For a number of years, French mathematicians, supported by the Société Mathamatique de France, have organized colloquia in the theory of numbers, to which they have always invited British mathematicians (and others). The London Mathematical Society returned their hospitality during the ‘Journées Arithmétiques’ held at the University of Exeter from 13th April to 19th April 1980.

This volume contains the texts of the invited addresses, in some cases considerably expanded, with the exception of expository papers by Dr B.J. Birch (a progress report on elliptic curves) and Professor J. Coates (on work of Mazur and Wiles on cyclotomic fields), who preferred not to publish their talks. In addition there are selected contributions, also sometimes in expanded versions, from the several splinter groups organized during the colloquium. Our gratitude is due to the lecturers for making the publication of this volume possible, and indeed for their original contribution to the success of the Journées’.

The first part of Professor Chudnovsky's article is a report of his talk. We are grateful to him for making available for publication a proof of his results, as presented here in the second part of his article. Because of the importance of the results, it was an editorial decision to publish them here, even though there was no time to submit the manuscript to the usual refereeing procedure.

We should like to express our thanks to the London Mathematical Society for supporting these ‘Journees1, to the organisers and especially to the secretary, Dr R.W.K. Odoni of the University of Exeter.