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Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties
 
Jörg Jahnel Universität Siegen, Germany
Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties
Hardcover ISBN:  978-1-4704-1882-3
Product Code:  SURV/198
List Price: $129.00
MAA Member Price: $116.10
AMS Member Price: $103.20
eBook ISBN:  978-1-4704-1962-2
Product Code:  SURV/198.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Hardcover ISBN:  978-1-4704-1882-3
eBook: ISBN:  978-1-4704-1962-2
Product Code:  SURV/198.B
List Price: $254.00 $191.50
MAA Member Price: $228.60 $172.35
AMS Member Price: $203.20 $153.20
Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties
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Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties
Jörg Jahnel Universität Siegen, Germany
Hardcover ISBN:  978-1-4704-1882-3
Product Code:  SURV/198
List Price: $129.00
MAA Member Price: $116.10
AMS Member Price: $103.20
eBook ISBN:  978-1-4704-1962-2
Product Code:  SURV/198.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Hardcover ISBN:  978-1-4704-1882-3
eBook ISBN:  978-1-4704-1962-2
Product Code:  SURV/198.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: 1982014; 267 pp
    MSC: Primary 11; 14; 16;

    The central theme of this book is the study of rational points on algebraic varieties of Fano and intermediate type—both in terms of when such points exist and, if they do, their quantitative density. The book consists of three parts. In the first part, the author discusses the concept of a height and formulates Manin's conjecture on the asymptotics of rational points on Fano varieties.

    The second part introduces the various versions of the Brauer group. The author explains why a Brauer class may serve as an obstruction to weak approximation or even to the Hasse principle. This part includes two sections devoted to explicit computations of the Brauer–Manin obstruction for particular types of cubic surfaces.

    The final part describes numerical experiments related to the Manin conjecture that were carried out by the author together with Andreas-Stephan Elsenhans.

    The book presents the state of the art in computational arithmetic geometry for higher-dimensional algebraic varieties and will be a valuable reference for researchers and graduate students interested in that area.

    Readership

    Graduate students and research mathematicians interested in computational arithmetic geometry.

  • Table of Contents
     
     
    • Chapters
    • 1. Introduction
    • Part A. Heights
    • Chapter I. The concept of a height
    • Chapter II. Conjectures on the asymptotics of points of bounded height
    • Part B. The Brauer group
    • Chapter III. On the Brauer group of a scheme
    • Chapter IV. An application: The Brauer–Manin obstruction
    • Part C. Numerical experiments
    • Chapter V. The Diophantine equation $x^4 + 2 y^4 = z^4 + 4 w^4$
    • Chapter VI. Points of bounded height on cubic and quartic threefolds
    • Chapter VII. On the smallest point on a diagonal cubic surface
    • Appendix
  • 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: 1982014; 267 pp
MSC: Primary 11; 14; 16;

The central theme of this book is the study of rational points on algebraic varieties of Fano and intermediate type—both in terms of when such points exist and, if they do, their quantitative density. The book consists of three parts. In the first part, the author discusses the concept of a height and formulates Manin's conjecture on the asymptotics of rational points on Fano varieties.

The second part introduces the various versions of the Brauer group. The author explains why a Brauer class may serve as an obstruction to weak approximation or even to the Hasse principle. This part includes two sections devoted to explicit computations of the Brauer–Manin obstruction for particular types of cubic surfaces.

The final part describes numerical experiments related to the Manin conjecture that were carried out by the author together with Andreas-Stephan Elsenhans.

The book presents the state of the art in computational arithmetic geometry for higher-dimensional algebraic varieties and will be a valuable reference for researchers and graduate students interested in that area.

Readership

Graduate students and research mathematicians interested in computational arithmetic geometry.

  • Chapters
  • 1. Introduction
  • Part A. Heights
  • Chapter I. The concept of a height
  • Chapter II. Conjectures on the asymptotics of points of bounded height
  • Part B. The Brauer group
  • Chapter III. On the Brauer group of a scheme
  • Chapter IV. An application: The Brauer–Manin obstruction
  • Part C. Numerical experiments
  • Chapter V. The Diophantine equation $x^4 + 2 y^4 = z^4 + 4 w^4$
  • Chapter VI. Points of bounded height on cubic and quartic threefolds
  • Chapter VII. On the smallest point on a diagonal cubic surface
  • Appendix
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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