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

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 |

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