At its simplest an elliptic curve is a non-singular curve in ℝ² with the equation

The non-singularity then requires that 4p³ + 27q² ≠ 0. But the study of elliptic curves over the complex numbers (including points at infinity, so studying the curves within the complex projective plane (see projective space)) has a rich history. Such a curve is then topologically a torus and as a Riemann surface can be identified with ℂ/Λ‎, where Λ‎ is some lattice aℤ+bℤ, where a/b is not real. Now ℂ/Λ‎ is an abelian group, and this group structure on the curve manifests geometrically via a theorem due to Abel.

Elliptic curves over the rational numbers are of great interest in number theory; the Birch-Swinnerton-Dyer conjecture (see appendix 22) concerns them. And over Galois fields, elliptic curves have applications in cryptography. See also degree-genus formula.