Page 24 -

Introduction to Rational Points on Plane Curves

This introduction is designed to bring up some of the main issues of the book in an
informal way so that the reader with only a minimal background in mathematics can
get an idea of the character and direction of the subject.
An elliptic curve, viewed as a plane curve, is given by a nonsingular cubic equa-
tion. We wish to point out what is special about the class of elliptic curves among all
plane curves from the point of view of arithmetic. In the process the geometry of the
curve also enters the picture.
For the first considerations our plane curves are defined by a polynomial equation
in two variables f (x, y) = 0 with rational coefficients. The main invariant of this f
is its degree, a natural number. In terms of plane analytic geometry there is a curve
C f which is the locus of this equation in the x, y-plane, that is, C f is defined as the
2
set of (x, y) ∈ R satisfying f (x, y) = 0. To emphasize that the locus consists of
2
points with real coordinates (so is in R ), we denote this real locus by C f (R) and
2
consider C f (R) ⊂ R .
2
2
Since some curves C f , like for example f (x, y) = x + y + 1, have an empty
real locus C f (R), it is always useful to work also with the complex locus C f (C)
2
contained in C even though it cannot be completely pictured geometrically. For
geometric considerations involving the curve, the complex locus C f (C) plays the
central role.
For arithmetic the locus of special interest is the set C f (Q) of rational points
2
(x, y) ∈ Q satisfying f (x, y) = 0, that is, points whose coordinates are rational
numbers. The fundamental problem of this book is the description of this set C f (Q).
An elementary formulation of this problem is the question whether or not C f (Q) is
finite or even empty.
This problem is attacked by a combination of geometric and arithmetic argu-
ments using the inclusions C f (Q) ⊂ C f (R) ⊂ C f (C).Alocus C f (Q) can be
compared with another locus C g (Q), which is better understood, as we illustrate for
lines where deg( f ) = 1 and conics where deg( f ) = 2. In the case of cubic curves
we introduce an internal operation.
In terms of the real locus, curves of degree 1, degree 2, and degree 3 can be
pictured respectively as follows.