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Fundamentals of Diophantine Geometry (Hardcover, 1983 ed.)
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Fundamentals of Diophantine Geometry (Hardcover, 1983 ed.)
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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.
General
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