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Berkeley Lectures on p-adic Geometry PDF

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”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 Copyright (cid:13)c 2020 by Princeton University Press Requests for permission to reproduce material from this work should be sent to [email protected] Published by Princeton University Press 41 William Street, Princeton, New Jersey 08540 6 Oxford Street, Woodstock, Oxfordshire OX20 1TR press.princeton.edu All Rights Reserved ISBN 978-0-691-20209-9 ISBN (pbk.) 978-0-691-20208-2 ISBN (e-book) 978-0-691-20215-0 British Library Cataloging-in-Publication Data is available Editorial: Susannah Shoemaker Production Editorial: Brigitte Pelner Production: Jacqueline Poirier Publicity: Matthew Taylor (US) and Katie Lewis (UK) This book has been composed in LATEX Thepublisherwouldliketoacknowledgetheauthorsofthisvolumeforproviding the print-ready files from which this book was printed. Printed in the United States of America 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 ”Berkeley Lectures” March 27, 2020 6.125x9.25 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 ”Berkeley Lectures” March 27, 2020 6.125x9.25 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 ”Berkeley Lectures” March 27, 2020 6.125x9.25 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 ”Berkeley Lectures” March 27, 2020 6.125x9.25 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

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