# Meromorphic functions on a modular curves of genus $0$ that take each value exactly once

Let $$\Gamma$$ be a congruence subgroup of $$\operatorname{SL}_2(\mathbb Z)$$, and let $$\mathfrak H$$ be the upper half-plane. Let $$X(\Gamma)$$ be the compactification of $$\Gamma\backslash\mathfrak H$$. Then $$X(\Gamma)$$ is a compact Riemann surface.

Suppose that $$X(\Gamma)$$ has genus $$0$$. Then also according to a general theorem there exists an analytic isomorphism $$X(\Gamma)\longrightarrow \mathbb P_\mathbb C^1$$. In the case of $$\Gamma=\operatorname{SL}_2(\mathbb Z)$$ this isomorphism is given by the $$j$$-invariant.

1. What is the simplest way to prove that there exists an analytic isomorphism $$X(\Gamma)\longrightarrow \mathbb P_\mathbb C^1$$ in this special case? That is, how much advantage can our knowledge that $$X(\Gamma)$$ is a modular curve give us?
2. Is there an algorithm to determine such isomorphism in terms of known functions (such as Dedekind $$\eta$$-function or $$j$$-invariant)?

For example, consider $$\Gamma=\bigg\lbrace\gamma\in \operatorname{SL}_2(\mathbb Z)\colon \gamma\equiv \begin{pmatrix}1 & 0 \\ 0 &1\end{pmatrix},\begin{pmatrix}0 & 1 \\ 1 &0\end{pmatrix}\text{mod }2\bigg\rbrace.$$ Then an isomorphism $$X(\Gamma)\longrightarrow \mathbb P_\mathbb C^1$$ is determined by the function $$j_\Gamma(\tau)=-\frac{\eta\left(\frac{\tau+1}{2}\right)^{24}}{\eta(\tau)^{24}}=q^{-1/2}\prod_{n=1}^{\infty}(1+q^{n-1/2}).$$

• Can't you just take the $j$ invariant to answer both questions? – Will Sawin Jun 7 '20 at 20:52
• @WillSawin Thank you, I will modify my question. – Shimrod Jun 7 '20 at 20:55
• probably what you are looking for is the notion of Hauptmodul. – Henri Cohen Jun 7 '20 at 21:25