Groups Attached to Elliptic Curves

Figure 10.1.1: The Elliptic Curve $y^2 = x^3 + x$ over ${\mathbf {Z}/7\mathbf {Z}{}}$

Figure 10.1.2: The Elliptic Curve $y^2 = x^3 + x$ over $\mathbf {R}$

Definition 10.1.1 (Elliptic Curve)   An elliptic curve over a field $K$ is a genus one curve $E$ over $K$ equipped with a point $\O\in E(K)$ defined over $K$.

We will not define genus in this book, except to note that a nonsingular curve over $K$ has genus one if and only if over  $\overline{K}$ it can be realized as a nonsingular plane cubic curve. Moreover, one can show (using the Riemann-Roch formula) that a genus one curve with a rational point can always be defined by a projective cubic equation of the form

$$Y^2 Z + a_1 XYZ + a_3 YZ^2 = X^3 + a_2 X^2Z + a_4 XZ^2 + a_6 Z^3.$$

In affine coordinates this becomes

$$y^2 +a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6.$$ (10.1)

Thus one presents an elliptic curve by giving a Weierstrass equation (10.1.1).

Figure 10.1.1 contains the graph of an elliptic curve over $\mathbf{F}_7$, and Figure 10.1.2 contains a graph of the real points on an elliptic curve defined over $\mathbf {Q}$.