# Capacity Theory with Local Rationality: The Strong Fekete-Szegö Theorem on Curves

American Mathematical Soc., Dec 26, 2013 - Mathematics - 437 pages

This book is devoted to the proof of a deep theorem in arithmetic geometry, the Fekete-Szegö 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 img style="width:6px;height:23px;vertical-align:-6px;margin-right:0.023em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Main/Regular/120/005B.png"img style="width:11px;height:9px;margin-right:0.023em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Math/Italic/120/0061.png"img style="width:5px;height:9px;vertical-align:-5px;margin-right:0.067em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Main/Regular/120/002C.png"img style="width:9px;height:14px;margin-right:0.007em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Math/Italic/120/0062.png"img style="width:4px;height:23px;vertical-align:-6px;margin-right:0.117em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Main/Regular/120/005D.png"  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 "universal 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.

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### Contents

 Variants 1 Examples and Applications 9 Preliminaries 61 Reductions 103 Archimedean Case 133 Nonarchimedean Case 159 The Approximation Theorems 160 Reduction to a Set Ev in a Single Ball 163
 Mass Bounds in the Archimedean Case 339 Description of μXs in the Nonarchimedean Case 349 Appendix B The Construction of Oscillating Pseudopolynomials 351 Weighted X sCapacity Theory 352 The Weighted Cheybshev Constant 356 The Weighted Transfinite Diameter 361 Comparisons 366 Particular Cases of Interest 370

 Generalized Stirling Polynomials 171 Proof of Proposition 6 5 174 Corollaries to the Proof of Theorem 6 3 186 The Global Patching Construction 191 The Uniform Strong Approximation Theorem 193 Sunits and Ssubunits 195 The Semilocal Theory 196 Gauss norm 412 199 Proof of Theorem 4 2 when charK 0 217 global patching when charK p 0 224 Proof of Theorem 4 2 when CharK p 0 231 Proof of Proposition 7 18 242 Local Patching when K v 249 Local Patching when K v 257 Local Patching for Nonarchimedean RLdomains 269 Local Patching for Nonarchimedean Kvsimple Sets 279 The Patching Lemmas 284 Stirling Polynomials when CharKv p 0 293 Proof of Theorems 11 1 and 11 2 318 Appendix A X sPotential Theory 1 X sPotential Theory for Compact Sets 331
 Chebyshev Pseudopolynomials for Short Intervals Oscillating Pseudopolynomials 378 3 400 4 404 6 405 Appendix C The Universal Function Appendix D The Local Action of the Jacobian 407 The Local Action of the Jacobian on 409 Lemmas on Power Series in Several Variables 411 Proof of the Local Action Theorem 414 Bibliography 423 Index 425 160 427 171 428 191 432 195 433 318 434 356 435 382 436 414 437 Copyright