Perfectoid Spaces: Lectures from the 2017 Arizona Winter School
About this Title
Bhargav Bhatt, University of Michigan, Ann Arbor, MI, Ana Caraiani, Imperial College, London, United Kingdom, Kiran S. Kedlaya, University of California, San Diego, La Jolla, CA, Peter Scholze, University of Bonn, Bonn, Germany and Jared Weinstein, Boston University, Boston, MA. Edited by Bryden Cais, University of Arizona, Tucson, AZ
Publication: Mathematical Surveys and Monographs
Publication Year: 2019; Volume 242
ISBNs: 978-1-4704-5015-1 (print); 978-1-4704-5411-1 (online)
MathSciNet review: 3970252
MSC: Primary 14G22; Secondary 11F80, 11G25, 14C30, 14F30, 14G35, 14G40, 32P99
Introduced by Peter Scholze in 2011, perfectoid spaces are a bridge between geometry in characteristic 0 and characteristic $p$, and have been used to solve many important problems, including cases of the weight-monodromy conjecture and the association of Galois representations to torsion classes in cohomology. In recognition of the transformative impact perfectoid spaces have had on the field of arithmetic geometry, Scholze was awarded a Fields Medal in 2018.
This book, originating from a series of lectures given at the 2017 Arizona Winter School on perfectoid spaces, provides a broad introduction to the subject. After an introduction with insight into the history and future of the subject by Peter Scholze, Jared Weinstein gives a user-friendly and utilitarian account of the theory of adic spaces. Kiran Kedlaya further develops the foundational material, studies vector bundles on Fargues–Fontaine curves, and introduces diamonds and shtukas over them with a view toward the local Langlands correspondence. Bhargav Bhatt explains the application of perfectoid spaces to comparison isomorphisms in $p$-adic Hodge theory. Finally, Ana Caraiani explains the application of perfectoid spaces to the construction of Galois representations associated to torsion classes in the cohomology of locally symmetric spaces for the general linear group.
This book will be an invaluable asset for any graduate student or researcher interested in the theory of perfectoid spaces and their applications.
Graduate students and researchers interested in new developments in algebraic geometry and algebraic number theory.
Table of Contents
- Adic spaces
- Sheaves, stacks, and shtukas
- The Hodge-Tate decomposition via perfectoid spaces
- Perfectoid Shimura varieties