# Moduli Space of Tori identified with $\mathbb{C}$

I am currently reading through the book An Introduction to Teichmüller Spaces by Imayoshi and Taniguchi. In Section 1.2, we see that $$M_1$$, the moduli space of tori, can be identified with $$\mathbb{H}/PSL(2,\mathbb{Z})$$. This is clear, since two tori $$R_\tau$$ and $$R_{\tau'}$$ - generated by normalized lattices having sides $$1, \tau$$ and $$1, \tau'$$, respectively - are biholomorphically equivalent if and only if $$\tau=\tau'$$ (where $$\tau \in \mathbb{H}$$), and in particular we can identify this with $$\mathbb{C}$$ using the $$j$$-invariant of an elliptic curve (associated to a torus generated by the normalized lattice with sides $$1, \tau$$).

We also know that for a given cross ratio $$\{z_1, z_2, z_3, z_4\} = \lambda$$, permutation of the $$z_i$$ by an element of $$S_4$$ results in the cross ratio being one of the following: $$\lambda, \frac{1}{\lambda}, 1-\lambda, \frac{1}{1-\lambda}, \frac{\lambda-1}{\lambda}, \frac{\lambda}{\lambda-1}$$, and we can put the 4-tuple having cross ratio $$\lambda$$ in "canonical" form by setting it to be $$\{0, 1, \lambda, \infty\}$$. Next, a torus $$S_\lambda$$ is given by the following equation (depending on $$\lambda$$): $$w^2 = z(z-1)(z-\lambda)$$. Then two tori $$S_\lambda$$ and $$S_{\lambda'}$$ are biholomorphically equivalent if and only if there is a linear fractional transformation taking $$\{0, 1, \lambda, \infty\}$$ to $$\{0, 1, \lambda', \infty\}$$, so then $$\lambda'$$ can only be one of the values after a permutation as above.

Let $$G$$ be the group (which is actually just $$S_3$$) generated by the two functions $$\lambda \mapsto \frac{1}{\lambda}$$ and $$\lambda \mapsto 1-\lambda$$ (note these functions are analytic automorphisms of $$D = \mathbb{C}-\{0, 1\}$$). In particular we have the $$S_4/V \approx S_3$$, where $$V$$ is the Klein 4-group (and is in fact the permutations that fix the cross ratio). The book goes on to say that this shows $$M_1 \approx D/G$$ and there is a biholomorphic map $$F: D/G \to \mathbb{C}$$ given by $$F([\lambda]) = f(\lambda) = \frac{{(\lambda^2-\lambda+1)}^3}{\lambda^2{(\lambda-1)}^2}$$.

I do not understand where exactly $$f(\lambda)$$ comes from or how the identification is clear, but the setup makes sense to me. Does this function have a special name? Does it appear anywhere else (in a significant way)?

• I don't see what you mean with cross ratio $\{z_1, z_2, z_3, z_4\} = \lambda$. Then $j,\lambda$ are complicated functions of $\tau$. It is more convenient to look at $f(\lambda) = j(E_\lambda)$ where $E_\lambda : w^2 = z(z-1)(z-\lambda)$ and $j(E)=j(E')$ iff $E \cong E'$. And $G$ is the group of biholomorphic functions $g:D \to D$ such that $f(g(\lambda)) = f(\lambda)$ thus $D/G \cong f(D)$, to identify it with $\mathbb{H}/PSL(2,\mathbb{Z})$ you need to make clear the isomorphism between isomorphism class of complex tori and isomorphism class of elliptic curves. – reuns Apr 15 at 23:05
• I don't really see how this answers the question. For the time being, I am just following the book, which presents the material in this manner. It's primarily focused on geometry, so the $j$-invariant isn't really crucial, or even explicitly mentioned - just that it gives a way to identify $\mathbb{H}/PSL(2,\mathbb{Z})$ and $\mathbb{C}$. – nilradical1 Apr 15 at 23:07
• Consider the field of invariants of $\mathbb{C}(\lambda)$ under the action of $G$. I believe that you will find that it is generated by $f(\lambda)$. As to why $f(\lambda)$ and not another generator, look at what the points $\{0,1,\infty\}$ have to go to. – Kapil Apr 16 at 1:58