Surjectivity of parameters for torus double cover of sphere

Question

Let $T=\mathbb{C}/\Lambda$ be a torus, viewed as a Riemann surface. Quotienting out by the relation $z\sim -z$ gives a double cover $\pi : T \to \hat{\mathbb{C}}$ of the Riemann sphere, ramified over 4 points $e_1,e_2,e_3,e_4$. This is essentially the Weierstrass $\wp$ function.

The four points will depend on the lattice $\Lambda$ (more precisely the cross ratio depends on the lattice, since we can always post compose the quotient map by a Mobius transformation.

Ahlfors claims (page 34, Lectures on Quasiconformal mappings) that the mapping $\Lambda$ to $e_1,e_2,e_3,e_4$ is surjective. See screenshot:

Now, given 4 points $e_1, e_2, e_3, e_4$ on the sphere, how do we know that there is a torus double cover which is ramified at exactly those 4 points? I would like a proof not involving the Weierstrass function, if possible.

Answer 1

You can just use standard branch-cut constructions to do this. Re-index the points so that the segments $\alpha = \overline{e_1 e_2}$ and $\beta = \overline{e_3 e_4}$ do not intersect. Using those two segments as branch cuts, you can then form the double branched cover:

• Cut open the sphere along $\alpha$ and $\beta$, leaving a surface $S$ with two boundary components $A = \alpha' \bar\alpha''$ and $B=\beta' \bar \beta''$, so that the sphere is recovered by identifying $\alpha'$ and $\alpha''$ and by identifying $\beta'$, and $\beta''$;
• Take two copies $S_1,S_2$ of $S$;
• Reglue the disjoint union $S_1 \coprod S_2$, gluing $\alpha'_1$ to $\alpha''_2$, $\alpha'_2$ to $\alpha''_1$, $\beta'_1$ to $\beta''_2$, and $\beta'_2$ to $\beta''_1$.

The result of that gluing is the surface you want.