Is the complex structure on the modular curve coming from the quotient $\mathbb{H}/\operatorname{PSL}(2,\mathbb{Z})$ unique? (Here $\mathbb{H}$ is the upper half plane in $\mathbb{C}$)
According to the first comment here, it should be, but the brief description of how this is proved is not so clear to me. I am looking to better understand this argument (or another), or locate a reference discussing this. Does it use the fact that $S^2$ has a unique complex structure?
Edit: Changed terminology from "modular surface" to "modular curve" -- thanks @reuns