Elliptic curve - Knowlpedia

An elliptic curve over a field $k$ is a smooth projective curve $E/k$ , geometrically integral and of genus $1$ , together with a specified point $O\in E(k)$ . Here genus $1$ means that the space of global regular differential one-forms has dimension $1$ over $k$ .

The point $O$ determines the identity in a canonical algebraic group law on $E$ ; it is part of the data.

Equations and characteristic

An elliptic curve admits a nonsingular generalized Weierstrass equation

y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6,

with its projective point at infinity as $O$ . If $\operatorname{char}k\ne2,3$ , a change of coordinates gives

y^2=x^3+Ax+B,\qquad 4A^3+27B^2\ne0.

The short equation must not replace the general one in characteristic $2$ or $3$ .

Morphisms

A homomorphism of elliptic curves over $k$ is a morphism of algebraic groups over $k$ . Endomorphisms therefore preserve $O$ ; arbitrary maps of the underlying point sets do not define the endomorphism ring. The base field matters: maps over an algebraic closure may outnumber maps over $k$ .

References

Andrew V. Sutherland, 18.783 Elliptic Curves, Fall 2023, Lecture 1, Definition 1.1 and §§1.1–1.2.
John Voight, Quaternion Algebras, §42.1, elliptic-curve and geometric-endomorphism conventions.