14 - The study of Diophantine equations over function fields

Summary

Introduction

In recent years there has been important progress in the study of Diophantine equations over function fields. This has aroused increasing interest in the subject, and has led to the re-examination of some old results, as well as the realisation that there is now a wide variety of areas to which the subject is applicable. We shall spend a little time on the history and present state of the subject before proving a specific result concerning decomposable form equations. This result forms the completion of a series of works which attack several general classes of equations by means of an important inequality on solutions of the multivariate unit equation.

There are now three distinct analytical approaches to the study of Diophantine equations over function fields. The first is that of algebraic geometry, the second that of differential equations, and the third that of Diophantine approximation.

The celebrated theorem of Manin and Grauert established the analogue of Mordell's conjecture (now Faltings' theorem) for function fields, on the finiteness of the number of solutions of equations in two variables. This confirmed the great progress made in algebraic geometry since the Second World War. Grauert's approach was heavily dependent on esoteric algebraic geometry, and his work led to further advances in that subject by Shafarevitch and Paršin. Although the Manin-Grauert result was extremely general, for many years it was believed that their methods would not lead to effective bounds on the actual solutions of equations, nor would it allow the solutions themselves to be determined explicitly.