Table Of Content”Berkeley Lectures” March 27, 2020 6.125x9.25
Annals of Mathematics Studies
Number 207
”Berkeley Lectures” March 27, 2020 6.125x9.25
”Berkeley Lectures” March 27, 2020 6.125x9.25
Berkeley Lectures on p-adic
Geometry
Peter Scholze and Jared Weinstein
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
2020
”Berkeley Lectures” March 27, 2020 6.125x9.25
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10 9 8 7 6 5 4 3 2 1
”Berkeley Lectures” March 27, 2020 6.125x9.25
Contents
Foreword ix
Lecture 1: Introduction 1
1.1 Motivation: Drinfeld, L. Lafforgue, and V. Lafforgue 1
1.2 The possibility of shtukas in mixed characteristic 4
Lecture 2: Adic spaces 7
2.1 Motivation: Formal schemes and their generic fibers 7
2.2 Huber rings 9
2.3 Continuous valuations 13
Lecture 3: Adic spaces II 17
3.1 Rational Subsets 17
3.2 Adic spaces 20
3.3 The role of A+ 20
3.4 Pre-adic spaces 21
Appendix: Pre-adic spaces 23
Lecture 4: Examples of adic spaces 27
4.1 Basic examples 27
4.2 Example: The adic open unit disc over Z 29
p
4.3 Analytic points 32
Lecture 5: Complements on adic spaces 35
5.1 Adic morphisms 35
5.2 Analytic adic spaces 36
5.3 Cartier divisors 38
Lecture 6: Perfectoid rings 41
6.1 Perfectoid Rings 41
6.2 Tilting 43
6.3 Sousperfectoid rings 47
Lecture 7: Perfectoid spaces 49
7.1 Perfectoid spaces: Definition and tilting equivalence 49
7.2 Why do we study perfectoid spaces? 50
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vi CONTENTS
7.3 The equivalence of ´etale sites 50
7.4 Almost mathematics, after Faltings 52
7.5 The ´etale site 55
Lecture 8: Diamonds 56
8.1 Diamonds: Motivation 56
8.2 Pro-´etale morphisms 57
8.3 Definition of diamonds 60
8.4 The example of SpdQ 62
p
Lecture 9: Diamonds II 64
9.1 Complements on the pro-´etale topology 64
9.2 Quasi-pro-´etale morphisms 67
9.3 G-torsors 68
9.4 The diamond SpdQ 69
p
Lecture 10:Diamonds associated with adic spaces 74
10.1 The functor X (cid:55)→X♦ 74
10.2 Example: Rigid spaces 77
10.3 The underlying topological space of diamonds 79
10.4 The ´etale site of diamonds 80
Appendix: Cohomology of local systems 82
Lecture 11:Mixed-characteristic shtukas 90
11.1 Theequalcharacteristicstory: Drinfeld’sshtukasandlocalshtukas 90
11.2 The adic space “S×SpaZ ” 91
p
11.3 Sections of (S×˙ SpaZ )♦ →S 94
p
11.4 Definition of mixed-characteristic shtukas 95
Lecture 12:Shtukas with one leg 98
12.1 p-divisible groups over O 98
C
12.2 Shtukas with one leg and p-divisible groups: An overview 100
12.3 Shtukas with no legs, and ϕ-modules over the integral Robba ring 103
12.4 Shtukas with one leg, and B -modules 105
dR
Lecture 13:Shtukas with one leg II 108
13.1 Y is an adic space 108
13.2 The extension of shtukas over x 109
L
13.3 Full faithfulness 109
13.4 Essential surjectivity 111
13.5 The Fargues-Fontaine curve 112
Lecture 14:Shtukas with one leg III 115
14.1 Fargues’ theorem 115
14.2 Extending vector bundles over the closed point of SpecA 116
inf
14.3 Proof of Theorem 14.2.1 119
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CONTENTS vii
14.4 Description of the functor “?” 121
Appendix: Integral p-adic Hodge theory 123
14.6 Cohomology of rigid-analytic spaces 124
14.7 Cohomology of formal schemes 124
14.8 p-divisible groups 126
14.9 The results of [BMS18] 127
Lecture 15:Examples of diamonds 131
15.1 The self-product SpdQ ×SpdQ 131
p p
15.2 Banach-Colmez spaces 133
Lecture 16:Drinfeld’s lemma for diamonds 140
16.1 The failure of π (X×Y)=π (X)×π (Y) 140
1 1 1
16.2 Drinfeld’s lemma for schemes 141
16.3 Drinfeld’s lemma for diamonds 143
Lecture 17:The v-topology 149
17.1 The v-topology on Perfd 149
17.2 Small v-sheaves 152
17.3 Spatial v-sheaves 152
17.4 Morphisms of v-sheaves 155
Appendix: Dieudonn´e theory over perfectoid rings 158
Lecture 18:v-sheaves associated with perfect and formal schemes 161
18.1 Definition 161
18.2 Topological spaces 162
18.3 Perfect schemes 163
18.4 Formal schemes 167
Lecture 19:The B+ -affine Grassmannian 169
dR
19.1 Definition of the B+ -affine Grassmannian 169
dR
19.2 Schubert varieties 172
19.3 The Demazure resolution 173
19.4 Minuscule Schubert varieties 176
Appendix: G-torsors 178
Lecture 20:Families of affine Grassmannians 182
20.1 The convolution affine Grassmannian 183
20.2 Over SpdQ 184
p
20.3 Over SpdZ 185
p
20.4 Over SpdQ ×...×SpdQ 186
p p
20.5 Over SpdZ ×...×SpdZ 189
p p
Lecture 21:Affine flag varieties 191
21.1 Over F 191
p
21.2 Over Z 192
p
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viii CONTENTS
21.3 Affine flag varieties for tori 194
21.4 Local models 194
21.5 D´evissage 196
Appendix: Examples 198
