Contents
Introduction ix
Some History xii
A Sketch of the Proof of the Fekete-Szeg¨ o Theorem xiii
The Definition of the Cantor Capacity xvi
Outline of the Book xix
Acknowledgments xxiv
Symbol Table xxv
Chapter 1. Variants 1
Chapter 2. Examples and Applications 9
1. Local Capacities and Green’s Functions of Archimedean Sets 9
2. Local Capacities and Green’s Functions of Nonarchimedean Sets 20
3. Global Examples on
P1
27
4. Function Field Examples concerning Separability 38
5. Examples on Elliptic Curves 40
6. The Fermat Curve 53
7. The Modular Curve X0(p) 57
Chapter 3. Preliminaries 61
1. Notation and Conventions 61
2. Basic Assumptions 62
3. The L-rational and
Lsep-rational
Bases 64
4. The Spherical Metric and Isometric Parametrizability 69
5. The Canonical Distance and the (X,)-Canonical s Distance 73
6. (X,)-Functions s and (X,)-Pseudopolynomials s 77
7. Capacities 78
8. Green’s Functions of Compact Sets 81
9. Upper Green’s Functions 85
10. Green’s Matrices and the Inner Cantor Capacity 91
11. Newton Polygons of Nonarchimedean Power Series 94
12. Stirling Polynomials and the Sequence ψw(k) 98
Chapter 4. Reductions 103
Chapter 5. Initial Approximating Functions: Archimedean Case 133
1. The Approximation Theorems 134
2. Outline of the Proof of Theorem 5.2 136
3. Independence 141
4. Proof of Theorem 5.2 144
vii
Previous Page Next Page