##
**Rigid analytic geometry and its applications.**
*(English)*
Zbl 1096.14014

Progress in Mathematics (Boston, Mass.) 218. Boston, MA: Birkhäuser (ISBN 0-8176-4206-4/hbk). xii, 296 p. (2004).

Publisher’s description: The theory of rigid (analytic) spaces, originally invented to describe degenerations, reductions, and moduli of algebraic curves and abelian varieties, has undergone significant growth in the last two decades; today the theory has applications to arithmetic algebraic geometry, number theory, and several complex variables. This new English edition, which was revised and greatly expanded from an earlier French text by the authors, provides a thorough introduction to the basics of the theory of rigid analytic spaces. In addition, the work presents important new results and applications to “points of rigid spaces”, étale topology, Drinfeld modular curves, Shimura varieties, and symmetric spaces.

A final chapter treats rigid methods, which led to the solution of Abhyankar’s problem, and other developments on the Galois theory of function fields. At the heart of the theory lies a complete non-archimedean field \(K\), such as the field of \(p\)-adic numbers or a field of Laurent series. Function theory is developed in one or more variables over \(K\) and rigid analytic spaces over \(K\). Further, the theory of rigid analytic spaces has much in common with complex analytic spaces. Its close relation with algebraic geometry is apparent from the building blocks, namely affinoid rings and affinoid spaces, which are similar to polynomial rings and affine varieties. A rigid space also can be interpreted as the “generic fiber” of a formal scheme over a complete valuation ring. Each of these interpretations and connections is carefully examined in the book’s eight chapters, which encompass topics ranging from cohomology to several complex variables. With its concise, self-contained exposition and many examples and exercises, this book will be useful to graduate students and researchers as an introductory course text, self-study volume, or reference.

A final chapter treats rigid methods, which led to the solution of Abhyankar’s problem, and other developments on the Galois theory of function fields. At the heart of the theory lies a complete non-archimedean field \(K\), such as the field of \(p\)-adic numbers or a field of Laurent series. Function theory is developed in one or more variables over \(K\) and rigid analytic spaces over \(K\). Further, the theory of rigid analytic spaces has much in common with complex analytic spaces. Its close relation with algebraic geometry is apparent from the building blocks, namely affinoid rings and affinoid spaces, which are similar to polynomial rings and affine varieties. A rigid space also can be interpreted as the “generic fiber” of a formal scheme over a complete valuation ring. Each of these interpretations and connections is carefully examined in the book’s eight chapters, which encompass topics ranging from cohomology to several complex variables. With its concise, self-contained exposition and many examples and exercises, this book will be useful to graduate students and researchers as an introductory course text, self-study volume, or reference.