$p$-adic Geometry: Lectures from the 2007 Arizona Winter School
About this Title
Matthew Baker, Georgia Institute of Technology, Atlanta, GA, Brian Conrad, University of Michigan, Ann Arbor, MI, Samit Dasgupta, Harvard University, Cambridge, MA, Kiran S. Kedlaya, Massachusetts Institute of Technology, Cambridge, MA and Jeremy Teitelbaum, University of Illinois at Chicago, Chicago, IL. Edited by David Savitt, University of Arizona, Tucson, AZ and Dinesh S. Thakur, University of Arizona, Tucson, AZ
Publication: University Lecture Series
Publication Year 2008: Volume 45
ISBNs: 978-0-8218-4468-7 (print); 978-1-4704-1836-6 (online)
MathSciNet review: MR2482343
MSC: Primary 14-06; Secondary 11F85, 14G22
In recent decades, $p$-adic geometry and $p$-adic cohomology theories have become indispensable tools in number theory, algebraic geometry, and the theory of automorphic representations. The Arizona Winter School 2007, on which the current book is based, was a unique opportunity to introduce graduate students to this subject.
Following invaluable introductions by John Tate and Vladimir Berkovich, two pioneers of non-archimedean geometry, Brian Conrad's chapter introduces the general theory of Tate's rigid analytic spaces, Raynaud's view of them as the generic fibers of formal schemes, and Berkovich spaces. Samit Dasgupta and Jeremy Teitelbaum discuss the $p$-adic upper half plane as an example of a rigid analytic space and give applications to number theory (modular forms and the $p$-adic Langlands program). Matthew Baker offers a detailed discussion of the Berkovich projective line and $p$-adic potential theory on that and more general Berkovich curves. Finally, Kiran Kedlaya discusses theoretical and computational aspects of $p$-adic cohomology and the zeta functions of varieties. This book will be a welcome addition to the library of any graduate student and researcher who is interested in learning about the techniques of $p$-adic geometry.
Graduate students and research mathematicians interested in number theory and algebraic geometry.
Table of Contents
- Vladimir Berkovich – Non-archimedean analytic geometry: first steps
- Chapter 1. Brian Conrad – Several approaches to non-archimedean geometry
- Chapter 2. Samit Dasgupta and Jeremy Teitelbaum – The $p$-adic upper half plane
- Chapter 3. Matthew Baker – An introduction to Berkovich analytic spaces and non-archimedean potential theory on curves
- Chapter 4. Kiran S. Kedlaya – $p$-adic cohomology: from theory to practice