Introduction

The abc-conjecture of Masser and Oesterlé is a typical example of a simple statement that can be used to unify and motivate many results in number theory, which otherwise would be scattered statements without a common link. As such, it deserves to be discussed, first by showing its power and then by generalizing it and showing how it fits into the much more general and coherent set of conjectures provided by Vojta in his thesis.

Although a pessimist may conclude that the ease with which the abc-conjecture may be applied to solve notoriously difficult problems is only a reflection of how difficult its proof is likely to be, we should keep in mind that its function field analogue is quite easy to prove and provides a unified method of attack for many problems in the arithmetic of function fields. Moreover, whatever its status in the classical case, it is likely that exceptions, if any, will be extremely rare and most of the conclusions obtained by its application are also likely to be valid and provable in some instances by different methods. The abc-conjecture is also a useful tool for guessing the right answer when analysing specific problems, hence its significance should not be too easily discounted.

The content of this chapter is as follows.