The Ball and Some Hilbert Problems: (Lectures in Mathematics. ETH Zürich)

As an interesting object of arithmetic, algebraic and analytic geometry the complex ball was born in a paper of the French Mathematician E. PICARD in 1883. In recent developments the ball finds great interest again in the framework of SHIMURA varieties but also in the theory of diophantine equations (asymptotic FERMAT Problem, see ch. VI). At first glance the original ideas and the advanced theories seem to be rather disconnected. With these lectures I try to build a bridge from the analytic origins to the actual research on effective problems of arithmetic algebraic geometry. The best motivation is HILBERT'S far-reaching program consisting of 23 prob­ lems (Paris 1900) " . . . one should succeed in finding and discussing those functions which play the part for any algebraic number field corresponding to that of the exponential function in the field of rational numbers and of the elliptic modular functions in the imaginary quadratic number field". This message can be found in the 12-th problem "Extension of KRONECKER'S Theorem on Abelian Fields to Any Algebraic Realm of Rationality" standing in the middle of HILBERTS'S pro­ gram. It is dedicated to the construction of number fields by means of special value of transcendental functions of several variables. The close connection with three other HILBERT problems will be explained together with corresponding advanced theories, which are necessary to find special effective solutions, namely: 7. Irrationality and Transcendence of Certain Numbers; 21.

Table of Contents:
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.