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

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