This volume grew out of the London Mathematical Society symposium on “Galois representations in arithmetic algebraic geometry” held in Durham from the 9th to the 18th of July 1996. We understood our title rather loosely and the symposium considered many recent developments on the interface between algebraic number theory and arithmetic algebraic geometry. There were six expository courses on

1. Galois module structure

2. Shimura varieties in mixed characteristic

3. p-adic comparison theorems

4. the work of Kato on the Birch-Swinnerton-Dyer conjecture

5. poly logarithms

6. rigid analysis and modular forms

We are very grateful to the organisers of each of theses courses (Chinburg, Oort, Fontaine, Kato, Goncharov and Coleman) as well as all the other lecturers who worked hard to make these courses highly successful. In addition to the short courses there were 14 research seminars. We would also like to thank these lecturers and particularly those who have contributed to this volume. The symposium received generous financial support from the EPSRC and from the EU (through the network on “automorphic forms and arithmetic algebraic geometry”). We were particularly pleased that this enabled a large number of young European researchers to attend. Finally we would like to thank Steve Wilson and the department of mathematics at Durham University for their help with the organisation of the meeting.

This volume contains both expository and research articles. We are particularly grateful to the authors (Erez, Mazur, Moonen and Schneider) who have put a lot of time into preparing what we feel will be a useful collection of expositions.