Brauer groups, Tamagawa measures, and rational points on algebraic varieties / Jörg Jahnel.

Format:

Book

Author/Creator:

Jahnel, Jörg, 1968- author.

Series:

Mathematical surveys and monographs ; Volume 198.

Mathematical surveys and monographs ; Volume 198

Language:

English

Subjects (All):

Brauer groups.

Algebraic varieties.

Geometry, Algebraic.

Rational points (Geometry).

Physical Description:

1 online resource (viii, 267 pages) : illustrations.

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2014]

Language Note:

English

Summary:

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.

Contents:

Cover

Title page

Contents

Preface

Introduction

Chapter I. The concept of a height

Chapter II. Conjectures on the asymptotics of points of bounded height

Chapter III. On the Brauer group of a scheme

Chapter IV. An application: The Brauer-Manin obstruction

Chapter V. The Diophantine equation ⁴+2 ⁴= ⁴+4 ⁴

Chapter VI. Points of bounded height on cubic and quartic threefolds

Chapter VII. On the smallest point on a diagonal cubic surface

Appendix

Bibliography

Index

Back Cover.

Notes:

Bibliographic Level Mode of Issuance: Monograph

Includes bibliographical references and index.

Description based on print version record.

Description based on publisher supplied metadata and other sources.

ISBN:

1-4704-1962-9

OCLC:

907357746