21.7 An EL case 203
21.8 A PEL case 204
Lecture 22:Vector bundles and G-torsors 207
22.1 Vector bundles 207
22.2 Semicontinuity of the Newton polygon 208
22.3 The ´etale locus 209
22.4 Classification of G-torsors 210
22.5 Semicontinuity of the Newton point 212
22.6 Extending G-torsors 213
Lecture 23:Moduli spaces of shtukas 215
23.1 Definition of mixed-characteristic local shtukas 216
23.2 The case of no legs 217
23.3 The case of one leg 218
23.4 The case of two legs 220
23.5 The general case 223
Lecture 24:Local Shimura varieties 225
24.1 Definition of local Shimura varieties 225
24.2 Relation to Rapoport-Zink spaces 226
24.3 General EL and PEL data 229
Lecture 25:Integral models of local Shimura varieties 232
25.1 Definition of the integral models 232
25.2 The case of tori 235
25.3 Non-parahoric groups 236
25.4 The EL case 237
25.5 The PEL case 238
Bibliography 241
Index 249
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Foreword
This is a revised version of the lecture notes for the course on p-adic geometry
given by P. Scholze in Fall 2014 at UC Berkeley. At a few points, we have ex-
pandedslightlyonthematerial,inparticularsoastoprovideafullconstruction
oflocalShimuravarietiesandgeneralmodulispacesofshtukas,alongwithsome
applications to Rapoport-Zink spaces, but otherwise we have tried to keep the
informal style of the lectures.
Let us give an outline of the contents:
In the first half of the course (Lectures 1–10) we construct the category
of diamonds, which are quotients of perfectoid spaces by so-called pro-´etale
equivalence relations. In brief, diamonds are to perfectoid spaces as algebraic
spaces are to schemes.
• Lecture1isanintroduction,explainingthemotivationcomingfromtheLang-
lands correspondence and moduli spaces of shtukas.
• In Lectures 2–5 we review the theory of adic spaces [Hub94].
• In Lectures 6–7 we review the theory of perfectoid spaces [Sch12].
• In Lectures 8–10 we review the theory of diamonds [Sch17].
Inthesecondhalfofthecourse(Lectures11–25), wedefinespacesofmixed-
characteristic local shtukas, which live in the category of diamonds. This re-
quires making sense of products like SpaQ ×S, where S is an adic space over
p
F .
p
• In Lecture 11 we give a geometric meaning to SpaZ × S, where S is a
p
perfectoid space in characteristic p, and we define the notion of a mixed-
characteristic local shtuka.
• In Lectures 12–15, we study shtukas with one leg, and their connection to
p-divisible groups and p-adic Hodge theory.
• In Lecture 16, we prove the analogue of Drinfeld’s lemma for the product
SpaQ ×SpaQ .
p p
• In Lectures 17–23, we construct a moduli space of shtukas for any triple
(G,b,{µ ,...,µ }), for any reductive group G/Q , any σ-conjugacy class
1 m p
”Berkeley Lectures” March 27, 2020 6.125x9.25
x FOREWORD
b, and any collection of cocharacters µ . This moduli space is a diamond,
i
whichisfiberedoverthem-foldproductofSpaQ . Provingthisissomewhat
p
technical; it requires the technology of v-sheaves developed in Lecture 17.
• In Lecture 24, we show that our moduli spaces of shtukas specialize (in
the case of one leg) to local Shimura varieties, which in turn specialize to
Rapoport-Zink spaces. For this we have to relate local shtukas to p-divisible
groups.
• In Lecture 25, we address the question of defining integral models for local
Shimura varieties.
Since 2014, some of the material of this course has found its way to other
manuscripts which discuss it in more detail, in particular [Sch17], and we will
oftenrefertothesereferences. Inparticular,theproperfoundationsondiamonds
can only be found in [Sch17]; here, we only survey the main ideas in the same
way as in the original lectures. In this way, we hope that this manuscript can
serve as an informal introduction to these ideas.
During the semester at Berkeley, Laurent Fargues formulated his conjecture
onthegeometrizationofthelocalLanglandsconjecture,[Far16],whichisclosely
relatedtothecontentsofthiscourse,butleadstoaradicalchangeofperspective.
We have kept the original perspective of the lecture course in these notes, so
that Fargues’ conjecture does not make an explicit appearance.
Acknowledgments. We thank the University of California at Berkeley for
theopportunitytogivetheselecturesandforhostingusinFall2014. Moreover,
wethankalltheparticipantsofthecoursefortheirfeedback, andwewouldlike
tothankespeciallyBrianConradandJo˜aoLourenc¸oforverydetailedcomments
and suggestions for improvements. Part of this work was done while the first
author was a Clay Research Fellow.
June 2019 Peter Scholze, Jared Weinstein
