Complex Multiplication and Lifting Problems

Mathematical Surveys and Monographs Volume 195

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. Dieudonnétheory,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. Dieudonné 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 counterexamples (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 gratefully 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.

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 DieudonnéringforaperfectfieldK ⊃ 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 Dieudonné module of Gλ, 139 Xét the maximal étale 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.

373 374 GLOSSARY OF NOTATIONS

X(0,1) the largest local-local part of a p-divisible group X, 140

Fr N the relative Frobenius homomorphism of a group scheme N in characteristic p, 142

Ver N the Verschiebung homomorphism of a group scheme N in characteristic p, 142 G[F, V ], X[F, V ] the unique maximal α-subgroup of a finite commutative group scheme G (respectively a p-divisible group X)overafieldK of characteristic p>0, 143 a(G)thea-number of a finite commutative group scheme G over a field K of characteristic p>0, 143 Xt the Serre dual of a p-divisible group X, 152 Tp(X•) the essential projective limit attached to a p-divisible group, 148 X a p-divisible group with a-number a(X) = 2, 156–166 Yb the αp-quotient of H given by b, 159–166 (F, Φ) p-adic CM type, decomposed as (F1, Φ1) ×···×(Fc, Φc), 170 E(Fi, Φi) reflex field of the p-adic CM type Φi, 170 E(F, Φ) the compositum of the reflex fields E(Fi, Φi), 170 × G Q F ResF/Qp ( m), the Weil restricted torus for a p-algebra with [F : Qp] < ∞, 171 σ × ∈ Q χ the geometric character of F attached to σ HomQp,ring(F, p), 171 × μΦ the geometric cocharacter of F attached to a p-adic CM type (F, Φ), 171

NμΦ reflex norm attached to a p-adic CM type (F, Φ), 171 ξσ the geometric character of F × attached to a p-adic embedding → Q σ : F p, 187 × νσ the geometric cocharacter of F attached to a p-adic embedding → Q σ : F p, 187 E(μ) the field of definition of a geometric p-adic cocharacter μ, 187 Nμ the reflex norm of a geometric p-adic cocharacter μ, 187 L aCMfield,195, 240–248 L+ the maximal totally real subfield of L, 206, 240–248 v a p-adic place of L+ above p, 206, 240–248 OL,v OL ⊗O O + , 206, 240–248 L+ Lv [Lie(Zv)] the Lie type of an OL,v -linear CM p-divisible group Zv, 206 + F F ΔL+ , Δ(L/L , q)) the set of all bad p-adic places of L0 with respect to q, 208 ΔL the set of all p-adic places of L above ΔL+ , 208 Dp,∞ the quaternion division algebra over Q ramified exactly at p and ∞, 221 E a finite extension field of Qp, 211–220 a commutative semisimple quadratic algebra over E+, 228–240 + E a finite extension field of Qp, 228–240 E0 the maximal unramified subextension of E/Qp, 211–220 + × + same as above when E is a field, equal to E0 E0 if E = E+ × E+, 228–240

κE residue field of E, 211–220 either the quadratic extension field of κ or κ × κ , 228–240 E+ E+ E+ f, fE dimF (κE ), 211–220, 228–240 p ∼ e, e(E/Qp) the ramification index of E, equals to e(E0/Qp)ifE = E0 × E0, 211–220, 228–240 ι the involution for E/E+, 228–240, 240–248 k an algebraically closed field of characteristic p>0, 211–248 GLOSSARY OF NOTATIONS 375

a perfect field of characteristic p>0, 343–370 O I HomZp-alg( E0 ,W(k)), 211–220, 228–240 κ a finite field contained in k, 211–248 O O ⊗ RF ( Lv ) the Grothendieck group of ( Lv ZpF )-modules of finite length, F ⊃ Fp, 206 O O ⊗ Rk( E ) the Grothendieck group of ( E Zpk)-modules of finite length, 211–228, 229–240 Rκ(OE ) κ-rational elements in Rk(OE ), 211–228, 229–240 + O + O O O Rk ( E ), Rκ ( E ) effective elements in Rk( E )andRκ( E ) respectively, 211–228, 229–240 Rk(OE ,i)thei-th component of Rk(OE ) for any i ∈ I, 212–228, 229–240 the degree map for Rk(OE ), 212–228, 229–240, i the i-th component of the degree map , 212–228, 229–240, [Lie(X0,α0)] the Lie type of an OE -linear CM p-divisible group (X0,α0), 212, 230 ξ reduction map, from p-adic CM types to Lie types, 212, 230 δunif the supersingular uniform Lie type, 234 F δstr,J the striped Lie type attached to a Gal(κE / p2 )-orbit J, 234 (Ytoy,βtoy)aCMp-divisible group with a striped Lie type, 236–239 w a p-adic place of L above a place v of L0, 240–248 Ow the ring of integers in the w-adic completion Lw of w, 240–248 Ow,0 the maximal absolutely unramified subfield of Lw, 240–248 κw the residue field of w, 240–248 O Iw HomZp-alg( Lw,0 ,W(k)), 241–248 O Iv HomZp-alg( Lv,0 ,W(k)), 241–248 O O ⊗ Rk( Lv ) the Grothendieck group of ( Lv Zp k)-modules of finite length, 240–248 O O ⊗ Rκ( Lv ) the Grothendieck group of ( Lv Zp k)-modules of finite length, O equal to the subgroup of all κ-rational elements in Rk( Lv ), 240–248 O O ⊗ Rk( L,p) the Grothendieck group of ( L,p Zpk)-modules of finite length, 240–248 w the degree map for Rk(Ow), 241–248 w,i the i-th component of w, 241–248 × F ResF/Q(Gm), the Weil restricted Q-torus attached to a number field F , 296 τ × ξ the character of F attached to an embedding τ ∈ Homring(F, Q), 297 ξf the character of F × attached to a Z-valued function on Homring(F, Q), 297 E(χ) the minimal subextension of F/L such that χ factors through NmF/E(χ), 299 ι a complex conjugation, 299–320 F wF the weight cocharacter of the connected Serre group S , 300 Fcm the subfield of F generated by all CM subfields of F , 301 μ(L) the group of all roots of 1 in a number field L, 305 Pic(OF ) the replete divisor class group of OF , 308 F (χ) the field of moduli of an admissible Q-homomorphism χ, 310 × ντ the geometric cocharacter of F attached to an embedding τ : F → Q, 312 E(μ) the field of definition of a geometric cocharacter μ of a Q-torus K×, 315 χ(μ) the homomorphism between Weil restricted tori attached to μ, 315 376 GLOSSARY OF NOTATIONS

