In the first decades of the development of the theory of p-adic numbers and p-adic analysis, people realized that the theory of analytic continuation had no easy p-adic analogue. This made it very hard to develop a p-adic version of the theory of complex manifolds. The problem was solved by John Tate in the 1950s, in a famous paper called "Rigid Analytic Geometry." Tate's idea has since been developed by many other mathematicians, and is now one an important part of number theory and algebraic geometry.
Rigid analytic geometry is notoriously hard to learn. A prominent mathematician once described the situation as follows. First, one didn't learn it until it was necessary to one's work. Then, one worked very hard to learn the theory. Finally, when writing up the work, one gave one's own "take" on the theory. As a result, there was no easy way to pick it up, and there were several conflicting accounts of the foundations in the literature.
The situation is still a bit messy, but this and other books have improved things a lot. When I was a graduate student, we used the original (French) version of this book in an informal seminar on rigid geometry. It was quite helpful then, and it is much better now. The authors have updated the material, added quite a bit on new applications and new results, and changed languages. Despite the competition it now has, this is still one of the best places in which to start learning this theory.
Fernando Q. Gouvêa is Professor of Mathematics at Colby College and the co-author, with William P. Berlinghoff, of Math through the Ages. He somehow finds time to also be the editor of MAA Reviews.
|Preface * Valued fields and normed spaces * The projective line * Affinoid algebras * Rigid spaces * Curves and their reductions * Abelian varieties * Points of rigid spaces, rigid cohomology * Etale cohomology of rigid spaces * Covers of algebraic curves * References * List of Notation * Index|