# Elliptic Curve as Sphere with 4 Punctures?

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?

• I believe something about this is at the start of Ch 6 of Silverman. – Hoot Dec 8 '15 at 11:58
• Yeah, turns out the link provided below was modeled almost identically off Silverman. I'm curious why it's necessary that $\infty$ be one of the ramification points? It seems like if you just had four points on a plane, you could still glue the two cuts, then add the point at infinity, and still get an elliptic curve. Is the idea maybe something along the lines of, given any four points on the Riemann sphere, there exists a change of coordinates to place three of them at 0,1, and $\infty$? – Benighted Dec 8 '15 at 18:18
• I think perhaps I figured this out. There appears to be something called the Jacobi form of an elliptic curve, which can be written $y^{2} = (1-\sigma^{2} x^{2})(1-x^{2}/\sigma^{2})$ which has the four ramification points at points of the plane. There also appears to be a simple recipe for converting Jacobi form to Legendre form and visa versa. – Benighted Dec 8 '15 at 20:50

You're looking for the "j-invariant" and the "Legendre form" of an elliptic curve.

Every elliptic curve (seen as a complex algebraic curve) can be written in the form $y^2=x(x-1)(x-\lambda)$ where $\lambda \in \mathbb{C}\setminus\{0,1\}$. This formula parametrizes all of them (varying $\lambda$) up to isomorphism, but different values of $\lambda$ might correspond to isomorphic curves. If you define the $j$-invariant of the elliptic curve to be $j(E)=256\cdot\frac{(\lambda^2-\lambda+1)^3}{\lambda^2(\lambda-1)^2}$, this parametrizes isomorphism classes of elliptic curves "faithfully", in the sense that $j(E)=j(E')$ exactly when they are isomorphic.

• Thanks a lot, I'll read up on those topics. How does this relate to thinking about gluing branch cuts on the Riemann sphere? I'm guessing the Legendre Form corresponds to branch points at 0,1, $\lambda$, and infinity, but do the branch cuts connecting these matter at all? Perhaps the cuts are themselves just irrelevant choices, giving rise to isomorphic elliptic curves. – Benighted Dec 8 '15 at 6:53
• I usually think about these from the algebraic POV, but you seem to be right: a quick google search turned up math.mcmaster.ca/ms/images/Arash_Islami.pdf that seems to explain the connection (look for 'branch cut'). I'm sure there are many other references online. – Mattia Talpo Dec 8 '15 at 7:03