× μΦ the cocharacter of L corresponding to a CM type Φ for a CM field L, 317 ℘ a finite place of a number field K above p, 313–318 v a finite place of a CM field L above p, 313–318 v the place of the maximal totally real subfield L+ in L below v, 313–318 Ψv the set of all p-adic embeddings of Lv, 313–318 Ψ℘ the set of all p-adic embeddings of K℘, 313–318 I identified with {1, 2, 3,...,4m}, 322–332, 333–342 J the subset {2, 4, 6,...,4m} of I, 322–342 J the subset {1, 3, 5,...,2m − 1, 2m +2, 2m +4,...,4m} of I, 322–342 O F W ( p4m ), 321–342 (Y0,β0)anO-linear CM p-divisible group over Fp with Lie type J, 322–342 m ρ0 an O-linear isogeny from X0 to Y0 of degree p , 322–342 (X0,γ0)anO-linear CM p-divisible group over Fp with Lie type J , 322–342 N,D∗(Y0)theO-liner Cartier module of (Y0,β0), 322–342 M,D∗(X0)theO-liner Cartier module of (X0,γ0), 324–342 m G0 ker(ρ0), a finite group scheme over Fp of order p 324–342 4m−1 HJ−{4m} a subgroup scheme of X0 isomorphic to αp with Lie type J −{4m}, 327–342 HS the subgroup scheme of HJ−{4m} with Lie type S ⊂ J −{4m}, 328–342 K0 the fraction field of W (Fp), 324–342 the fraction field W (k)[1/p]ofW (k), 360–370 4m K the tame totally ramified extension field of K0 of degree p − 1, 324–342 a totally ramified finite extension field of K0 = W (k)[1/p], 360–370 K1/K0 the subextension of K/K0 such that [K : K1]=p − 1 X X O F , Λ0 the -linear lifting of X0 over W ( p), 324–342, 336–342 X F M0,φM,0 ,φM,0 the Kisin module for [p]overW ( p), 336–342 X O (M,φM ,φM ) the Kisin module for the base change of [p]to K , 336–342 ∗ E∗( · ), E ( · ) the covariant and contravariant crystal in crystalline Dieudonné theory, 354–359 G a finite commutative group scheme over the perfect base field k, 343–342 X a p-divisible group over k, 343–342 C∗(X) the Cartier module of a smooth formal group X over k, 352–359 M∗(G), M∗(X) covariant Dieudonné module for G and X, 354–359 M∗(G), M∗(X) contravariant Dieudonné module for G and X, 354–359 G • the co-Lie complex of a finite group scheme G (over a perfect field k), 356–359 −1 G nG H ( ), 356–359 1 G∨ νG H ( ), 356–359 E(u) the Eisenstein polynomial over K0 for an local parameter π of K, 360–370 S W (k)[[u]], 334–342, 360–370 S0 W (Fp)[[t]], 334–342, n S the p-adic completion of W [u, E(u) /n!]n∈N, 360 (Mod/S)c the category of connected Kisin modules for finite flat group schemes over OK , 334, 360 c (Mod/S0) the category of connected Kisin modules for finite flat group schemes over W (Fp), 334 GLOSSARY OF NOTATIONS 377

(p-Gr/k)c the category of connected commutative finite group schemes of p-power order over k, 361 c (Mod/Dk) Dieudonné modules for commutative finite groups schemes over k of local type, 361 f (p-div/OK ) the category of formal p-divisible groups over OK , 361 BTφ,f the category of Kisin modules for formal p-divisible groups over /S0 W (Fp), 334 φ,f BT/S the category of Kisin modules for formal p-divisible groups over OK , 362–370 (Win/S) the category of S-windows, 362–370 (Dsp/R) the category of displays over a p-adic ring R, 362–370

