Elliptic Curves, An Introduction

Introduction

An elliptic curve can be drawn in the xy plane, but it has nothing to do with an ellipse. An ellipse is the graph of a quadratic equation, such as 2x² + 3y² = 11. In contrast, an elliptic curve is the graph of a cubic equation, such as x³ = y²-26. In this regard, elliptic curves are (perhaps) poorly named. But that's what they're called, so on we go.

As you explore the properties of elliptic curves, you might wonder how these structures came to be. "How did anybody ever think of that?" Well they come from algebraic geometry, as a special case of a more general, abstract shape. In this context, an elliptic curve is a projective nonsingular curve of genus 1, with a base point p₀. We may explore the deep connection to algebraic geometry in another section, but not here. That would only serve to confuse. Besides, you don't need an advanced course in algebraic geometry to appreciate the beauty and the power of the elliptic curve.

It would be difficult to overstate the importance of elliptic curves in pure and applied mathematics. They are used in primality testing, factorization, public key cryptography, and the proof of Fermat's last theorem, to name a few. No wonder so much is written about this subject.