I saw an informal comment somewhere about taking a Riemann sphere with four punctures and generating all possible elliptic curves. I was hoping someone could describe for me how this construction goes? Or possibly point me to online literature?
If I had to guess, I feel like given four points on the sphere, you could connect them pairwise with branch cuts, and glue these two cuts together to get the torus/elliptic curve. I know that any two Riemann spheres with three punctures are biholomorphic, so is the idea that the fourth puncture treated as a moduli which turns into the modular parameter $\tau$ of the elliptic curve? Kind of like a change of variables, or so?