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14 Introduction to Rational Points on Plane Curves

This means that P + Q is the third intersection point on the line through O and PQ.

Clearly we have the commutative law P + Q = Q + P since PQ = QP. From
the fact that O, PO,and P are the three intersection points of the cubic with the
line through O and P, we see that P = PO = P + O, and thus O is the zero
element. To find −P given P, we use the tangent line to the cubic at O and its third
intersection point OO. Then we join P to O(OO) = 0 with a line and −P is the
third intersection point.

Note that P + (−P) = O(OO) which is O in the above figure. The associative law
is more complicated and is taken up in 2. It results from intersection theory for plane
curves. Observe that if a line intersects the cubic in three rational points P, Q,and
R, then we have P + Q + R = OO. We will be primarily interested in cubics where
O = OO, i.e., the tangent to the cubic O has a triple intersection point. These points
are called flexes of the cubic and are considered in 2.
In the next definition we formulate the notion of an elliptic curve over any field
k, but, in keeping with the ideas of the introduction, we have in mind the rational
field Q, the real field R, or the complex field C.

(5.1) Definition. An elliptic curve E over a field k is a nonsingular cubic curve E
over k together with a given point O ∈ E(k). The group law on E(k) is defined as
above by O and the chord-tangent law of composition PQ with the relation P+Q =
O(PQ).