**Mathematical Surveys and Monographs**

Volume: 198;
2014;
267 pp;
Hardcover

MSC: Primary 11; 14; 16;

Print ISBN: 978-1-4704-1882-3

Product Code: SURV/198

List Price: $100.00

Individual Member Price: $80.00

**Electronic ISBN: 978-1-4704-1962-2
Product Code: SURV/198.E**

List Price: $100.00

Individual Member Price: $80.00

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#### Supplemental Materials

# Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties

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*Jörg Jahnel*

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.

#### Table of Contents

# Table of Contents

## Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Preface vii8 free
- Introduction 110 free
- Chapter I. The concept of a height 1524 free
- Chapter II. Conjectures on the asymptotics of points of bounded height 3544
- Chapter III. On the Brauer group of a scheme 8392
- Chapter IV. An application: The Brauer–Manin obstruction 119128
- Chapter V. The Diophantine equation 𝑥⁴+2𝑦⁴=𝑧⁴+4𝑤⁴ 165174
- Chapter VI. Points of bounded height on cubic and quartic threefolds 185194
- Chapter VII. On the smallest point on a diagonal cubic surface 205214
- Appendix 239248
- Bibliography 247256
- Index 261270 free
- Back Cover Back Cover1280

#### Readership

Graduate students and research mathematicians interested in computational arithmetic geometry.