Berkeley Lectures on p-adic Geometry

Name: Berkeley Lectures on p-adic Geometry | (AMS-207)
Brand: Princeton University Press
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(AMS-207)

Peter Scholze Jared Weinstein(Autor*in)

Princeton University Press

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Erschienen am 26. Mai 2020

264 Seiten

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Inhalt

Cover
Title
Copyright
Contents
Foreword
Lecture 1: Introduction
1.1 Motivation: Drinfeld, L. Lafforgue, and V. Lafforgue
1.2 The possibility of shtukas in mixed characteristic
Lecture 2: Adic spaces
2.1 Motivation: Formal schemes and their generic fibers
2.2 Huber rings
2.3 Continuous valuations
Lecture 3: Adic spaces II
3.1 Rational Subsets
3.2 Adic spaces
3.3 The role of A^+
3.4 Pre-adic spaces
Appendix: Pre-adic spaces
Lecture 4: Examples of adic spaces
4.1 Basic examples
4.2 Example: The adic open unit disc over Zp
4.3 Analytic points
Lecture 5: Complements on adic spaces
5.1 Adic morphisms
5.2 Analytic adic spaces
5.3 Cartier divisors
Lecture 6: Perfectoid rings
6.1 Perfectoid Rings
6.2 Tilting
6.3 Sousperfectoid rings
Lecture 7: Perfectoid spaces
7.1 Perfectoid spaces: Definition and tilting equivalence
7.2 Why do we study perfectoid spaces?
7.3 The equivalence of étale sites
7.4 Almost mathematics, after Faltings
7.5 The étale site
Lecture 8: Diamonds
8.1 Diamonds: Motivation
8.2 Pro-étale morphisms
8.3 Definition of diamonds
8.4 The example of Spd Qp
Lecture 9: Diamonds II
9.1 Complements on the pro-étale topology
9.2 Quasi-pro-étale morphisms
9.3 G-torsors
9.4 The diamond Spd Qp
Lecture 10: Diamonds associated with adic spaces
10.1 The functor X ? X^?
10.2 Example: Rigid spaces
10.3 The underlying topological space of diamonds
10.4 The étale site of diamonds
Appendix: Cohomology of local systems
Lecture 11: Mixed-characteristic shtukas
11.1 The equal characteristic story: Drinfeld's shtukas and local shtukas
11.2 The adic space "S × Spa Zp"
11.3 Sections of (S × Spa Zp)^? S
11.4 Definition of mixed-characteristic shtukas
Lecture 12: Shtukas with one leg
12.1 p-divisible groups over OC
12.2 Shtukas with one leg and p-divisible groups: An overview
12.3 Shtukas with no legs, and f-modules over the integral Robba ring
12.4 Shtukas with one leg, and BdR-modules
Lecture 13: Shtukas with one leg II
13.1 Y is an adic space
13.2 The extension of shtukas over xL
13.3 Full faithfulness
13.4 Essential surjectivity
13.5 The Fargues-Fontaine curve
Lecture 14: Shtukas with one leg III
14.1 Fargues' theorem
14.2 Extending vector bundles over the closed point of Spec Ainf
14.3 Proof of Theorem 14.2.1
14.4 Description of the functor "?"
Appendix: Integral p-adic Hodge theory
14.6 Cohomology of rigid-analytic spaces
14.7 Cohomology of formal schemes
14.8 p-divisible groups
14.9 The results of [BMS18]
Lecture 15: Examples of diamonds
15.1 The self-product Spd Qp × Spa Qp
15.2 Banach-Colmez spaces
Lecture 16: Drinfeld's lemma for diamonds
16.1 The failure of p1(X × Y) = p1(X) × p1(Y)
16.2 Drinfeld's lemma for schemes
16.3 Drinfeld's lemma for diamonds
Lecture 17: The v-topology
17.1 The v-topology on Perfd
17.2 Small v-sheaves
17.3 Spatial v-sheaves
17.4 Morphisms of v-sheaves
Appendix: Dieudonné theory over perfectoid rings
Lecture 18: v-sheaves associated with perfect and formal schemes
18.1 Definition
18.2 Topological spaces
18.3 Perfect schemes
18.4 Formal schemes
Lecture 19: The B^+dR-affine Grassmannian
19.1 Definition of the B^+dR-affine Grassmannian
19.2 Schubert varieties
19.3 The Demazure resolution
19.4 Minuscule Schubert varieties
Appendix: G-torsors
Lecture 20: Families of affine Grassmannians
20.1 The convolution affine Grassmannian
20.2 Over Spd Qp
20.3 Over Spd Zp
20.4 Over Spd Qp × . . . × Spd Qp
20.5 Over Spd Zp × . . . × Spd Zp
Lecture 21: Affine flag varieties
21.1 Over Fp
21.2 Over Zp
21.3 Affine flag varieties for tori
21.4 Local models
21.5 Dévissage
Appendix: Examples
21.7 An EL case
21.8 A PEL case
Lecture 22: Vector bundles and G-torsors
22.1 Vector bundles
22.2 Semicontinuity of the Newton polygon
22.3 The étale locus
22.4 Classification of G-torsors
22.5 Semicontinuity of the Newton point
22.6 Extending G-torsors
Lecture 23: Moduli spaces of shtukas
23.1 Definition of mixed-characteristic local shtukas
23.2 The case of no legs
23.3 The case of one leg
23.4 The case of two legs
23.5 The general case
Lecture 24: Local Shimura varieties
24.1 Definition of local Shimura varieties
24.2 Relation to Rapoport-Zink spaces
24.3 General EL and PEL data
Lecture 25: Integral models of local Shimura varieties
25.1 Definition of the integral models
25.2 The case of tori
25.3 Non-parahoric groups
25.4 The EL case
25.5 The PEL case
Bibliography
Index

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