I saw a proposition in page 60-61 in https://math.berkeley.edu/~teleman/math/Riemann.pdf
Proposition: $\mathfrak{H} / S L(2, \mathbb{Z}) \cong \mathbb{C}$, and the bijection is implemented by the (holomorphic) elliptic modular function $J$, $$ J(\tau)=\frac{g_{2}^{3}}{g_{2}^{3}-27 g_{3}^{2}}, $$ ......
where $\mathfrak{H}=\{z|Im(z)>0\}$.
I am confused about the statement of this proposition, since I do not understand why $\mathfrak{H} / S L(2, \mathbb{Z})$ is a Riemann surface. I only know that if a group $G$ acts on a Riemann surface $X$ with no fixed point (i.e. $gx\ne x$ for any $1\ne g\in G$ and $x\in X$), then the quotient space $X/G$ has a induced complex structure. However, the action of $\begin{pmatrix}0&1\\-1&0\end{pmatrix}\in SL(2,\mathbb{Z})$ on $\mathfrak{H}$, which is $z\mapsto-\frac{1}{z}$, has a fixed point $i$.
A similar example is $\mathbb{C}/\{\pm 1\}$, which seems like a cone. Does it have an induced complex structure?