Formal groups and applications

602 Pages·2012·23.504 MB·English

by  Hazewinkel, Michiel

F G
ORMAL ROUPS
A
AND PPLICATIONS

M H
ICHIEL AZEWINKEL
AMS CHELSEA PUBLISHING
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F G
ORMAL ROUPS
A
AND PPLICATIONS

http://dx.doi.org/10.1090/chel/375
F G
ORMAL ROUPS
A
AND PPLICATIONS

M H
ICHIEL AZEWINKEL
AMS CHELSEA PUBLISHING
American Mathematical Society • Providence, Rhode Island
ERMIACAN MΑΓΕΩΜΕAΤTΡHΗEΤΟMΣ AΜTΗIΕΙΣΙΤΩCALSOYCTIE
FOUNDED 1888
2010 Mathematics Subject Classification. Primary 14L05; Secondary 16Txx, 05Exx,
11Fxx, 11Sxx, 12Fxx, 13F35, 55N22.
For additional informationand updates on this book, visit
www.ams.org/bookpages/chel-375
Library of Congress Cataloging-in-Publication Data
Hazewinkel,Michiel.
Formalgroupsandapplications/MichielHazewinkel.
pagescm
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-5349-8(alk.paper)
1.Formalgroups. I.Title.
QA177.4.H39 2012
512(cid:2).2—dc23
2012028054
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To Marijke, Maarten, and Annette

PrefacetotheCorrectedPrinting xi
Preface xiii
LeitfadenandIndicien xv
Introduction xix
Vlll CONTENTS
Chapter IV Homomorphisms, Endomorphisms, and the
Classification of Formal Groups by Power Series Methods
18 Definitions and Preliminary Elementary Results. Survey of Chapter IV 147
19 Universal Isomorphisms 162
20 Existence and Nonexistence of Homomorphisms and Isomorphisms 173
21 Formal A-Modules 199
22 Lifting and Reducing Formal Group Laws. Formal Moduli 230
23 Rings of Endomorphisms of Formal Group Laws 245
24 Classification of One Dimensional Formal Group Laws over Finite
Fields 253
25 Rings of Curves and Artin-Hasse-Like Exponential Mappings 269
E.3 Bibliographical and Other Notes 310
Chapter V Cartier-Dieudonne Modules
26 Basic Definitions and Reminders. Survey of the Results of Chapter V 312
27 Cartier-Dieudonne Modules for Formal Group Laws 320
28 On the Classification of Commutative Formal Group Laws over an
Algebraically Closed Field of Characteristic p > 0 354
29 Cartier-Dieudonne Theory for Formal A-Modules 374
30 "Le Tapis de Cartier" for Formal A-Modules, or Lifting Formal Group
Laws and Formal A-Modules Revisited 395
E.4 Bibliographical and Other Notes 422
Chapter VI Applications of Formal Groups in Algebraic
Topology, Number Theory, and Algebraic Geometry
31 Basic Definitions and Survey of the Results of Chapter VI 427
32 Local Class Field Theory 433
33 Zeta Functions of Elliptic Curves over Q and Atkin-Swinnerton-Dyer
Conjectures 441
34 On Complex Cobordism and B.rown-Peterson Cohomology 446
35 Tate Modules (for One Dimensional Formal Group Laws) 460
E.5 Bibliographical and Other Notes 475
Chapter VII Formal Groups and Bialgebras
36 Basic Definitions and Survey of the Results of Chapter VII 478
37 Formal Groups and Bialgebras 485
38 Curves in Noncommutative Formal Groups 504
E.6 Bibliographical and Other Notes 514
Appendix A On Power Series Rings
A.l Power Series Rings 517
A.2 Filtration and Topology 518
A.3 Formal Weierstrass Preparation Theorem 519
A.4 Homomorphisms and Isomorphisms. Formal Inverse Function
and Implicit Function Theorems 520
CONTENTS IX
Appendix B Brief Notes on Further Applications
of Formal Group (Law) Theory
B.I More on Formal Groups in Number Theory 523
B.2 More on Formal Groups in Algebraic Geometry 524
B.3 More on Formal Groups in Arithmetical Algebraic Geometry 525
B.4 More on Formal Groups in Algebraic Topology 527
Bibliography 531
Notation 551
Index 567