| Hardcover ISBN: | 978-1-4704-1882-3 |
| Product Code: | SURV/198 |
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| 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 |
| 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 |
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Book DetailsMathematical Surveys and MonographsVolume: 198; 2014; 267 ppMSC: 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.
ReadershipGraduate students and research mathematicians interested in computational arithmetic geometry.
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Table of Contents
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Chapters
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1. Introduction
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Part A. Heights
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Chapter I. The concept of a height
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Chapter II. Conjectures on the asymptotics of points of bounded height
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Part B. The Brauer group
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Chapter III. On the Brauer group of a scheme
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Chapter IV. An application: The Brauer–Manin obstruction
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Part C. Numerical experiments
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Chapter V. The Diophantine equation $x^4 + 2 y^4 = z^4 + 4 w^4$
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Chapter VI. Points of bounded height on cubic and quartic threefolds
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Chapter VII. On the smallest point on a diagonal cubic surface
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Appendix
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Additional Material
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RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
- Book Details
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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.
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 $x^4 + 2 y^4 = z^4 + 4 w^4$
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Chapter VI. Points of bounded height on cubic and quartic threefolds
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Chapter VII. On the smallest point on a diagonal cubic surface
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Appendix
