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

  • $\begingroup$ You meant the modular curve (modular surfaces are quotients of $\mathbb{H} \times \mathbb{H} $). If you change the complex structure on $\mathbb{H}$ by the one of $\rho(\mathbb{H}), \rho(x+iy)=-x+iy$ you get another complex structure on $\mathbb{H}/\operatorname{PSL}(2,\mathbb{Z})$ (making $\overline{f}$ meromorphic for $\rho$ iff $f$ is meromorphic for $\rho^2$). Your linked post says given a complex structure on $\mathbb{H}$ you have some holomorphic charts of the modular curves so a unique complex structure on it. $\endgroup$ – reuns Jan 15 at 5:41
  • $\begingroup$ Thanks. By unique complex structure on $\Omega$ I mean that if $(\Omega, J_1)$ and $(\Omega, J_2)$ are complex structures on $\Omega$, then there exists a biholomorphism $g:(\Omega, J_1)\rightarrow(\Omega, J_2)$. The map $\rho$ above doesn't seem to give a different complex structure in that sense, since $(x,y)\mapsto(-x,y)$ gives a biholomorphism between the complex structures induced by $\rho$ and $\rho^2$. $\endgroup$ – Paul W. Jan 15 at 6:57
  • $\begingroup$ You are right that the discussion in the linked post also involves a holomorphic quotient map; here I am just interested in uniqueness of the complex structure on the specific quotient space $\mathbb{H}/PSL(2,\mathbb{Z})$ without any restrictions on the quotient map. $\endgroup$ – Paul W. Jan 15 at 6:58
  • $\begingroup$ $\rho$ is a complex conjugaison, it is not a biholomorphism, but yes $\mathbb{H}/PSL(2,\mathbb{Z}),\rho^2$ and $\mathbb{H}/PSL(2,\mathbb{Z}),\rho$ are isomorphic as Riemann surfaces. The complex torus $\mathbb{C}/(\mathbb{Z}+i\mathbb{Z})$ when changing its complex structure becomes any complex torus $\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})$ (this time the two are not isomorphic). $\endgroup$ – reuns Jan 15 at 8:37
  • $\begingroup$ The same happen should happen with the compactified level $n$ modular curve $\mathbb{H}/\Gamma_0(n)$, changing its complex structure will give any curve of the same genus. The full modular curve has genus 0, so it is isomorphic to every curves of genus 0. The modular curve without the point at $\infty$ is isomorphic to $\mathbb{C}$, topologically it is just a simply connected domain, changing its complex structure may give something isomorphic to the unit disk, by the Riemann mapping theorem all the others will be isomorphic to one of them. $\endgroup$ – reuns Jan 15 at 8:37

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