Elliptic Curves
Elliptic Curves: Basic Definitions
What an elliptic curve is, why the group structure exists, and the three distinct roles curves play in this collection.
Engineering / MathematicsElliptic Curves2 min readKV-MATH-0633
An elliptic curve is a smooth cubic curve with a distinguished point. The remarkable fact is that its points form an abelian group, and that group is what every application exploits.
Definition
Over a field of characteristic not two or three, an elliptic curve is given by a short Weierstrass equation together with a point at infinity serving as the identity.
Three roles in this collection
Objects of study
Rank, torsion, L-functions and the Birch-Swinnerton-Dyer conjecture. See L-functions and BSD.
Factoring engine
The group order varies with the curve, which is what Pollard's p-1 method lacks. See ECM.
Primality engine
Point counting and complex multiplication give primality proofs. See ECPP.
Over different fields
| Base field | Structure of the group |
|---|---|
| Complex numbers | A torus; the quotient of the plane by a lattice |
| Real numbers | One or two connected components |
| Finite field | Finite abelian, of order near the field size |
| Rational numbers | Finitely generated, by Mordell's theorem |
| Ring modulo a composite | Not a group; the failure is exploited by ECM |
Why genus one matters
Elliptic curves are exactly the smooth projective curves of genus one with a rational point. Genus zero curves are conics and have no group law of this kind; genus two and above have finitely many rational points by Faltings' theorem and no group structure on the curve itself.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 7.1.1. Structural reference unverified: the source file was not available during authoring; chapter and section numbers are taken from the published edition and have not been checked against a physical copy.