Annals of Mathematics Studies

Peter Scholze, Jared Weinstein

• Publisher: Princeton University Press
• Year: 2020
• ISBN:
  9780691202150
• Topic: Mathematics
• Subtopic: Algebraic Geometry
• Index: Mathematics

Berkeley Lectures on p-adic Geometry presents an important breakthrough in arithmetic geometry. In 2014, leading mathematician Peter Scholze delivered a series of lectures at the University of California, Berkeley, on new ideas in the theory of p -adic geometry. Building on his discovery of perfectoid spaces, Scholze introduced the concept of "diamonds, " which are to perfectoid spaces what algebraic spaces are to schemes. The introduction of diamonds, along with the development of a mixed-characteristic shtuka, set the stage for a critical advance in the discipline. In this book, Peter Scholze and Jared Weinstein show that the moduli space of mixed-characteristic shtukas is a diamond, raising the possibility of using the cohomology of such spaces to attack the Langlands conjectures for a reductive group over a p -adic field.This book follows the informal style of the original Berkeley lectures, with one chapter per lecture. It explores p- adic and perfectoid spaces before laying out the newer theory of shtukas and their moduli spaces. Points of contact with other threads of the subject, including p -divisible groups, p -adic Hodge theory, and Rapoport-Zink spaces, are thoroughly explained. Berkeley Lectures on p-adic Geometry will be a useful resource for students and scholars working in arithmetic geometry and number theory.

Table of contents

• Cover
• Title
• Copyright
• Contents
• Foreword
• Lecture 1: Introduction
• Lecture 2: Adic spaces
• Lecture 3: Adic spaces II
• Lecture 4: Examples of adic spaces
• Lecture 5: Complements on adic spaces
• Lecture 6: Perfectoid rings
• Lecture 7: Perfectoid spaces
• Lecture 8: Diamonds
• Lecture 9: Diamonds II
• Lecture 10: Diamonds associated with adic spaces
• Lecture 11: Mixed-characteristic shtukas
• Lecture 12: Shtukas with one leg
• Lecture 13: Shtukas with one leg II
• Lecture 14: Shtukas with one leg III
• Lecture 15: Examples of diamonds
• Lecture 16: Drinfeld's lemma for diamonds
• Lecture 17: The v-topology
• Lecture 18: v-sheaves associated with perfect and formal schemes
• Lecture 19: The B^+dR-affine Grassmannian
• Lecture 20: Families of affine Grassmannians
• Lecture 21: Affine flag varieties
• Lecture 22: Vector bundles and G-torsors
• Lecture 23: Moduli spaces of shtukas
• Lecture 24: Local Shimura varieties
• Lecture 25: Integral models of local Shimura varieties
• Bibliography
• Index