Diophantine problems represent some of the strongest aesthetic
attractions to algebraic geometry. They consist in giving criteria for
the existence of solutions of algebraic equations in rings and fields,
and eventually for the number of such solutions. The fundamental ring of
interest is the ring of ordinary integers Z, and the fundamental field
of interest is the field Q of rational numbers. One discovers rapidly
that to have all the technical freedom needed in handling general
problems, one must consider rings and fields of finite type over the
integers and rationals. Furthermore, one is led to consider also finite
fields, p-adic fields (including the real and complex numbers) as
representing a localization of the problems under consideration. We
shall deal with global problems, all of which will be of a qualitative
nature. On the one hand we have curves defined over say the rational
numbers. Ifthe curve is affine one may ask for its points in Z, and
thanks to Siegel, one can classify all curves which have infinitely many
integral points. This problem is treated in Chapter VII. One may ask
also for those which have infinitely many rational points, and for this,
there is only Mordell's conjecture that if the genus is: 2, then there
is only a finite number of rational points.
