Author | Holzapfel, Rolf-Peter. author |
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Title | The Ball and Some Hilbert Problems [electronic resource] / by Rolf-Peter Holzapfel |
Imprint | Basel : Birkhรคuser Basel, 1995 |
Connect to | http://dx.doi.org/10.1007/978-3-0348-9051-9 |
Descript | 160 p. 3 illus. online resource |
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 Parshinโs 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 -- 7.5 Projective 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 Kroneckerโs 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