**Mathematical Surveys and Monographs**

Volume: 193;
2013;
437 pp;
Hardcover

MSC: Primary 11; 14;
Secondary 31

Print ISBN: 978-1-4704-0980-7

Product Code: SURV/193

List Price: $119.00

Individual Member Price: $95.20

**Electronic ISBN: 978-1-4704-1446-7
Product Code: SURV/193.E**

List Price: $119.00

Individual Member Price: $95.20

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#### Supplemental Materials

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

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*Robert Rumely*

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 \([a,b]\) 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
“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.

#### Table of Contents

# Table of Contents

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

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Introduction ix10 free
- Variants 128 free
- Examples and applications 936
- Preliminaries 6188
- Reductions 103130
- Initial approximating functions: Archimedean case 133160
- Initial approximating functions: Nonarchimedean case 159186
- The global patching construction 191218
- Local patching when 𝐾ᵥ≅ℂ 249276
- Local patching when 𝐾ᵥ≅ℝ 257284
- Local patching for nonarchimedean RL-domains 269296
- Local patching for nonarchimedean 𝐾ᵥ-simple sets 279306
- (𝔛,⃗𝔰)-Potential theory 331358
- The construction of oscillating pseudopolynomials 351378
- The universal function 389416
- The local action of the Jacobian 407434
- Bibliography 423450
- Index 427454 free
- Back Cover Back Cover1466

#### Readership

Graduate students and research mathematicians interested in arithmetic geometry, number theory, potential theory, algebraic geometry, and dynamics.