The Ball and Some Hilbert Problems

Preface.- 1 Elliptic Curves, the Finiteness Theorem of Shafarevi?.- 1.1 Elliptic Curves over ?.- 1.2 Elliptic Curves over Arbitrary Fields.- 2 Picard Curves.- 2.1 The Moduli Space of Picard Curves.- 2.2The 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 $$mathbb{U}$$(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.- 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.- 7.5Projective Isomorphism 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.- 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.