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24 1. Elementary Properties of the Chord-Tangent Group Law on a Cubic Curve

infinity which we call O, and this O is the third point of intersection of the vertical
line with the locus of the cubic equation in normal form in the projective plane.
(1.1) Definition. The elliptic curve E corresponding to the cubic equation in normal
2
form is the locus of all solutions (x, y) ∈ k of the equation
2 3 2
y + a 1 xy + a 3 y = x + a 2 x + a 4 x + a 6
together with the point O which is on every vertical line.

When we wish to emphasize that we are looking at solutions (x, y) with x and y
in k,wewrite E(k) for E. Usually we use the term elliptic curve only for nonsingular
cubics. We return to criteria for nonsingularity in the next chapter. The choice of O
on cubic makes the nonsingular curve into an elliptic curve.
In the context of the normal form of a cubic equation for an elliptic curve E(k)
we give rules for the group law. Then O is the zero element in E(k) andadditionis
carried out as in §5 of the Introduction.
(1.2) Assertion. Let E be an elliptic curve defined by an equation in normal form.
∗ ∗
If P = (x, y) is a point on E(k), then the negative −P is (x, y ), where y + y =
−a 1 x − a 3 or, in other words, −(x, y) = (x, −y − a 1 x − a 3 ).
Observe that O, (x, y),and (x, y ) are the points of intersection of the vertical
∗
∗
line through (x, 0) with E(k). As seen above, y and y are two roots of a quadratic
equation over k where the sum of the roots is −(a 1 x + a 3 ) in k and so, if y is in k,
then y is also in k.
∗
The operation P → −P defines a map E(k) → E(k) which is an involution of
the curve onto itself, i.e., −(−P) = P. Also it shows that the curve has a vertical
reflection symmetry with respect to the line

a 1 x + a 3
y =−
2
in the plane. For this we require 2 = 0in k, that is, the characteristic of k = 2.

(1.3) Example. For E given by the equation
2 3
y + y − xy = x

we have −(x, y) = (x, −y − 1 + x) and the curve is vertically symmetric about the
line y = (1/2)x − 1/2.
In the diagram we have included for future reference two tangent lines to the
curve T at (1, 1) and T at (1, −1). The slopes of tangent lines are computed by

implicit differentiation of the equation of the curve
2
(2y + 1 − x)y = 3x + y.