Quotient of the upper-half plane as a projective line

Question

In the discussion of the Belyi theorem here one pointed out that $\mathbb{H}^{*}/SL(2, \mathbb{Z})$ is isomorphic to the (complex) projective line $\mathbb{P}^1$, where $\mathbb{H}^{*}=\mathbb{H} \cup \mathbb{P^1(\mathbb{Q})}$.

Could anyone give an explanation of this fact, please?

Answer 1

First of all, a definition

Definiton (Extended Upper Half Plane) We define $\mathbb{H}^*$ as $\mathbb{H} \cup \mathbb{P}^1(\mathbb{Q})$, where $\mathbb{P}^1(\mathbb{Q})$ is looked as points on the real line plus a point to infinity

We can define the action of $SL_2(\mathbb{Z})$ on $\mathbb{P}^1(\mathbb{Q})$ sending $[x:y]$ to $[ax+by : cx+dy]$.

Now I have to cite two facts, one very simple and the other very complicated that are foundamental:

1. The action defined above on $\mathbb{P}^1(\mathbb{Q})$ is transitive so in the quotient the image of $\mathbb{P}^1(\mathbb{Q})$ is a single point.
2. You can define a topology on $\mathbb{H}^*$ such that $\mathbb{H}^*/SL_2(\mathbb{Z})$ is a compact connected Riemann surface.

What we have done? Well, considering $\mathbb{H}^*$ we have just made a sort compactification of the quotient of $\mathbb{H}$, the only that adjoin just a single point to $\mathbb{H}/SL_2(\mathbb{Z})$ in a sensated way that allow us to define a topology on $\mathbb{H}^*$ compatible with the action of $SL_2(\mathbb{Z})$ and the Riemann Surface structure.

Now we want to construct an isomorphism between $\mathbb{H}^*/SL_2(\mathbb{Z})$ and $\mathbb{P}^1(\mathbb{C})$.

We can use the function $J$ that sends a complex number $\tau \in \mathbb{H}$ to the $j$-invariat of the Elliptic Curve $\mathbb{C}/\Lambda_{\tau}$:

• It is holomorphic on $\mathbb{H} $
• It constant on the orbits of the action of $SL_2(\mathbb{Z})$ so induces a map from $\mathbb{H}/SL_2(\mathbb{Z})$ to $\mathbb{C}$
• It has a simple pole at the infinity so it extends to an olomorphic map $\tilde{J}:\mathbb{H}^*/SL_2(\mathbb{Z}) \rightarrow \mathbb{P}^1(\mathbb{C})$.

Now to conclude just observe that $\tilde{J}$ is an holomorphic map between compact connected Riemann Surface so it is surjective (It's an open and closed map) and is injective by properties of the $j$-invariant.

There are other several ways to verify this isomorphism (using modular forms, using algebraic geometry, calculating the genus of the quotient of $ \mathbb{H}^*$, there's also a complete elementary way using just residue theorem) but in my opinion this one is the most "topologically" clear.