_{ell} \rightarrow \mbox{Spec}(\mathbb{Z})$ along $\mbox{Spec}(\mathbb{C}) \rightarrow \mbox{Spec}(\mathbb{Z})$; for instance, I might expect this pullback to look something like $SL_2(\mathbb{Z}) \backslash \! \! \backslash \mathfrak{h}$. However, I'm pretty sure that $\mathfrak{h}$ isn't actually a complex variety (basically by the Riemann mapping theorem), so at best this would admit a map from the analytification of the actual algebro-geometric pullback.
The answer might be bound up in the $j$-invariant; over $\mathbb{C}$, this is a "biholomorphism" $SL_2(\mathbb{Z}) \backslash \mathfrak{h} \rightarrow \mathbb{C}$, i.e. it is a holomorphic bijection of complex orbifolds. (Around the cone points $i \in \mathfrak{h}$ and $\omega=e^{2 \pi i /3}\in \mathfrak{h}$, the map is locally modeled by $z \mapsto z^2$ and by $z \mapsto z^3$, respectively.) This has always been sort of mysterious to me, but I think the point is just that "biholomorphism" is the wrong notion of equivalence for complex orbifolds; it seems somehow besides the point to me that we happen to have such an equivalence.