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Capacity Theory with Local Rationality: The Strong Fekete-Szegö Theorem on Curves
 
Robert Rumely University of Georgia, Athens, GA
Capacity Theory with Local Rationality
Hardcover ISBN:  978-1-4704-0980-7
Product Code:  SURV/193
List Price: $129.00
MAA Member Price: $116.10
AMS Member Price: $103.20
eBook ISBN:  978-1-4704-1446-7
Product Code:  SURV/193.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Hardcover ISBN:  978-1-4704-0980-7
eBook: ISBN:  978-1-4704-1446-7
Product Code:  SURV/193.B
List Price: $254.00 $191.50
MAA Member Price: $228.60 $172.35
AMS Member Price: $203.20 $153.20
Capacity Theory with Local Rationality
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Capacity Theory with Local Rationality: The Strong Fekete-Szegö Theorem on Curves
Robert Rumely University of Georgia, Athens, GA
Hardcover ISBN:  978-1-4704-0980-7
Product Code:  SURV/193
List Price: $129.00
MAA Member Price: $116.10
AMS Member Price: $103.20
eBook ISBN:  978-1-4704-1446-7
Product Code:  SURV/193.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Hardcover ISBN:  978-1-4704-0980-7
eBook ISBN:  978-1-4704-1446-7
Product Code:  SURV/193.B
List Price: $254.00 $191.50
MAA Member Price: $228.60 $172.35
AMS Member Price: $203.20 $153.20
  • Book Details
     
     
    Mathematical Surveys and Monographs
    Volume: 1932013; 437 pp
    MSC: Primary 11; 14; Secondary 31;

    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.

    Readership

    Graduate 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 RL-domains
    • 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1932013; 437 pp
MSC: Primary 11; 14; Secondary 31;

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.

Readership

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 RL-domains
  • 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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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