Bibliography

[1] A. A. Albert, On the construction of Riemann matrices, I, II. Ann. Math. 35 (1934), 1–28; 36 (1935), 376–394. [2]A.A.Albert,A solution of the principal problem in the theory of Riemann matrices. Ann. Math. 35 (1934), 500–515. [3]A.A.Albert,Involutorial simple algebras and real Riemann matrices. Ann. Math. 36 (1935), 886–964. [4] E. Artin & J. Tate, Class field theory. AMS Chelsea Publishing, 2009. [5] M. Artin, Algebraization of formal moduli: I.InGlobal Analysis (Papers in honor of K. Kodaira), Univ. Tokyo Press, 1969. [6] P. Berthelot, L. Breen, & W. Messing, ThéoriedeDieudonné cristalline II. Lect. Notes Math. 930, Springer-Verlag, 1982. [7] P. Berthelot, A. Grothendieck, & L. Illusie, Théorie des intersections et théorème de Riemann-Roch. Lect. Notes Math. 225, Springer-Verlag, 1971. [8] P. Berthelot & A. Ogus, F -isocrystals and de Rham cohomology I.Inv.Math.72 (1983), 159–199. [9]S.Bosch,U.Güntzer, & R. Remmert, Non-archimedean analysis. Grundlehren Math. Wiss. 261, Springer-Verlag, 1983. [10] S. Bosch, W. Lütkebohmert, & M. Raynaud, Néron models. Ergebnisse der Mathematik 21, Springer-Verlag, 1990. [11] N. Bourbaki, Algèbre, Chapitre 8, Modules et anneaux semi-simples. Actualités Scientifiques et Industrielles 1261, Hermann, 1958. [12] L. Breen, Rapport sur la théorie de Dieudonné.Astérisque 63 (1979), 39–66. [13] C. Breuil, Groupes p-divisibles, groupes finis et modules filtrés. Ann. Math. 152 (2000), 489–549. [14] P. Cartier, Modules associés à un groupe formel commutatif. Courbes typiques.C.R.Acad. Sci. 265 (1967), 129–132. [15] J. Cassels, Global fields.InAlgebraic number theory (J.Cassels&A.Fröhlich, ed.), Aca- demic Press, 1967. [16] C.-L. Chai & P. Norman, Bad reduction of the Siegel moduli scheme of genus two with Γ0(p)-level structure.Amer.J.Math.112 (1990), 1003–1071. [17]C.-L.Chai&F.Oort,Hypersymmetric abelian varieties. Quat. J. Pure Appl. Math. 2 (Coates Special Issue) (2006), 1–27. [18] C.-L. Chai & F. Oort, Moduli of abelian varieties and p-divisible groups. In Arithmetic algebraic geometry,(H.Darmon,D.A.Ellwood,B.Hassett&Y.Tschinkel,ed.),Clay Math. Proc. 8, 2009, 441–536. [19] C. Chevalley, Deux théorèmes d’arithmétique. J. Math. Soc. Japan 3 (1951), 36–44. [20] P. Colmez & J-M. Fontaine, Construction des représentations p-adiques semi-stables.Inv. Math. 140 (2000), 1–43. [21] B. Conrad, Finite-order automorphisms of a certain torus. Michigan Math. J. 52 (2004), 423–434. [22] B. Conrad, Gross-Zagier revisited. In Heegner points and Rankin L-series, MSRI Publ. 49, Cambridge Univ. Press, 2004. [23] B. Conrad, Chow’s K/k-image and K/k-trace, and the Lang-Néron theorem. Enseign. Math. (2) 52, no. 1–2 (2006), 37–108.

