how to show a curve is elliptic

Question

I'm trying to show that a curve is an elliptic curve. I'm looking at the example

$\{(z,w)\in\mathbb{C}^2:z^3+w^3=1\}$.

What is the best way to do this? And in a more general case? Thanks!

Answer 1

Recall the definition of an elliptic curve: Let $f(z,w)$ be a cubic polynomial in two variables. Then $\{(z,w) \in \mathbb{C}^2 : f(z,w) = 0\}$ constitutes an elliptic curve if $f$ has no singular point in the projective plane $\mathbb P_2(\mathbb C)$.

For your example, $f(z,w) = z^3 + w^3 - 1$ which in the projective plane is $$F(Z,W,Y) = Z^3 + W^3 - Y^3.$$ To determine whether $F$ has a singular point, first calculate the partial derivatives: $$\frac{\partial F}{\partial Z} = 3Z^2 \qquad \frac{\partial F}{\partial W} = 3W^2 \qquad \frac{\partial F}{\partial Y} = -3Y^2.$$ Next, find the points at which those partial derivatives are all zero. In this case, there is only one point $0:0:0,$ which is not a member of the projective plane. Thus, $F$ has no singular points, so $\{(z,w) \in \mathbb{C}^2 : z^3 + w^3 = 1\}$ is an elliptic curve.

You can use that procedure for other cases as well.