2 introduction

shows that the second point where this line meets the unit circle is given by

1 − t2

1 + t2

,

2t

1 + t2

. (‡)

As every point on the unit circle may be connected with (1, 0) by a line, one sees that

the parametrization (‡) yields every rational point on the circle (except for (−1, 0),

for which the form of line equation given is not adequate).

Consequently, formula (†) delivers essentially every solution of equation (∗), a fact

which was seemingly not known to the ancient mathematicians. The morphism

P1

−→

P2,

(p : q) →

(

(p2

−

q2)

: 2pq :

(p2

+

q2)

)

provides a rational parametrization of the plane conic C given by the equation

x2 +y2 = z2 in P2. More or less the same method works for every conic in the plane.

Further, it may be extended to several classes of singular curves of higher degree.

Every Diophantine equation defines an algebraic variety X in an aﬃne or projec-

tive space. There is a one-to-one correspondence between solutions of the Diophan-

tine equation and -rational points on X. We will prefer geometric language to

number theoretic throughout this book.

The cases in which there is an obvious rational parametrization are, in some sense,

the best possible. But even when there is nothing like that, algebraic geometry

often yields a guideline of which behaviour to expect—whether there will be no, a

few, or many solutions.

The Kodaira classification distinguishes between Fano varieties, varieties of interme-

diate type, and varieties of general type (at least under the additional assumption

that X is non-singular). It does not use any specifically arithmetic information, but

only information about X as a complex variety. Nevertheless, there is overwhelm-

ing evidence for a strong connection between the classification of X according to

Kodaira and its set of rational points.

To make a vague statement, on a Fano variety, there are infinitely many rational

points expected while, on a variety of general type, there are only finitely many

rational points or even none at all. More precisely, there is the conjecture that,

on a Fano variety, there are always infinitely many rational points after a suitable

finite extension of the ground field. On the other hand, for varieties of general type,

there is the conjecture of Lang. It states that there are only finitely many rational

points outside the union of all closed subvarieties that are not of general type.

Another method to analyze a Diophantine equation is given by congruences.

Kurt Hensel provided a more formal framework for this method by his invention

of the p-adic numbers. As one is working over local fields, this might be called the

local method.

Consider, for example, the Diophantine equation

x3

+

7y3

+

49z3

+

2u3

+

14v3

+

98w3

= 0 . (§)