379 380 BIBLIOGRAPHY

[24] B. Conrad, Deligne’s notes on Nagata compactifications. J. Ramanujan Math. Soc. 22 (2007), 205–257. [25] B. Conrad, O. Gabber & G. Prasad, Pseudo-reductive groups. New Math. Monogr. 17, Cambridge Univ. Press, 2010. [26] B. Conrad, Lifting global representations with local properties. Preprint, 2012. [27] P. Deligne, Application de la formule des traces aux sommes trigonométriques.InCohomolo- 1 gie étales,Séminaire de Géométrie Algébrique du Bois-Marie SGA4 2 , Lect. Notes Math. 569, Springer-Verlag 1977, 168–232. [28] P. Deligne, Motifs et groupes de Taniyama.InHodge cycles, motives, and Shimura varieties, Lect. Notes Math. 900, Springer-Verlag, 1982, 261–279. [29] M. Demazure, Lectures on p-divisible Groups. Lect. Notes Math. 302, Springer-Verlag, 1972. [30] M. Demazure & A. Grothendieck, Schémas en groupes I, II, III, Lect. Notes Math. 151, 152, 153, Springer-Verlag, 1970. [31] M. Demazure & A. Grothendieck, Schémas en groupes, I, Documents mathématiques 7, Soc. Math. France, 2011. [32] M. Demazure & P. Gabriel, Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs. Masson & Cie éditeur, North-Holland Publishing Company, 1970. [33] M. Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Semin. Univ. Hambg. 14 (1941), 197–272. [34] J. Dieudonné & A. Grothendieck, El´´ ements de Géométrie Algébrique. III. Publ. Math. IHES 11, 1961. [35] T. Ekedahl, An effective version of Hilbert’s irreducibility theorem. Séminaire de Théorie des Nombres, Paris 1988–1989, Progr. Math. 91, Birkhäuser, 1990, 241–249. [36] G. Faltings, Finiteness theorems for abelian varieties over number fields.InArithmetic Geometry (G. Cornell & J. Silverman, ed.), Springer-Verlag, 1986, 9–27. [37] G. Faltings, Erratum Endlichkeitssätze für abelsche Varietätenuber ¨ Zahlkörpern.Inv.Math 75 (1984), 381. [38] G. Faltings & C-L. Chai, Degeneration of abelian varieties. Ergeb. Math. Grenzgeb. 22, Springer-Verlag, 1990. [39] B. Fantechi, L. Göttsche, L. Illusie, S. Kleiman, N. Nitsure, & A. Vistoli, Fundamental algebraic geometry. Math. Surveys Monogr. 123, Amer. Math. Soc., 2005. [40] D. Ferrand, Conducteur, descent, et pincement. Bull. Soc. Math. France 131 (2003), 553– 585. [41] J-M. Fontaine, Groupes p-divisible sur les corps locaux.Astérisque 47–48 (1977). [42] J-M. Fontaine, Module galoisiens, modules filtrés et anneaux de Barsotti-Tate.Astérisque 65 (1979), 3–80. [43] J-M. Fontaine, Sur certains types de représentations p-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate. Ann. Math. 115, 1982, 529–577. [44] J-M. Fontaine, Représentations p-adiques semi-stables. Astérisque 223 (1994), 113–184. [45] G. van der Geer & B. Moonen, Abelian varieties. In preparation. [46] L. Gerritzen, On multiplication algebras of Riemann matrices. Math. Ann. 194 (1971), 109–122. [47] W.-D. Geyer, Galois groups of intersections of local fields. Israel J. Math. 30 (1978), 382– 396. [48] A. Grothendieck, Groupes de Barsotti-Tate et cristaux de Dieudonné.Sém. Math. Supé. 45, Presses Univ. Montreal, 1970. [49] M. Hazewinkel, Formal groups and applications. Academic Press, 1978. [50] T. Honda, Isogeny classes of abelian varieties over finite fields. J. Math. Soc. Japan 20 (1968), 83–95. [51] L. Illusie, Déformation des groupes de Barsotti-Tate.Astérisque 127 (1985), 151–198. [52] A. Ivić, The Riemann zeta-function: the theory of the Riemann zeta-function with applications. John Wiley & Sons, 1985. [53] N. Jacobson, Basic algebra II. W.H. Freeman & Co., 1989. [54] A. J. de Jong, Crystalline Dieudonné module theory via formal and rigid geometry. Publ. Math. IHES 82, 1995, 5–96. [55] A. J. de Jong. Homomorphisms of Barsotti-Tate groups and crystals in positive characteristic.Inv.Math.134 (1998), 301–333. BIBLIOGRAPHY 381

