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4 Introduction to Rational Points on Plane Curves

projective plane. In particular the points a : b : c in the projective plane can be used
to parametrize the lines in the projective plane.
From the theory of elimination of variables in beginning algebra we have the
following geometric assertions of projective geometry whose verification is left to
the reader.

(1.4) Assertion. Two distinct points P and P in P 2 (C) lie on a unique line L in

the projective plane, and, further, if P and P are rational points, then the line L

is rational. Two distinct lines L and L in P 2 (C) intersect at a unique point P,and

further, if L and L are rational lines, then the intersection point P is rational.
The projective line L with equation L : aw + bx + cy = 0 determines the line
a + bx + cy = 0 in the Cartesian plane. Two projective lines L : aw + bx + cy = 0

and L : a w + b x + c y = 0 intersect on the line at infinity w = 0 if and only

if b : c = b : c , that is, the pairs (b, c) and (b , c ) are proportional. Hence the
corresponding lines in the x, y-plane given by

a + bx + cy = 0and a + b x + c y = 0
have the same slope or are parallel exactly when the projective lines intersect at
infinity. Now the reader is invited to reconsider the correspondence between rational

points on two rational lines L and L which arises by intersecting L and L with all

rational lines through a fixed point P 0 not on either L or L .
To define more general plane curves in projective space, we use nonzero homo-
geneous polynomials F(w, x, y) ∈ C[w, x, y]. Then we have the relation
d
F(qw, qx, qy) = q F(w, x, y),
where q ∈ C and d is the degree of the homogenous polynomial F(w, x, y).The
locus C F is the set of all w : x : y in the projective plane such that F(w, x, y) = 0.
The homogeneity of F(w, x, y) is needed for F(w, x, y) = 0 to be independent
of the scale for w : x : y ∈ P 2 . Again the complex points of C F are denoted by
C F (C) ⊂ P 2 (C), and, moreover, C F (C) = C F (C) if and only if F(w, x, y) and

F (w, x, y) are proportional with a nonzero complex number. This assertion is not

completely evident and is taken up again in Chapter 2.
(1.5) Definition. A rational (resp. real) plane curve in P 2 is one of the form C F
where F(w, x, y) has rational (resp. real) coefficients.
As in the x, y-plane for a rational plane curve C F , we have rational, real, and
complex points C F (Q) ⊂ C F (R) ⊂ C F (C).
(1.6) Remark. The above definition of a rational plane curve is an arithmetic notion,
and it means the curve can be defined over Q. There is a geometric concept of rational
curve (genus = 0) which should not be confused with (1.5).

§2. Rational Points on Conics

Now we study rational points on rational plane curves of degree 2 which in x, y-
coordinates are given by