It is possible to write endlessly on elliptic curves. (This is not
a threat.) We deal here with diophantine problems, and we lay the
foundations, especially for the theory of integral points. We
review briefly the analytic theory of the Weierstrass function, and
then deal with the arithmetic aspects of the addition formula, over
complete fields and over number fields, giving rise to the theory
of the height and its quadraticity. We apply this to integral
points, covering the inequalities of diophantine approximation both
on the multiplicative group and on the elliptic curve directly.
Thus the book splits naturally in two parts. The first part deals
with the ordinary arithmetic of the elliptic curve: The
transcendental parametrization, the p-adic parametrization, points
of finite order and the group of rational points, and the reduction
of certain diophantine problems by the theory of heights to
diophantine inequalities involving logarithms. The second part
deals with the proofs of selected inequalities, at least strong
enough to obtain the finiteness of integral points.
General
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