[56] A. J. de Jong & F. Oort, Purity of the stratification by Newton polygons. J. Amer. Math. Soc. 13 (2000), 209–241. [57] N. Katz, Serre–Tate local moduli. Lect. Notes Math. 868, Springer-Verlag 1981, 138–202. [58] M. Kisin, Crystalline representations and F -crystals.InAlgebraic geometry and number theory, Progr. Math., 253, Birkhäuser, 2006, 459–496. [59] M. Kisin, Modularity of 2-adic Barsotti-Tate representations.Inv.Math.178 (2009), 587– 634. [60] D. Knutson, Algebraic spaces, Lect. Notes Math. 203, Springer-Verlag, 1971. [61] A. Kresch, Hodge-theoretic obstruction to the existence of quaternion algebras. Bull. London Math. Soc. 35 (2003), 109–116. [62] J. Kollár, Quotients by finite equivalence relations. Preprint, 2009. [63] J. Lagarias, H. Montgomery & A. Odlyzko, A bound for the least prime ideal in the Cheb- otarev density theorem.Inv.Math.54 (1979), 271–296. [64] J. C. Lagarias & A. M. Odlyzko, Effective versions of the Chebotarev density theorem,in Algebraic number fields (A. Fröhlich, ed.), Academic Press, 1977, 409–464. [65] S. Lang, Complex multiplication. Grundlehren Math. Wiss. 255, Springer-Verlag, 1983. [66] S. Lang, Elliptic functions (2nd ed.). Graduate Texts in Math. 112, Springer-Verlag, 1987. [67] S. Lang, Algebra (revised 3rd ed.). Graduate Texts in Math. 211, Springer-Verlag, 2002. [68] E. Lau, Displays and formal p-divisible groups,Inv.Math.171 (2008), 617–628. [69] M. Lazard, Commutative formal groups. Lect. Notes Math. 443, Springer-Verlag, 1975. [70] H. W. Lenstra Jr. & F. Oort, Simple abelian varieties having a prescribed formal isogeny type. J. Pure Appl. Algebra 4 (1974), 47–53. [71] Y. Manin, The theory of commutative formal groups over fields of finite characteristic.Usp. Math. 18 (1963), 3–90; Russ. Math. Surveys 18 (1963), 1–80. [72] H Matsumura, Commutative algebra. Benjamin, 1969; 2nd ed., 1980. [73] H. Matsumura, Commutative ring theory. Cambridge Stud. Adv. Math. 8, Cambridge Univ. Press, 1989. [74] B. Mazur & W. Messing, Universal extensions and one-dimensional crystalline cohomology. Lect. Notes Math. 370, Springer-Verlag, 1974. [75] W. Messing, The Crystals associated to Barsotti-Tate Groups: with applications to abelian schemes. Lect. Notes Math. 264, Springer-Verlag, 1972. [76] J. Milne, Abelian varieties.InArithmetic geometry (G. Cornell & J. Silverman, ed.), Springer-Verlag, 1986. [77] J. Milne, Canonical models of (mixed) Shimura varieties and automorphic vector bundles. In Automorphic forms, Shimura varieties, and L-functions, Vol. I, Perspect. Math. 10, Academic Press, 1990, 283–414. [78] J. Milne & K.-Y. Shih, Langlands’ construction of the Taniyama group.InHodge cycles, motives, and Shimura varieties. Lect. Notes Math. 900, Springer-Verlag, 1982, 229–260. [79] J. Milne & W. Waterhouse, Abelian varieties over finite fields.In1969 number theory institute, Proc. Sympos. Pure Math. 20, SUNY Stony Brook, NY, 1969), Amer. Math. Soc., 1971, 53–64. [80] L. Moret-Bailly, Pinceaux de variétés abéliennes.Astérisque 129, 1985. [81] L. Moret-Bailly, Construction de revêtements de courbes pointées.J.Algebra240 (2001), 505–534. [82] D. Mumford, Abelian varieties. TIFR Stud. Math. 5, 1st ed. 1970; 2nd ed. TIFR and Oxford Univ. Press 1974; corrected reprint TIFR and Hindustan Book Agency 2008. [83] D. Mumford, Geometric invariant theory. Ergeb. Math. Grensgeb. 34, 1st ed. Springer- Verlag, 1965; 3rd enlarged ed. by D. Mumford, J. Fogarthy & F. Kirwan, Springer-Verlag, 1994. [84] J. Neukirch, Algebraic number theory. Grundlehren Math. Wiss. 322, Springer, 1999. [85] P. Norman & F. Oort, Moduli of abelian varieties. Ann. Math., 112 (1980), 413–439. [86] T. Oda The first de Rham cohomology group and Dieudonnémodules, Ann. Sci. E.N.S. (4) 2 (1969), 63–135. [87] F. Oort, Commutative group schemes. Lect. Notes in Math. 15, Springer-Verlag, 1966. [88] F. Oort, Finite group schemes, local moduli for abelian varieties, and lifting problems. Compos. Math. 23 (1971), 265–296. [89] F. Oort, The isogeny class of a CM-type abelian variety is defined over a finite extension of the prime field. J. Pure Appl. Algebra 3 (1973), 399–408. 382 BIBLIOGRAPHY

