The Ball and Some Hilbert Problems

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Contents Preface 1. Elliptic Curves, the Finiteness Theorem of Shafarevič 1.1 Elliptic Curves over ℂ 1.2 Elliptic Curves over Arbitrary Fields 2. Picard Curves 2.1 The Moduli Space of PICARD Curves 2.2 The Relative SCHOTTKY Problem for PICARD Curves 2.3 Typical Period Matrices 2.4 Metrization 2.5 Arithmetization 2.6 A Retrospect to Elliptic Curves 2.7 Rough Solution of the Relative SCHOTTKY Problem for PICARD Curves 3. Uniformizations and Differential Equations of Euler-Picard Type 3.1 Ball Uniformization of Algebraic Surfaces 3.2 Special Fuchsian Systems and GAUSS-MANIN Connection 3.3 PICARD Modular Forms 3.4 PICARD Modular Forms as Theta Constants 4. Algebraic Values of Picard Modular Theta Functions 4.1 Introduction 4.2 Complex Multiplication on Abelian Varieties 4.3 Types of Complex Multiplication 4.4 Transformation of Constants 4.5 SHIMURA Class Fields 4.6 Moduli Fields 4.7 The Main Theorem of Complex Multiplication 4.8 SHIMURA Class Fields by Special Values 4.9 Special Points on SHIMURA Varieties of 𝕌(2, 1) 5. Transcendental Values of Picard Modular Theta Constants 5.1 Transcendence at Non-Singular Simple Algebraic Moduli 5.2 Transcendence at Non-Singular Non-Simple Algebraic Moduli 5.3 Some More History 6. Arithmetic Surfaces of Kodaira-Picard Type and some Diophantine Equations 6.1 Introduction 6.2 Arithmetic Surfaces and Curves of KODAIRA-PICARD Type 6.3 Heights 6.4 Conjectures of VOJTA and PARSHIN'S Problem 6.5 KUMMER Maps 6.6 Proof of the Main Implication 7. Appendix I: A Finiteness Theorem for Picard Curves With Good Reduction (by J. Estrada-Sarlabous) 7.1 Some Definitions and Known Results 7.2 Affine Models of n-gonal Cyclic Curves 7.3 Normal Forms of PICARD Curves 7.4 Conditions for Smoothness 7.5 Projective Isomorphy Classification in Characteristic > 3 7.6 Minimal Normal Forms for PICARD Curves 7.7 Good Reduction of PICARD Curves 8. Appendix* II. The Hilbert Problems 7, 12, 21 and 22 8.1 Irrationality and Transcendence of Certain Numbers 8.2 Extension of KRONECKER'S Theorem on Abelian Fields to any Algebraic Realm of Rationality 8.3 Proof of the Existence of Linear Differential Equations Having a Prescribed Monodromic Group 8.4 Uniformization of Analytic Relations by Means of Automorphic Functions Basic Notations Bibliography [18] [36] [56] [75] Additional References for Appendix I Index