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2 Introduction to Rational Points on Plane Curves

or

degree 1 degree 2 degree 3

§1. Rational Lines in the Projective Plane

Plane curves C f can be defined for any nonconstant complex polynomial with com-
plex coefficients f (x, y) ∈ C[x, y] by the equation f (x, y) = 0. For a nonzero con-
stant k the equations f (x, y) = 0and kf (x, y) = 0 have the same solutions and de-
fine the same plane curve C f = C kf . When f has complex coefficients, there is only
a complex locus defined. If f has real coefficients or if f differs from a real poly-
nomial by a nonzero constant, then there is also a real locus with C f (R) ⊂ C f (C).
Such curves are called real curves.

(1.1) Definition. A rational plane curve or a curve defined over Q is one of the form
C f where f (x, y) is a polyomial with rational coefficients.
This is an arithmetic definition of rational curve, and it should not be confused
with the geometric definition of rational curve or variety. We will not use the geo-
metric concept.
In the case of a rational plane curve C f we have rational, real, and complex points
C f (Q) ⊂ C f (R) ⊂ C f (C) or loci.
A polynomial of degree 1 has the form f (x, y) = a + bx + cy. We assume the
coefficients are rational numbers and begin by describing the rational line C f (Q).
For c nonzero we can set up a bijective correspondence between rational points on
the line C f andonthe x-axis using intersections with vertical lines.

The rational point (x, 0) on the x-axis corresponds to the rational point

(x, −(1/c)(a + bx))

on C f . When b is nonzero, the points on the rational line C f (Q) can be put in
bijective correspondence with the rational points on the y-axis using intersections
with horizontal lines. Observe that the vertical or horizontal lines relating rational
points are themselves rational lines.