The Definition

Definition 6.1 (Elliptic Curve)   An elliptic curve over a field $K$ is a curve defined by an equation of the form

$$y^2 = x^3 + ax+b,$$

where  $a, b\in{}K$ and $-16(4a^3+27b^2)\neq 0$ .

The condition that $-16(4a^3+27b^2)\neq 0$ implies that the curve has no ``singular points'', which will be essential for the applications we have in mind (see Exercise 6.1).

In Section 6.2 we will put a natural abelian group structure on the set

$$E(K) = \{ (x,y)\in K\times K : y^2 = x^3 + ax +b \} \cup \{\O\}$$

of $K$ -rational points on an elliptic curve $E$ over $K$ . Here $\O$ may be thought of as a point on $E$ ``at infinity''. In Figure 6.1 we graph $y^2 = x^3 + x$ over the finite field $\mathbb {Z}/7\mathbb {Z}{}$ , and in Figure 6.2 we graph $y^2 = x^3 + x$ over the field $K=\mathbb{R}$ of real numbers.

Remark 6.1   If $K$ has characteristic $2$ (e.g., $K=\mathbb{Z}/2\mathbb{Z}{}$ ), then for any choice of $a, b$ , the quantity $-16(4a^3+27b^2)\in K$ is 0 , so according to Definition 6.1.1 there are no elliptic curves over $K$ . There is a similar problem in characteristic $3$ . If we instead consider equations of the form

$$y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6,$$

we obtain a more general definition of elliptic curves, which correctly allows for elliptic curves in characteristic $2$ and $3$ ; these elliptic curves are popular in cryptography because arithmetic on them is often easier to efficiently implement on a computer.

Figure 6.2: The Elliptic Curve $y^2 = x^3 + x$ over $\mathbb {R}$