Holzapfel R.-P. The Ball and Some Hilbert Problems

Basel: Birkhäuser, 1994. — 167 p.Elliptic Curves, the Finiteness Theorem of Shafarevič
Elliptic Curves over ℂ
Elliptic Curves over Arbitrary Fields
Picard Curves
The Moduli Space of PICARD Curves
The Relative SCHOTTKY Problem for PICARD Curves
Typical Period Matrices
Metrization
Arithmetization
A Retrospect to Elliptic Curves
Rough Solution of the Relative SCHOTTKY Problem for PICARD Curves
Uniformizations and Differential Equations of Euler-Picard Type
Ball Uniformization of Algebraic Surfaces
Special Fuchsian Systems and GAUSS-MANIN Connection
PICARD Modular Forms
PICARD Modular Forms as Theta Constants
Algebraic Values of Picard Modular Theta FunctionsComplex Multiplication on Abelian Varieties
Types of Complex Multiplication
Transformation of Constants
SHIMURA Class Fields
Moduli Fields
The Main Theorem of Complex Multiplication
SHIMURA Class Fields by Special Values
Special Points on SHIMURA Varieties of ?(2, 1)
Transcendental Values of Picard Modular Theta Constants
Transcendence at Non-Singular Simple Algebraic Moduli
Transcendence at Non-Singular Non-Simple Algebraic Moduli
Some More History
Arithmetic Surfaces of Kodaira-Picard Type and some Diophantine EquationsArithmetic Surfaces and Curves of KODAIRA-PICARD Type
Heights
Conjectures of VOJTA and PARSHIN'S Problem
KUMMER Maps
Proof of the Main Implication
Appendix I: A Finiteness Theorem for Picard Curves With Good Reduction (by J. Estrada-Sarlabous)
Some Definitions and Known Results
Affine Models of n-gonal Cyclic Curves
Normal Forms of PICARD Curves
Conditions for Smoothness
Projective Isomorphy Classification in Characteristic > 3
Minimal Normal Forms for PICARD Curves
Good Reduction of PICARD Curves
Appendix* II. The Hilbert Problems 7, 12, 21 and 22
Irrationality and Transcendence of Certain Numbers
Extension of KRONECKER'S Theorem on Abelian Fields to any Algebraic Realm of Rationality
Proof of the Existence of Linear Differential Equations Having a Prescribed Monodromic Group
Uniformization of Analytic Relations by Means of Automorphic Functions
Basic NotationsAdditional References for Appendix I