[90] F. Oort, Isogenies of formal groups. Indag. Math., 37 (1975) 391–400. [91] F. Oort, Lifting algebraic curves, abelian varieties and their endomorphisms to characteristic zero. In Algebraic geometry, Bowdoin 1985 (S. J. Bloch, ed.), Proc. Sympos. Pure Math. 46, Part 2, Amer. Math. Soc. 1987, 165–195. [92] F. Oort, Endomorphism algebras of abelian varieties.InAlgebraic geometry and commutative algebra: in honor of M. Nagata, Vol. II, (H. Hijikata et al., ed.), Kinokuniya, 1988, 469–502. [93] F. Oort, CM-liftings of abelian varieties.J.Alg.Geom.1 (1992), 131–146. [94] F. Oort, Newton polygon strata in the moduli space of abelian varieties. In Moduli of abelian varieties (C. Faber, G. van der Geer & F. Oort, ed.), Progr. Math. 195, Birkhäuser, 2001, 417–440. [95] F. Oort, Abelian varieties over finite fields. Summer School on ”Varieties over finite fields”, Göttingen, 25-VI – 6-VII-2007. Higher-dimensional geometry over finite fields,Proc.NATO Adv. Stud. Inst. 2007 (Dmitry Kaledin & Yuri Tschinkel, ed.), IOS Press, 2008, 123–188. [96] J. Oesterlé, Versions effectives du théorème de Chetotarev sous l’hypothèsedeRiemann généralisée,Astérisque 61 (1979), 165–167. [97] R. Pink, The Mumford–Tate conjecture for Drinfeld modules. Publ. RIMS, Kyoto Univ. 33 (1997), 393–425. [98] M. Raynaud, Passage au quotient par une relation d’équivalence plate. 1967 Proc. Conf. Local Fields (Driebergen), Springer-Verlag, 78–85. [99] M. Raynaud, Schémas en groupes de type (p,...,p). Bull. Soc. math. France 102 (1974), 241–280. [100] D.S. Rim, Formal deformation theory In Groupes de monodromie en géométrie algébrique. I., Lect. Notes Math. 288, Springer-Verlag, 1972, 32–132. [101] D. Rydh, Compactification of tame Deligne-Mumford stacks. Preprint, 2012. [102] M. Schlessinger, Functors of Artin rings.Trans.Amer.Math.Soc.103 (1968), 208–222. [103] J-P. Serre, Local class field theory.InAlgebraic number theory (J.Cassels&A.Fröhlich, ed.). Academic Press, 1967, 129–161. [104] J-P. Serre, Complex multiplication.InAlgebraic number theory (J. Cassels & A. Fröhlich, ed.), Academic Press, 1967, 292–296. [105] J-P. Serre, Abelian -adic representations and elliptic curves. W. A. Benjamin, 1968. [106] J-P. Serre, Linear representations of finite groups. Grad. Texts in Math. 42, Springer-Verlag, 1977. [107] J-P. Serre, Local fields. Grad. Texts in Math. 67, Springer-Verlag, 1979. [108] J-P. Serre, Groupes algébriques associés aux modules de Hodge-Tate.Astérisque 65 (1979) 155–188. [109] J-P. Serre & J. Tate, Good reduction of abelian varieties. Ann. Math. 88 (1965), 492–517. [110] S. Shatz, Group schemes, formal groups, and p-divisible groups.InArithmetic geometry (G. Cornell & J. Silverman, ed.), Springer-Verlag, 1986. [111] G. Shimura & Y. Taniyama, Complex multiplication of abelian varieties and its applications to number theory. Publ. Math. Soc. Japan 6, 1961. [112] G. Shimura, On analytic families of polarized abelian varieties and automorphic functions. Ann. Math. 78 (1963), 149–193. [113] G. Shimura, Introduction to the arithmetic theory of automorphic functions. Princeton Univ. Press, 1971; reprinted 1994. [114] G. Shimura, On the zeta-function of an abelian variety with complex multiplication. Ann. Math. 94 (1971), 504–533. [115] G. Shimura, Abelian varieties with complex multiplication and modular functions. Princeton Univ. Press, 1998. [116] L. Szpiro, Degrés, intersections, hauteurs. In Astérisque 127 (1985), 11–28. [117] J. Tate, Algebraic cycles and poles of zeta functions.InArithmetical algebraic geometry (Proc. Conf. Purdue Univ., 1963). Harper & Row, 1965, 93–110. [118] J. Tate, Endomorphisms of abelian varieties over finite fields.Inv.Math.2 (1966), 134–144. [119] J. Tate, p-divisible groups.InProc. conference on local fields (T. Springer, ed.), Springer- Verlag, 1967, 158–183. [120] J. Tate, Global class field theory.InAlgebraic number theory (J. Cassels & A. Fröhlich, ed.), Academic Press, 1967, 163–203. BIBLIOGRAPHY 383

[121] J. Tate, Class d’isogenie des variétés abéliennes sur un corps fini (d’après T. Honda). Séminaire Bourbaki, 1968/69, no. 352. Lect. Notes Math. 179, Springer-Verlag, 1971, 95– 110. [122] E. Titchmarsh & D. Heath-Brown, The theory of the Riemann zeta-function,2nded., Clarendon Press, Oxford, 1986. [123] J. Tate & F. Oort, Group schemes of prime order. Ann. Sc. Ec.´ Norm. Supé 3 (1970), 1–21. [124] H. Umemura, Formal moduli for p-divisible groups. Nagoya Math. J. 42 (1971), 1–7. [125] A. Vasiu & T. Zink, Breuil’s classification of p-divisible groups over regular local rings of arbitrary dimension.InProceeding of Algebraic and Arithmetic Structures of Moduli Spaces, Hokkaido University, Sapporo, Japan, 2007, Adv. Stud. Pure Math. 58 (2010), 461–479. [126] W. Waterhouse, Affine group schemes. Grad. Texts in Math. 66 Springer-Verlag, 1979. [127] A. Weil, On a certain type of character of the idèle-class group of an algebraic number field. In Proc. intern. symp. on algebraic number theory, Tokyo-Nikko, 1955, 1–7. [128] A. Weil, Basic number theory. Grundlehren Math. 144, Springer-Verlag, 1967; 2nd ed. 1973. [129] A. Wiles, Modular elliptic curves and Fermat’s Last Theorem. Annals of Math. 141 (1995), 443–551. [130] C.-F. Yu, Lifting abelian varieties with additional structure.Math.Z.242 (2002), 427–441. [131] C.-F. Yu, On the supersingular locus in Hilbert-Blumenthal 4-folds.J.Alg.Geom.12 (2003), 653–698. [132] C.-F. Yu, On reduction of Hilbert-Blumenthal varieties. Annales de l’Institut Fourier (Greno- ble) 53 (2003), 2105–2154. [133] C.-F. Yu, The isomorphism classes of abelian varieties of CM-type. J. Pure Appl. Algebra 187 (2004), 305–319. [134] Y. Zarhin, Abelian varieties in characteristic p.Mat.Zametki19 (1976), no. 3, 393–400. [135] T. Zink, Isogenieklassen von Punkten von Shimura-Mannigfaltigkeiten mit Werten in einem endlichen Körper.Math.Nachr.112 (1983), 103–124. [136] T. Zink, Cartiertheorie kommutative formaler Gruppen. Teubner, 1984. [137] T. Zink, Windows for displays of p-divisible groups.InModuli of Abelian Varieties, Progr. Math. 195, Birkhäuser, 2001, 491–518. [138] T. Zink, On the slope filtration. Duke Math. J. 109 (2001), 79–95. [139] T. Zink, The display of a formal p-divisible group. Astérisque 278 (2002), 127–278.

