This book is devoted to the proof of a deep theorem in arithmetic geometry, the Fekete-Szego theorem with local rationality conditions. The prototype for the theorem is Raphael Robinson's theorem on totally real algebraic integers in an interval, which says that if $[a,b]$ is a real interval of length greater than 4, then it contains infinitely many Galois orbits of algebraic integers, while if its length is less than 4, it contains only finitely many. The theorem shows this phenomenon holds on algebraic curves of arbitrary genus over global fields of any characteristic, and is valid for a broad class of sets. The book is a sequel to the author's work Capacity Theory on Algebraic Curves and contains applications to algebraic integers and units, the Mandelbrot set, elliptic curves, Fermat curves, and modular curves. A long chapter is devoted to examples, including methods for computing capacities. Another chapter contains extensions of the theorem, including variants on Berkovich curves. The proof uses both algebraic and analytic methods, and draws on arithmetic and algebraic geometry, potential theory, and approximation theory. It introduces new ideas and tools which may be useful in other settings, including the local action of the Jacobian on a curve, the ouniversal function" of given degree on a curve, the theory of inner capacities and Green's functions, and the construction of near-extremal approximating functions by means of the canonical distance.
Introduction ix
Some History xii
A Sketch of the Proof of the Fekete-Szego xiii
Theorem
The Definition of the Cantor Capacity xvi
Outline of the Book xix
Acknowledgments xxiv
Symbol Table xxv
Chapter 1 Variants 1 (8)
Chapter 2 Examples and Applications 9 (52)
1 Local Capacities and Green's Functions 9 (11)
of Archimedean Sets
2 Local Capacities and Green's Functions 20 (7)
of Nonarchimedean Sets
3 Global Examples on P1 27 (11)
4 Function Field Examples concerning 38 (2)
Separability
5 Examples on Elliptic Curves 40 (13)
6 The Fermat Curve 53 (4)
7 The Modular Curve Xo(p) 57 (4)
Chapter 3 Preliminaries 61 (42)
1 Notation and Conventions 61 (1)
2 Basic Assumptions 62 (2)
3 The L-rational and Lsep-rational Bases 64 (5)
4 The Spherical Metric and Isometric 69 (4)
Parametrizability
5 The Canonical Distance and the (x, 73 (4)
s)-Canonical Distance
6 (x, s)-Functions and (x, 77 (1)
s)-Pseudopolynomials
7 Capacities 78 (3)
8 Green's Functions of Compact Sets 81 (4)
9 Upper Green's Functions 85 (6)
10 Green's Matrices and the Inner Cantor 91 (3)
Capacity
11 Newton Polygons of Nonarchimedean 94 (4)
Power Series
12 Stirling Polynomials and the Sequence 98 (5)
ψw(k)
Chapter 4 Reductions 103 (30)
Chapter 5 Initial Approximating Functions: 133 (26)
Archimedean Case
1 The Approximation Theorems 134 (2)
2 Outline of the Proof of Theorem 5.2 136 (5)
3 Independence 141 (3)
4 Proof of Theorem 5.2 144 (15)
Chapter 6 Initial Approximating Functions: 159 (32)
Nonarchimedean Case
1 The Approximation Theorems 160 (2)
2 Reduction to a Set Ev in a Single Ball 162 (9)
3 Generalized Stirling Polynomials 171 (3)
4 Proof of Proposition 6.5 174 (12)
5 Corollaries to the Proof of Theorem 6.3 186 (5)
Chapter 7 The Global Patching Construction 191 (58)
1 The Uniform Strong Approximation Theorem 193 (2)
2 S-units and 5-subunits 195 (1)
3 The Semi-local Theory 196 (3)
4 Proof of Theorem 4.2 when char(K) = 0 199 (24)
5 Proof of Theorem 4.2 when Char(K) =p 223 (19)
> 0
6 Proof of Proposition 7.18 242 (7)
Chapter 8 Local Patching when Kv C 249 (8)
Chapter 9 Local Patching when Kv R 257 (12)
Chapter 10 Local Patching for 269 (10)
Nonarchimedean RL-domains
Chapter 11 Local Patching for 279 (52)
Nonarchimedean Kv-simple Sets
1 The Patching Lemmas 284 (9)
2 Stirling Polynomials when Char(Kv) = p 293 (1)
> 0
3 Proof of Theorems 11.1 and 11.2 294 (24)
4 Proofs of the Moving Lemmas 318 (13)
Appendix A (x, s)-Potential Theory 331 (20)
1 (x, s)-Potential Theory for Compact Sets 331 (8)
2 Mass Bounds in the Archimedean Case 339 (2)
3 Description of μx,s in the 341 (10)
Nonarchimedean Case
Appendix B The Construction of Oscillating 351 (38)
Pseudopolynomials
1 Weighted (x, s)-Capacity Theory 353 (3)
2 The Weighted Cheybshev Constant 356 (5)
3 The Weighted Transfinite Diameter 361 (5)
4 Comparisons 366 (4)
5 Particular Cases of Interest 370 (8)
6 Chebyshev Pseudopolynomials for Short 378 (4)
Intervals
7 Oscillating Pseudopolynomials 382 (7)
Appendix C The Universal Function 389 (18)
Appendix D The Local Action of the Jacobian 407 (16)
1 The Local Action of the Jacobian on Cgv 409 (2)
2 Lemmas on Power Series in Several 411 (3)
Variables
3 Proof of the Local Action Theorem 414 (9)
Bibliography 423 (4)
Index 427
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