I'm learning the Shimura curve. When I reading the note of Pete L. Clark (SC2-Fuchsian.pdf (uga.edu)), I was stuck on a thinking question.
First, there is a theorem (Uniformization Theorem) about Riemann surfaces (page 3):
Let $Y$ be a connected Riemann surface.
a) Its universal cover $\tilde{Y}$ is naturally a Riemann surface.
b) We have $Y=\Pi \backslash \tilde{Y}$, where $\Pi$ is a discrete group of fixed-point free holomorphic automorphisms of $\tilde{Y}$, isomorphic to the fundamental group $\pi_1(Y)$.
c) There are, up to isomorphism, only three possibilities for $\tilde{Y}: \mathbb{C P}^1, \mathbb{C}$ and $\mathcal{H}$.
I know that we can get the modular curves $\Gamma \backslash \mathcal{H}$ from the Fuchsian groups $\Gamma$, and the question comes from page 4:
We saw at the beginning of the course that $P S L_2(\mathbb{Z}) \backslash \mathcal{H} \cong \mathbb{A}^1$; here the quotient map $J: \mathcal{H} \rightarrow \mathbb{A}^1$ is clearly not a uniformization map in the above sense. As we embark upon a a general study of Fuchsian groups and then try to wend our way back to moduli of elliptic curves (and other things), this is a good example to keep in mind: exactly what is preventing $J$ from being a uniformization map, and what is the modular interpretation of this?
My question is how to understand this phenomenon——the quotient $P S L_2(\mathbb{Z}) \backslash \mathcal{H} \cong \mathbb{A}^1$ is not a uniformization. What's the obstacle here? Is there any way to explain it via modular or moduli theory? Or is it a problem of geometry?