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12 Introduction to Rational Points on Plane Curves

manner onto a line to obtain a description of the rational points. Under projection
two points on the cubic correspond to one point on the line, and one rational point
on the line does not necessarily correspond to a pair of rational points on the cubic.
This leads to a new approach to the description of the rational points. Observe
that given two rational points on a rational nonsingular cubic C, we can construct a
third one. Namely, you draw the line connecting the two points P and Q.Thisisa
rational line since P and Q are rational, and this line meets the cubic at one more
point, denoted PQ, which must be rational by (4.1). The formation of PQ from P
and Q is some kind of law of composition for the rational points on a cubic.

Even if you have only one rational point P, you can still find another, in general,
because you draw a tangent to that point, i.e., you join the point to itself.

The tangent line meets the cubic twice at P, that is, it corresponds to a double
root in the equation of the x-coordinate. By the above argument the third intersection
point is rational. Thus, from a few rational points, one can, by forming compositions
successively, generate lots of other rational points. The function which associates to
a pair P and Q the point PQ is called the chord-tangent composition law.
(4.3) Primitive Form of Mordell’s Theorem. On a nonsingular rational cubic curve
there exists a finite set of rational points such that all rational points on the curve
are generated from these using iterates of the chord-tangent law of composition.

In other words there is a finite set X of rational points on the nonsingular rational
cubic such that every rational point P can be decomposed in the form,