Elliptic curve group law

John Voight

When can a number be written as the sum of two cubes? To answer this question, you want to know about rational points on an elliptic curve. Elliptic curves are almost too beautiful to exist, and they are deceptively simple: somehow, they manage to embody simultaneously many different kinds of mathematical objects. On the one hand, the set of points of an elliptic curve naturally forms a group, so you can add two points on the curve to make a third. This group law is provided by algebraic equations (shown here), but it is also described geometrically: three points sum to zero when they lie on a line. From an analytic point of view, an elliptic curve is a donut. And on top of all of this, the mysterious nature of elliptic curves make them ideal for use in cryptography. One would be hard pressed to find a more beautiful structure in mathematics than the group law on an elliptic curve.