Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties
About this Title
Jörg Jahnel, Universität Siegen, Germany
Publication: Mathematical Surveys and Monographs
Publication Year 2014: Volume 198
ISBNs: 978-1-4704-1882-3 (print); 978-1-4704-1962-2 (online)
MathSciNet review: MR3242964
MSC: Primary 14F22; Secondary 11G35, 11G50, 14G05
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.
Table of Contents
- 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