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Hardcover ISBN:  9781470409807 
Product Code:  SURV/193 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
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Product Code:  SURV/193.E 
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AMS Member Price:  $100.00 
Hardcover ISBN:  9781470409807 
eBook ISBN:  9781470414467 
Product Code:  SURV/193.B 
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Book DetailsMathematical Surveys and MonographsVolume: 193; 2013; 437 ppMSC: Primary 11; 14; Secondary 31
This book is devoted to the proof of a deep theorem in arithmetic geometry, the FeketeSzegö 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 nearextremal approximating functions by means of the canonical distance.
ReadershipGraduate students and research mathematicians interested in arithmetic geometry, number theory, potential theory, algebraic geometry, and dynamics.

Table of Contents

Chapters

1. Variants

2. Examples and applications

3. Preliminaries

4. Reductions

5. Initial approximating functions: Archimedean case

6. Initial approximating functions: Nonarchimedean case

7. The global patching construction

8. Local patching when $K_v \cong \mathbb {C}$

9. Local patching when $K_v \cong \mathbb {R}$

10. Local patching for nonarchimedean RLdomains

11. Local patching for nonarchimedean $K_v$simple sets

Appendix A. $(\mathfrak {X},\vec {s})$Potential theory

Appendix B. The construction of oscillating pseudopolynomials

Appendix C. The universal function

Appendix D. The local action of the Jacobian


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This book is devoted to the proof of a deep theorem in arithmetic geometry, the FeketeSzegö 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 nearextremal approximating functions by means of the canonical distance.
Graduate students and research mathematicians interested in arithmetic geometry, number theory, potential theory, algebraic geometry, and dynamics.

Chapters

1. Variants

2. Examples and applications

3. Preliminaries

4. Reductions

5. Initial approximating functions: Archimedean case

6. Initial approximating functions: Nonarchimedean case

7. The global patching construction

8. Local patching when $K_v \cong \mathbb {C}$

9. Local patching when $K_v \cong \mathbb {R}$

10. Local patching for nonarchimedean RLdomains

11. Local patching for nonarchimedean $K_v$simple sets

Appendix A. $(\mathfrak {X},\vec {s})$Potential theory

Appendix B. The construction of oscillating pseudopolynomials

Appendix C. The universal function

Appendix D. The local action of the Jacobian