Berkeley Lectures on p-adic Geometry

Contents
Foreword
1 Introduction
	1.1 Motivation: Drinfeld, L. Lafforgue, and V. Lafforgue
	1.2 The possibility of shtukas in mixed characteristic
2 Adic spaces
	2.1 Motivation: Formal schemes and their generic fibers
	2.2 Huber rings
	2.3 Continuous valuations
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
4 Examples of adic spaces
	4.1 Basic examples
	4.2 Example: The adic open unit disc over Zp
	4.3 Analytic points
5 Complements on adic spaces
	5.1 Adic morphisms
	5.2 Analytic adic spaces
	5.3 Cartier divisors
6 Perfectoid rings
	6.1 Perfectoid Rings
	6.2 Tilting
	6.3 Sousperfectoid rings
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
8 Diamonds
	8.1 Diamonds: Motivation
	8.2 Pro-étale morphisms
	8.3 Definition of diamonds
	8.4 The example of `39`42`"613A``45`47`"603ASpdQp
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 `39`42`"613A``45`47`"603ASpdQp
10 Diamonds associated with adic spaces
	10.1 The functor XX
	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
11 Mixed-characteristic shtukas
	11.1 The equal characteristic story: Drinfeld's shtukas and local shtukas
	11.2 The adic space ``S`39`42`"613A``45`47`"603ASpaZp''
	11.3 Sections of (S `39`42`"613A``45`47`"603ASpaZp)S
	11.4 Definition of mixed-characteristic shtukas
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 -modules over the integral Robba ring
	12.4 Shtukas with one leg, and BdR-modules
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
14 Shtukas with one leg III
	14.1 Fargues' theorem
	14.2 Extending vector bundles over the closed point of `39`42`"613A``45`47`"603ASpecAinf
	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 BMS
15 Examples of diamonds
	15.1 The self-product `39`42`"613A``45`47`"603ASpdQp`39`42`"613A``45`47`"603ASpdQp
	15.2 Banach-Colmez spaces
16 Drinfeld's lemma for diamonds
	16.1 The failure of 1(XY)=1(X)1(Y)
	16.2 Drinfeld's lemma for schemes
	16.3 Drinfeld's lemma for diamonds
17 The v-topology
	17.1 The v-topology on `39`42`"613A``45`47`"603APerfd
	17.2 Small v-sheaves
	17.3 Spatial v-sheaves
	17.4 Morphisms of v-sheaves
	Appendix: Dieudonné theory over perfectoid rings
18 v-sheaves associated with perfect and formal schemes
	18.1 Definition
	18.2 Topological spaces
	18.3 Perfect schemes
	18.4 Formal schemes
19 The BdR+-affine Grassmannian
	19.1 Definition of the BdR+-affine Grassmannian
	19.2 Schubert varieties
	19.3 The Demazure resolution
	19.4 Minuscule Schubert varieties
	Appendix: G-torsors
20 Families of affine Grassmannians
	20.1 The convolution affine Grassmannian
	20.2 Over `39`42`"613A``45`47`"603ASpdQp
	20.3 Over `39`42`"613A``45`47`"603ASpdZp
	20.4 Over `39`42`"613A``45`47`"603ASpdQp…`39`42`"613A``45`47`"603ASpdQp
	20.5 Over `39`42`"613A``45`47`"603ASpdZp…`39`42`"613A``45`47`"603ASpdZp
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
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
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
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
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