Index

a-number, 143 Barsotti-Tate group, see also p-divisible abelian scheme, 14, 46, 47 group dual, 36 truncated, see also truncated isogeny, 37 Barsotti-Tate group polarization, 37 truncated at level 1, 144 abelian variety Brauer group, 18 absolutely simple, 15 of a global field, 20 and primary extension of fields, 15 order of an element of, 20 isotypic, 22, 23, 31 of a local field, 19 lifting to characteristic 0, 86 Poincaré reducibility, 16 Cartier dual, 34, 326 simple over a field, 15 Cartier theory, 351–353 abelian variety with sufficiently many Cartier module, 322, 352 complex multiplication, 24, 25, 77 V -flat, 352 Grothendieck’s theorem in char. p>0, V -reduced, 352 79 Cartier ring, 351 refinement, 86 and the smooth formal group W, 351 admissible algebraic homomorphism, 301 central simple algebra, 16 and Weil numbers, 315 Wedderburn’s theorem, 17 construction of, 316 CM abelian variety, 23, 32 field of moduli, 310 descent to a number field, 67 primitive, 299 existence over field of moduli, 311 weight of, 303 isotypic, 31 Albert algebra, 28 L-function of, 272 classification of, 29 potential good reduction, 78 algebraic Hecke character, 118 CM algebra, 26 algebraic part of, 119 and complex multiplication, 127, 128 CM field, 26 compatible system of -adic characters CM formal abelian scheme attached to, 121 criterion for algebraicity, 106 construction by surgery procedure, 124 CM lifting equivalent definition, 120 of a p-divisible group, 169 existence over field of moduli, 311 CM lifting questions, 86 existence theorem, 305 after finite residue field extension (R), 88 formula for, see Shimura Taniyama CM lifting (CML), 87 formula, for an algebraic Hecke sufficient condition, 102 character Lie type of the closed fibers of CM weight of, 300 abelian schemes isogenous to a CM lift α-group scheme, 142 (LTI), 210 Dieudonné module of, 358–359 strong CM lifting (sCML), 88, 210 to normal domains up to isogeny (IN), 88 bad local method, 178 p-adic place, 208, 208, 209, 210, 246 necessary and sufficient condition, 100, pair, 209, 235, 236, 238 128–136

385 386 INDEX

to normal domains up to isogeny after of a cocharacter, 315 finite residual field extension (RIN) field of definition as obstruction to CML, Honda-Tate theorem on (RIN), 75 181–185 to normal domains up to isogeny after field of moduli finite residue field extension (RIN), 88 of an admissible algebraic up to isogeny (I), 88 homomorphism, 310 existence, 195 CM order, 65, 68, 80, 89 good CM p-divisible group, 167 p-adic place, 208, 208, 209, 247 existence, 172, 178 pair, 209, 235, 238 Galois representation of, 176 Grothendieck group, 207, 208, 211, 212, uniqueness up to isogeny, 173 227, 241, 324 CM structure Hodge-Tate decomposition, 168, 185, 189 dual, 67, see also CMtype of the dual of Honda-Tate theorem, 71 a CM abelian variety and CM lifting, 3 for abelian varieties, 2, 32 CM type Kisin modules, 360–362 for a CM algebra, 66 for a CM field, 66 level structure of a CM abelian variety, 66 finite étale, 13, 57 determines the isogeny class, 67 Lie type, 212, 229 of the dual of a CM abelian variety, 67 and Galois descent, 207, 211, 217 p-adic, see p-adic CM type rational over a field, 215, 236, 237, 242 valued in a field, 66 self-dual, 230, 235, 236, 239, 242, 242 co-Lie complex, 356 striped, 208, 234, 234, 236, 238, 239 counterexample local deformation space to (CML) and (R), 183–184 for a CM structure, see deformation ring, to (IN), 101, 110 for a CM structure with two slopes, 111–114 of a p-divisible group, 169 to CM lifting with action by the full ring of integers in the CM field, 198 main theorem of complex multiplication, crystalline Dieudonné theory, 354–359 127, 257–292 algebraic form, 266 deformation ring analytic form, 288 for a CM structure, 202 converse to, 128, 292–296 of a CM structure, 92 multiplicities of a Lie type, 233 of a p-divisible groups, 169 deformation ring argument, 91, 106, 169, Newton polygon, 114 233 and (IN), 114–116 deformation theory for abelian schemes, 196 order lattice, 286 for p-divisible groups, 196 p-adic abelian crystalline representation, Dieudonné theory, 38, 137–138, 347–351 174, 189 basic differential invariants, 356–359 algebraic on the inertia subgroup, 190 comparison of Dieudonné theories, 356 p-adic CM type, 168 Dieudonné ring, 38 compatible with a given CM structure, Dieudonné-Manin classification, 139 170 dimension of a Lie type, 212 of a CM p-divisible group, 168, 206 display, 354, 362–368 self-dual, 206, 208, 209, 230, 232, 235, duality theorem 238, 239, 242 for abelian varieties, 37 p-adic Hodge theory, 333, 359 for p-divisible groups, 152 p-divisible group, 39 effective elements a-number of, 143 and deformation of abelian varieties, 59 in Rk(OF )andRκ(OF ), 211, 212, 229 extended Lubin-Tate type, 156 connected, 40, 41 Dieudonné-Manin classification up to field of definition isogeny, 139 of a p-adic cocharacter, 187 étale, 40, 140 INDEX 387

