| Preface |
|
vii | |
| Acknowledgments |
|
xiii | |
|
1 Group Varieties and Transcendence |
|
|
1 | (30) |
|
|
|
2 | (6) |
|
1.2 Doubly Periodic Functions |
|
|
8 | (7) |
|
|
|
15 | (6) |
|
1.4 Transcendence of Vectors in a Polarized Lattice |
|
|
21 | (2) |
|
1.5 Linear Relations between Periods |
|
|
23 | (8) |
|
2 Transcendence Results for Exponential and Elliptic Functions |
|
|
31 | (24) |
|
2.1 Some Prerequisites from Algebra |
|
|
32 | (4) |
|
2.1.1 Size of an Algebraic Integer |
|
|
32 | (1) |
|
|
|
33 | (3) |
|
2.2 Some Prerequisites from Analysis |
|
|
36 | (1) |
|
2.3 Transcendence of Values of the Exponential Function |
|
|
36 | (8) |
|
2.3.1 Construction of the Auxiliary Function |
|
|
37 | (3) |
|
2.3.2 Existence and Choice of a Non-zero F(κ)(μα) |
|
|
40 | (1) |
|
2.3.3 Bound F(κ)(μα) from Above Using Zeros of F(z) |
|
|
41 | (1) |
|
2.3.4 Bound F(κ)(μα) from Below Using Size Estimate |
|
|
42 | (1) |
|
2.3.5 Obtaining a Contradiction |
|
|
43 | (1) |
|
2.3.6 Checking the Parameters and Concluding |
|
|
43 | (1) |
|
2.4 The Schneider-Lang Theorem and its Corollaries |
|
|
44 | (11) |
|
3 Modular Functions and Criteria for Complex Multiplication |
|
|
55 | (12) |
|
3.1 The Modular Group PSL(2, Z) |
|
|
55 | (3) |
|
3.2 The Elliptic Modular Function |
|
|
58 | (3) |
|
3.3 A Transcendence Criterion for CM on Abelian Varieties |
|
|
61 | (6) |
|
4 Periods of 1-forms on Complex Curves and Abelian Varieties |
|
|
67 | (42) |
|
4.1 The Fundamental Group and the Universal Cover |
|
|
67 | (2) |
|
4.2 The First Singular Homology Group |
|
|
69 | (3) |
|
4.3 The First de Rham Cohomology Space over R |
|
|
72 | (5) |
|
4.4 The First de Rham Cohomology Space over C |
|
|
77 | (5) |
|
4.5 Dolbeault Cohomology and Holomorphic 1-Forms |
|
|
82 | (2) |
|
4.6 Complex Kahler Manifolds and their First Cohomology |
|
|
84 | (3) |
|
4.7 Integrating Forms and the de Rham Theorem |
|
|
87 | (2) |
|
4.8 Normalized Periods of Tori and de Rham Cohomology |
|
|
89 | (5) |
|
4.9 The Mumford-Tate Group of a Complex Torus |
|
|
94 | (4) |
|
4.10 Level 1 Hodge Structures and Transcendence |
|
|
98 | (1) |
|
4.11 Algebraic 1-Forms on Riemann Surfaces |
|
|
99 | (4) |
|
4.12 Jacobian of a Riemann Surface |
|
|
103 | (3) |
|
4.13 Explicit Periods on some Curves with CM |
|
|
106 | (3) |
|
5 Transcendence of Special Values of Hypergeometric Functions |
|
|
109 | (34) |
|
5.1 Series, Differential Equation, and Euler Integral |
|
|
109 | (3) |
|
5.2 Hypergeometric Periods |
|
|
112 | (2) |
|
5.3 Monodromy of the Gauss Hypergeometric Function |
|
|
114 | (1) |
|
5.4 Deligne-Mostow's Condition INT and Triangle Groups |
|
|
115 | (2) |
|
5.5 Arithmetic Triangle Groups |
|
|
117 | (4) |
|
5.6 Special Values of Hypergeometric Functions and CM |
|
|
121 | (4) |
|
5.7 Exceptional Set and the Edixhoven-Yafaev Theorem |
|
|
125 | (3) |
|
5.8 Transcendence Results for Appell-Lauricella Functions |
|
|
128 | (12) |
|
5.8.1 Lattice and Arithmeticity Criteria |
|
|
132 | (3) |
|
5.8.2 The Exceptional Set and CM |
|
|
135 | (3) |
|
5.8.3 Shimura Varieties and Hypergeometric Functions |
|
|
138 | (1) |
|
5.8.4 Proof of Theorem 5.5 |
|
|
139 | (1) |
|
5.9 Ball (n + 3)-tuples and the Exceptional Set |
|
|
140 | (3) |
|
6 Transcendence Criterion for Complex Multiplication on K3 Surfaces |
|
|
143 | (28) |
|
6.1 Mumford-Tate Group of a Level 2 Hodge Structure |
|
|
144 | (3) |
|
|
|
147 | (1) |
|
6.3 The Kuga-Satake Correspondence |
|
|
148 | (3) |
|
6.4 The Kuga-Satake Variety |
|
|
151 | (5) |
|
6.5 Transcendence and CM for K3 Hodge Structures |
|
|
156 | (1) |
|
6.6 Transcendence Criterion for CM on K3 Surfaces |
|
|
157 | (2) |
|
6.7 Kuga-Satake for Hodge Numbers (1, 0, 1) |
|
|
159 | (2) |
|
6.8 Examples of Periods on K3 Surfaces |
|
|
161 | (10) |
|
7 Hodge Structures of Higher Level |
|
|
171 | (28) |
|
|
|
172 | (1) |
|
7.2 Hodge Theory of Complex Projective Manifolds |
|
|
173 | (2) |
|
7.3 Hodge Filtrations defined over Q and CM |
|
|
175 | (9) |
|
7.4 Question 7.2 for Borcea-Voisin Towers |
|
|
184 | (8) |
|
7.5 Example of a Viehweg-Zuo Tower |
|
|
192 | (4) |
|
7.6 Periods on Fermat Hypersurfaces |
|
|
196 | (3) |
| Bibliography |
|
199 | (8) |
| Index |
|
207 | |
