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Mathematical Surveys and Monographs Volume 195 Complex Multiplication and Lifting Problems Ching-Li Chai "RIAN#ONRAD Frans Oort American Mathematical Society http://dx.doi.org/10.1090/surv/195 Complex Multiplication and Lifting Problems http://dx.doi.org/10.1090/surv/195 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 MichaelI.Weinstein MichaelA.Singer 2010 Mathematics Subject Classification. Primary 11G15, 14K02, 14L05, 14K15, 14D15. For additional information and updates on this book, visit www.ams.org/bookpages/surv-195 Library of Congress Cataloging-in-Publication Data Chai, Ching-Li, author. Complex multiplication and lifting problems / Ching-Li Chai, Brian Conrad, Frans Oort. pages cm — (Mathematical surveys and monographs ; volume 195) Includes bibliographical references and index. ISBN 978-1-4704-1014-8 (alk. paper) 1. Multiplication, Complex. 2. Abelian varieties. I. Conrad, Brian, 1970– author. II. Oort, Frans, 1935– author. III. Title. QA564.C44 2014 516.353—dc23 2013036892 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2014 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://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´etheory,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 authors of this book formulated two refined versions of a problem concerning lifting into characteristic 0 for abelian varieties over a finite field. These problems address 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, and see 4.1.2 for a more thorough analysis. To our surprise, the same counterexam- ples (typical among toy models as defined in 4.1.3) play a crucial role in the general 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 May 2006 Spring School on Abelian Varieties which concluded with the workshop in Amsterdam. Support by NSF grants DMS 0400482, DMS 0901163, and DMS120027 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- fully acknowledges support by NSF grants DMS 0093542, DMS 0917686, and DMS 1100784. Frans Oort thanks the University of Pennsylvania for hospitality and stimulating environment during several visits. We are also grateful to Burcu Baran, Bas Edixhoven, Ofer Gabber, Johan de Jong, Bill Messing, Ben Moonen, James Parson, Ren´e Schoof, and Jonathan Wise for insightful and memorable discussions. ix Notes on Quotes Sources of quotes at the beginning of each chapter. (0) The Zariski quote on page 1 is from the Preface of Oscar Zariski: Collected Papers, page xii of volumes I, II and page xiv of volumes III, IV. (1) The Hilbert quote on page 13 is from page 182 of Hilbert’s obituary by Olga Taussky in Nature,vol.152, pp. 182–183. Taussky, who heard Hilbert’s com- ment, recorded it as follows. It is interesting to recall that, in connexion with a lecture by Prof. R. Fueter at the 1932 Zurich Congress, Hilbert asserted that the theory of complex multiplication (of elliptic modular functions) which forms a powerful link between number theory and analysis, is not only the most beautiful part of mathematics but also of all science. (2) The Igusa quote on page 91 is from page 614 of Igusa’s article Arithmetic variety of moduli for genus two,AnnalsofMath.vol.72, 1960, pp. 612–649. (3) The Tate quote on page 137 is from page 158 of Tate’s original article [119] on p-divisible groups. (4) The Grothendieck quote on page 195 is in a letter from Grothendieck to Mum- ford dated September 4, 1968. The letter is included in David Mumford Se- lected Papers vol. II, Springer, 2010, pp. 735–737. The passage quoted is on page 736. (5) The Shimura quote on page 249 is from a footnote on p. 96 of Shimura’s book [113], the first page of Chapter 4. (6) The Mumford quote on page 321 is from the Preface of David Mumford Selected Papers, On the Classification of Varieties and Moduli Spaces, Springer, 2004. 371 Glossary of Notations Hom(A, B) K-rational homomorphisms for abelian varieties A, B over a field K, 1018 End(A)ringofK-rational endomorphisms for an abelian variety A over a field K, 10 0 Hom (A, B) Q ⊗Z Hom(A, B), 10 0 End (A) Q ⊗Z End(A), the endomorphism algebra of an abelian variety A, 10 At the dual of an abelian variety A, 10 A ∼ B abelian varieties A, B over K are isogenous over K, 10 AL the adele ring for a number field L, 10 A the adele ring for Q, 10 AL,f the ring of finite adeles for a number field L, 10 Af the ring of finite adeles for Q, 10 recL the reciprocity law map for a number field L, 10 recF the Artin map for F , F either a number field or a local field, 10 Br(K) the Brauer group of a field K, 18 Trd D/Z the reduced trace of a central simple algebra D over a field K, 18 inv ,inv the invariant of an element of the Brauer group over a local field K K Lv or Lv, 19 DK the Dieudonn´eringforaperfectfieldK ⊃ Fp,38 N D the Cartier dual of a commutative finite locally free group scheme, 34 NΦ the reflex norm of a CM type Φ, 94 ResL/Q(Gm) Weil’s restriction of scalars for L/Q applied to Gm/L, 95 hΦ the cocharacter of ResL/Q(Gm) attached to a CM type Φ for L, 95 A× → × alg the algebraic part of an algebraic Hecke character : F K , 118 Sm the Serre group with conductor m for a number field F , 119 Tm the connected Serre group with conductor m attached to a number field F , 119 ψ the abelian -adic representation attached to an algebraic Hecke character , 121 SF the connected Serre group attached to a number field F , 119, 299 SF the Serre group attached to a number field F , 120 Fp an algebraic closure of Fp, 137–248 Gλ asimplep-divisible group with slope λ over a perfect base field of characteristic p, 139 Mλ the contravariant Dieudonn´e module of Gλ, 139 X´et the maximal ´etale quotient of a p-divisible group X, 140 Xmult the largest p-divisible subgroup of multiplicative type for a p-divisible group X, 140 18A bold-faced page number indicates where a general notation is first defined or introduced.
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