height of, 39 and residual reflex condition, 100 isoclinic, 139 for a CM abelian variety, 98 isogeny of, 148–152 for an algebraic Hecke character, 313 local-local, 140 short exact sequence, 34 local-local part, 140 singular j-invariant, 1 of multiplicative type, 140 Skolem–Noether theorem, 18 ordinary, 140 slopes quasi-polarization of, 60 of a Lie type, 214, 217, 233, 233 Serre dual of, 39 of a p-divisible group, 139, 139, 207, 215, slopes of, see slopes, of a p-divisible 237, 322 group of an abelian variety, 98 with sufficiently many complex striped Lie type, see Lie type, striped multiplication, 169 supersingular j-values, 1 polarization, 37 Tate’s theorem reciprocity laws and (RIN), 75 sign conventions of, 10 CM structure for abelian varieties with reduction map from p-adic CM types to Lie sufficiently many complex types, 212, 230 multiplication, 26 reflex field extending homomorphisms between of a CM type, 93 p-divisible groups, 58 of a p-adic cocharacter, 187 Hodge-Tate decomposition for p-divisible of a p-adic CM type, 170, 172, 182, 183 groups, 58 reflex norm homomorphisms between abelian of a CM type, 94,95 varieties over a finite field, 70 of a p-adic cocharacter, 187, 189 toy model, 110 of a p-adic CM type, 171 CM abelian surface, 196, 203 of a p-adic cocharacter, 188 classification of, 220 replete divisor, 308, 309 CM p-divisible group, 204, 209, 236–239 degree of, 308 higher dimensional, 321, 333 principle, 308 truncated Barsotti-Tate group residual reflex condition, 100 BT1 group scheme, 144 necessary and sufficient condition for BTn group, 143 (IN), 100, 128–136 uniform Lie type, 233, 235, 242, 242, 244 self-dual Lie type, 230, 232, 235, 236, 239, 242, 242 Weil q-integer, 70 self-dual p-adic CM type, 206, 208, 209, Weil q-number 230, 232, 235, 238, 239, 242 of weight w, 70, 304, 313, 315 self-dual up to isogeny, 243, 244 existence, 318 self-duality slopes of, 315 and algebraization, 243 Weil restriction of scalars, 22 and reduction modulo p, 231 windows, 362 condition, 199, 206, 208, 209, 228, 230, Witt covectors, 345, 345–347 243, 244, 248 Serre group attached to a number field, 120 neutral component, 119, 299 character group of, 299 weight cocharacter of, 303 Serre tensor product construction and Lie types, 227 for abelian varieties, 80, 83, 224 for p-divisible groups, 238 for p-divisible groups, 224, 236 Serre-Tate canonical lifting, 60 Serre-Tate theorem on deformation of abelian varieties, 59, 206 Shimura–Taniyama formula, 98 and (IN), 100

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For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/survseries/.

Abelian varieties with complex multiplication lie at the origins of class field theory, and they play a central role in the contemporary theory of Shimura varieties. They are special in characteristic 0 and ubiquitous over finite fields. This book explores the relationship between such abelian varieties over finite fields and over arithmetically interesting fields of characteristic 0 via the study of several natural CM lifting problems which had previously been solved only in special cases. In addition to giving complete solutions to such questions, the authors provide numerous examples to illustrate the general theory and present a detailed treatment of many fundamental results and concepts in the arithmetic of abelian varieties, such as the Main Theorem of Complex Multiplication and its generalizations, the finer aspects of Tate’s work on abelian varieties over finite fields, and deformation theory. This book provides an ideal illustration of how modern techniques in arithmetic geometry (such as descent theory, crystalline methods, and group schemes) can be fruitfully combined with class field theory to answer concrete questions about abelian varieties. It will be a useful reference for researchers and advanced graduate students at the inter- face of number theory and algebraic geometry.

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