Preface ....................................................... vii
Acknowledgments .............................................. xiii
1 Group Varieties and Transcendence ............................ 1
1.1 Group Varieties ......................................... 2
1.2 Doubly Periodic Functions ............................... 8
1.3 Abelian Varieties ...................................... 15
1.4 Transcendence of Vectors in a Polarized Lattice ........ 21
1.5 Linear Relations between Periods ....................... 23
2 Transcendence Results for Exponential and Elliptic
Functions ................................................... 31
2.1 Some Prerequisites from Algebra ........................ 32
2.1.1 Size of an Algebraic Integer .................... 32
2.1.2 Siegel's Lemma .................................. 33
2.2 Some Prerequisites from Analysis ....................... 36
2.3 Transcendence of Values of the Exponential Function .... 36
2.3.1 Construction of the Auxiliary Function .......... 37
2.3.2 Existence and Choice of a Non-zero F(k)(μα) ..... 40
2.3.3 Bound F(k)(μα) from Above Using Zeros of
F(z) ............................................ 41
2.3.4 Bound F(k)(μα) from Below Using Size Estimate ... 42
2.3.5 Obtaining a Contradiction ....................... 43
2.3.6 Checking the Parameters and Concluding .......... 43
2.4 The Schneider-Lang Theorem and its Corollaries ......... 44
3 Modular Functions and Criteria for Complex Multiplication ... 55
3.1 Tlie Modular Group PSL(2, ) ............................ 55
3.2 The Elliptic Modular Function .......................... 58
3.3 A Transcendence Criterion for CM on Abelian Varieties .. 61
4 Periods of 1-forms on Complex Curves and Abelian Varieties .. 67
4.1 The Fundamental Group and the Universal Cover .......... 67
4.2 The First Singular Homology Group ...................... 69
4.3 The First de Rham Cohomology Space over  .............. 72
4.4 The First de Rham Cohomology Space over  .............. 77
4.5 Dolbeault Cohomology and Holomorphic 1-Forms ........... 82
4.6 Complex Kahler Manifolds and their First Cohomology .... 84
4.7 Integrating Forms and the de Rham Theorem .............. 87
4.8 Normalized Periods of Tori and de Rham Cohomology ...... 89
4.9 The Mumford-Tate Group of a Complex Torus .............. 94
4.10 Level 1 Hodge Structures and Transcendence ............. 98
4.11 Algebraic 1-Forms on Riemann Surfaces .................. 99
4.12 Jacobian of a Riemann Surface ......................... 103
4.13 Explicit Periods on some Curves with CM ............... 106
5 Transcendence of Special Values of Hypergeometric
Functions .................................................. 109
5.1 Series, Differential Equation, and Euler Integral ..... 109
5.2 Hypergeometric Periods ................................ 112
5.3 Monodromy of the Gauss Hypergeometric Function ........ 114
5.4 Deligne-Mostow's Condition INT and Triangle Groups .... 115
5.5 Arithmetic Triangle Groups ............................ 117
5.6 Special Values of Hypergeometric Functions and CM ..... 121
5.7 Exceptional Set and the Edixhoven-Yafaev Theorem ...... 125
5.8 Transcendence Results for Appell-Lauricella
Functions ............................................. 128
5.8.1 Lattice and Arithmeticity Criteria ............. 132
5.8.2 The Exceptional Set and CM ..................... 135
5.8.3 Shimura Varieties and Hypergeometric
Functions ...................................... 138
5.8.4 Proof of Theorem 5.5 ........................... 139
5.9 Ball (n + 3)-tuples and the Exceptional Set ........... 140
6 Transcendence Criterion for Complex Multiplication on
Surfaces ................................................... 143
6.1 Mumford-Tate Group of a Level 2 Hodge Structure ....... 144
6.2 Chfford Algebras ...................................... 147
6.3 The Kuga-Satake Correspondence ........................ 148
6.4 The Kuga-Satake Variety ............................... 151
6.5 Transcendence and CM for К3 Hodge Structures .......... 156
6.6 Transcendence Criterion for CM on К3 Surfaces ......... 157
6.7 Kuga-Satake for Hodge Numbers (1,0,1) ................. 159
6.8 Examples of Periods on КЗ Surfaces .................... 161
7 Hodge Structures of Higher Level ........................... 171
7.1 Singular Homology ..................................... 172
7.2 Hodge Theory of Complex Projective Manifolds .......... 173
7.3 Hodge Filtrations defined over  and CM ............... 175
7.4 Question 7.2 for Borcea-Voisin Towers ................. 184
7.5 Example of a Viehweg-Zuo Tower ........................ 192
7.6 Periods on Fermat Hypersurfaces ....................... 196
Bibliography .................................................. 199
Index ......................................................... 207
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