Complex multiplication and lifting problems

402 Pages·2014·2.85 MB·English

by  Ching-Li Chai

Mathematical
Surveys
and
Monographs
Volume 195
Complex Multiplication
and Lifting Problems
Ching-Li Chai
(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:6)(cid:3)(cid:5)(cid:10)

Frans Oort
American Mathematical Society
Complex Multiplication
and Lifting Problems

Mathematical
Surveys
and
Monographs
Volume 195
Complex Multiplication
and Lifting Problems
Ching-Li Chai
Brian Conrad
Frans Oort
American Mathematical Society
Providence, Rhode Island
EDITORIAL COMMITTEE
Ralph L. Cohen, Chair Benjamin Sudakov
Robert Guralnick Michael I. Weinstein
Michael A. Singer
2010 Mathematics Subject Classification. Primary 11G15, 14K02, 14L05, 14K15, 14D15.
For additional informationand updates on this book, visit
www.ams.org/bookpages/surv-195
Library of Congress Cataloging-in-Publication Data
Chai,Ching-Li,author.
Complexmultiplicationandliftingproblems/Ching-LiChai,BrianConrad,FransOort.
pagescm—(Mathematicalsurveysandmonographs;volume195)
Includesbibliographicalreferencesandindex.
ISBN978-1-4704-1014-8(alk.paper)
1.Multiplication,Complex. 2.Abelianvarieties. I.Conrad,Brian,1970–author. II.Oort,
Frans,1935–author. III.Title.
QA564.C44 2014
516.3(cid:2)53—dc23
2013036892
Copying and reprinting. Individual readers of this publication, and nonprofit libraries
actingforthem,arepermittedtomakefairuseofthematerial,suchastocopyachapterforuse
in teaching or research. Permission is granted to quote brief passages from this publication in
reviews,providedthecustomaryacknowledgmentofthesourceisgiven.
Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication
is permitted only under license from the American Mathematical Society. Requests for such
permissionshouldbeaddressedtotheAcquisitionsDepartment,AmericanMathematicalSociety,
201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by
[email protected].
(cid:2)c 2014bytheAmericanMathematicalSociety. Allrightsreserved.
TheAmericanMathematicalSocietyretainsallrights
exceptthosegrantedtotheUnitedStatesGovernment.
PrintedintheUnitedStatesofAmerica.
(cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines
establishedtoensurepermanenceanddurability.
VisittheAMShomepageathttp://www.ams.org/
10987654321 191817161514
This book is dedicated to
John Tate
for what he taught us, and for inspiring us

Contents
Preface ix
Introduction 1
References 8
Notation and terminology 9
Chapter 1. Algebraic theory of complex multiplication 13
1.1. Introduction 13
1.2. Simplicity, isotypicity, and endomorphism algebras 15
1.3. Complex multiplication 23
1.4. Dieudonn´e theory, p-divisible groups, and deformations 33
1.5. CM types 65
1.6. Abelian varieties over finite fields 70
1.7. A theorem of Grothendieck and a construction of Serre 76
1.8. CM lifting questions 86
Chapter 2. CM lifting over a discrete valuation ring 91
2.1. Introduction 91
2.2. Existence of CM lifting up to isogeny 102
2.3. CM lifting to a normal domain up to isogeny: counterexamples 109
2.4. Algebraic Hecke characters 117
2.5. Theory of complex multiplication 127
2.6. Local methods 130
Chapter 3. CM lifting of p-divisible groups 137
3.1. Motivation and background 137
3.2. Properties of a-numbers 143
3.3. Isogenies and duality 146
3.4. Some p-divisible groups with small a-number 156
3.5. Earlier non-liftability results and a new proof 161
3.6. A lower bound on the field of definition 164
3.7. Complex multiplication for p-divisible groups 166
3.8. An upper bound for a field of definition 182
3.9. Appendix: algebraic abelian p-adic representations of local fields 185
3.10. Appendix: questions and examples on extending isogenies 191
Chapter 4. CM lifting of abelian varieties up to isogeny 195
4.1. Introduction 195
4.2. Classification and Galois descent by Lie types 211
vii
viii CONTENTS
4.3. Tensor construction for p-divisible groups 224
4.4. Self-duality and CM lifting 228
4.5. Striped and supersingular Lie types 233
4.6. Complex conjugation and CM lifting 240
Appendix A. Some arithmetic results for abelian varieties 249
A.1. The p-part of Tate’s work 249
A.2. The Main Theorem of Complex Multiplication 257
A.3. A converse to the Main Theorem of Complex Multiplication 292
A.4. Existence of algebraic Hecke characters 296
Appendix B. CM lifting via p-adic Hodge theory 321
B.1. A generalization of the toy model 321
B.2. Construct CM lifting by p-adic Hodge theory 333
B.3. Dieudonn´e theories over a perfect field of characteristic p 343
B.4. p-adic Hodge theory and a formula for the closed fiber 359
Notes on Quotes 371
Glossary of Notations 373
Bibliography 379
Index 385
Preface
During the Workshop on Abelian Varieties in Amsterdam in May 2006, the three
authorsofthisbookformulatedtworefinedversionsofaproblemconcerninglifting
intocharacteristic0forabelianvarietiesover afinitefield. These problemsaddress
the phenomenon of CM lifting: the lift into characteristic 0 is required to be a CM
abelian variety (in the sense defined in 1.3.8.1). The precise formulations appear
at the end of Chapter 1 (see 1.8.5), as problems (I) and (IN).
Abelian surface counterexamples to (IN) were found at that time; see 2.3.1–2.3.3,
andsee4.1.2foramorethoroughanalysis. Tooursurprise, thesamecounterexam-
ples(typicalamongtoy models asdefinedin4.1.3)playacrucialroleinthegeneral
solution to problems (I) and (IN). This book is the story of our adventure guided
by CM lifting problems.
Ching-Li Chai thanks Hsiao-Ling for her love and support during all these years.
He also thanks Utrecht University for hospitality during many visits, including the
May2006SpringSchoolonAbelianVarietieswhichconcludedwiththeworkshopin
Amsterdam. SupportbyNSFgrantsDMS0400482,DMS0901163,andDMS120027
is gratefully acknowledged.
Brian Conrad thanks the many participants in the “CM seminar” at the Univer-
sity of Michigan for their enthusiasm on the topic of complex multiplication, as
well as Columbia University for its hospitality during a sabbatical visit, and grate-
fullyacknowledgessupport byNSFgrantsDMS0093542, DMS0917686, andDMS
1100784.
Frans Oort thanks the University of Pennsylvania for hospitality and stimulating
environment during several visits.
Weare alsogratefultoBurcuBaran, BasEdixhoven, OferGabber, JohandeJong,
Bill Messing, Ben Moonen, James Parson, Ren´e Schoof, and Jonathan Wise for
insightful and memorable discussions.
ix