Formal Groups and Applications (eBook)

Front Cover 1
Formal Groups and Applications 4
Copyright Page 5
Contents 8
Preface 12
Leitfaden and Indicien 14
Introduction 18
Chapter I. Methods for Constructing One Dimensional Formal Groups 26
1 Definition and Elementary Properties of Formal Groups Survey of the Results of Chapter I 26
2 The Functional Equation–Integrality Lemma 33
3 The Formal Group Laws Fv(X, Y ), FV.T(X, Y ), and Fs ( X , Y) 41
4 Some Binomial Coefficient Arithmetic 45
5 A Universal One Dimensional Commutative Formal Group Law 49
6 Most One Dimensional Formal Group Laws Are Commutative 63
7 Honda's Method for Constructing Formal Group Laws 67
8 The Lubin–Tate Formal Group Laws 68
Chapter II. Methods for Constructing Higher Dimensional Formal Group Laws 76
9 Definitions and Elementary Properties. Survey of the Results of Chapter II 76
10 The Higher Dimensional Functional Equation Lemma 84
11 The Universal n-Dimensional Commutative Formal Group Laws H u( X , Y ) 88
12 Curvilinear Formal Group Laws 93
13 Higher Dimensional Honda Formal Group Laws and Higher Dimensional Lubin--Tate Formal Group Laws 98
14 Lie Theory 104
E.1 Bibliographical and Other Notes 112
Chapter III. Curves, p-Typical Formal Group Laws, and Lots of Witt Vectors 116
15 Definitions. Survey of Results 116
16 Curves and p-Typical Formal Groups 127
17 Lots of Witt Vectors 140
E.2 Bibliographical and Other Notes 169
Chapter IV. Homomorphisms, Endomorphisms, and the Classification of Formal Groups by Power Series Methods 172
18 Definitions and Preliminary Elementary Results. Survey of Chapter IV 172
19 Universal Isomorphisms 187
20 Existence and Nonexistence of Homomorphisms and Isomorphisms 198
21 Formal A-Modules 224
22 Lifting and Reducing Formal Group Laws. Formal Moduli 255
23 Rings of Endomorphisms of Formal Group Laws 270
24 Classification of One Dimensional Formal Group Laws over Finite Fields 278
25 Rings of Curves and Artin-Hasse-Like Exponential Mappings 294
E.3 Bibliographical and Other Notes 335
Chapter V. Cartier–Dieudonné Modules 337
26 Basic Definitions and Reminders. Survey of the Results of Chapter V 337
27 Cartier–Dieudonné Modules for Formal Group Laws 345
28 On the Classification of Commutative Formal Group Laws over an Algebraically Closed Field of Characteristic p > 0
29 Cartier–Dieudonné Theory for Formal A -Modules 399
30 "Le Tapis de Cartier" for Formal A-Modules, or Lifting Formal Group Laws and Formal A-Modules Revisited 420
E.4 Bibliographical and Other Notes 447
Chapter VI. Applications of Formal Groups in Algebraic Topology, Number Theory, and Algebraic Geometry 452
31 Basic Definitions and Survey of the Results of Chapter VI 452
32 Local Class Field Theory 458
33 Zeta Functions of Elliptic Curves over Q and Atkin--Swinnerton–Dyer Conjectures 466
34 On Complex Cobordism and Brown–Peterson Cohomology 471
35 Tate Modules (for One Dimensional Formal Group Laws) 485
E.5 Bibliographical and Other Notes 500
Chapter VII. Formal Groups and Bialgebras 503
36 Basic Definitions and Survey of the Results of Chapter VII 503
37 Formal Groups and Bialgebras 510
38 Curves in Noncommutative Formal Groups 529
E.6 Bibliographical and Other Notes 539
Appendix A. On Power Series Rings 542
A.1 Power Series Rings 542
A.2 Filtration and Topology 543
A.3 Formal Weierstrass Preparation Theorem 544
A.4 Homomorphisms and Isomorphisms. Formal Inverse Function and Implicit Function Theorems 545
Appendix B. Brief Notes on Further Applications of Formal Group (Law) Theory 548
B.1 More on Formal Groups in Number Theory 548
B.2 More on Formal Groups in Algebraic Geometry 549
B.3 More on Formal Groups in Arithmetical Algebraic Geometry 550
B.4 More on Formal Groups in Algebraic Topology 552
Bibliography 556
Notation 576
Index 592
Pure and Applied Mathematics 599