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