Number Theory III

Diophantine problems concern the solutions of equations in integers, rational numbers, or various generalizations. The book is an encyclopaedic survey of diophantine geometry. For the most part no proofs are given, but references are given where proofs may be found. There are some exceptions, notably the proof for a large part of Faltings' theorems is given. The survey puts together, from a unified point of view, the field of diophantine geometry which has developed since the early 1950s, after its origins in Mordell, Weil and Siegel's papers in the 1920s. The basic approach is that of algebraic geometry, but examples are given which show how this approach deals with (and sometimes solves!) classical problems phrased in very elementary terms. For instance, the Fermat problem is not solved, but it is shown to fit in to two great structural approaches, so that it is not an isolated problem any more. This monograph on number theory, algebraic geometry, several complex variables and differential geometry is intended for graduate students and researchers.