Torsors and Rational Points

Torsors and Rational Points
Author :
Publisher : Cambridge University Press
Total Pages : 197
Release :
ISBN-10 : 9780521802376
ISBN-13 : 0521802377
Rating : 4/5 (76 Downloads)

Synopsis Torsors and Rational Points by : Alexei Skorobogatov

This book, first published in 2001, is a complete and coherent exposition of the theory and applications of torsors to rational points.

Torsors, Étale Homotopy and Applications to Rational Points

Torsors, Étale Homotopy and Applications to Rational Points
Author :
Publisher : Cambridge University Press
Total Pages : 470
Release :
ISBN-10 : 9781107616127
ISBN-13 : 1107616123
Rating : 4/5 (27 Downloads)

Synopsis Torsors, Étale Homotopy and Applications to Rational Points by : Alexei Skorobogatov

Lecture notes and research articles on the use of torsors and étale homotopy in algebraic and arithmetic geometry.

Rational Points on Varieties

Rational Points on Varieties
Author :
Publisher : American Mathematical Soc.
Total Pages : 358
Release :
ISBN-10 : 9781470437732
ISBN-13 : 1470437732
Rating : 4/5 (32 Downloads)

Synopsis Rational Points on Varieties by : Bjorn Poonen

This book is motivated by the problem of determining the set of rational points on a variety, but its true goal is to equip readers with a broad range of tools essential for current research in algebraic geometry and number theory. The book is unconventional in that it provides concise accounts of many topics instead of a comprehensive account of just one—this is intentionally designed to bring readers up to speed rapidly. Among the topics included are Brauer groups, faithfully flat descent, algebraic groups, torsors, étale and fppf cohomology, the Weil conjectures, and the Brauer-Manin and descent obstructions. A final chapter applies all these to study the arithmetic of surfaces. The down-to-earth explanations and the over 100 exercises make the book suitable for use as a graduate-level textbook, but even experts will appreciate having a single source covering many aspects of geometry over an unrestricted ground field and containing some material that cannot be found elsewhere.

Torsors, Étale Homotopy and Applications to Rational Points

Torsors, Étale Homotopy and Applications to Rational Points
Author :
Publisher : Cambridge University Press
Total Pages : 470
Release :
ISBN-10 : 9781107245266
ISBN-13 : 1107245265
Rating : 4/5 (66 Downloads)

Synopsis Torsors, Étale Homotopy and Applications to Rational Points by : Alexei N. Skorobogatov

Torsors, also known as principal bundles or principal homogeneous spaces, are ubiquitous in mathematics. The purpose of this book is to present expository lecture notes and cutting-edge research papers on the theory and applications of torsors and étale homotopy, all written from different perspectives by leading experts. Part one of the book contains lecture notes on recent uses of torsors in geometric invariant theory and representation theory, plus an introduction to the étale homotopy theory of Artin and Mazur. Part two of the book features a milestone paper on the étale homotopy approach to the arithmetic of rational points. Furthermore, the reader will find a collection of research articles on algebraic groups and homogeneous spaces, rational and K3 surfaces, geometric invariant theory, rational points, descent and the Brauer–Manin obstruction. Together, these give a state-of-the-art view of a broad area at the crossroads of number theory and algebraic geometry.

Rational Points and Arithmetic of Fundamental Groups

Rational Points and Arithmetic of Fundamental Groups
Author :
Publisher : Springer
Total Pages : 257
Release :
ISBN-10 : 9783642306747
ISBN-13 : 3642306748
Rating : 4/5 (47 Downloads)

Synopsis Rational Points and Arithmetic of Fundamental Groups by : Jakob Stix

The section conjecture in anabelian geometry, announced by Grothendieck in 1983, is concerned with a description of the set of rational points of a hyperbolic algebraic curve over a number field in terms of the arithmetic of its fundamental group. While the conjecture is still open today in 2012, its study has revealed interesting arithmetic for curves and opened connections, for example, to the question whether the Brauer-Manin obstruction is the only one against rational points on curves. This monograph begins by laying the foundations for the space of sections of the fundamental group extension of an algebraic variety. Then, arithmetic assumptions on the base field are imposed and the local-to-global approach is studied in detail. The monograph concludes by discussing analogues of the section conjecture created by varying the base field or the type of variety, or by using a characteristic quotient or its birational analogue in lieu of the fundamental group extension.

Arithmetic Geometry

Arithmetic Geometry
Author :
Publisher : American Mathematical Soc.
Total Pages : 570
Release :
ISBN-10 : 9780821844762
ISBN-13 : 0821844768
Rating : 4/5 (62 Downloads)

Synopsis Arithmetic Geometry by : Clay Mathematics Institute. Summer School

Based on survey lectures given at the 2006 Clay Summer School on Arithmetic Geometry at the Mathematics Institute of the University of Gottingen, this tile is intended for graduate students and recent PhD's. It introduces readers to modern techniques and conjectures at the interface of number theory and algebraic geometry.

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

Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties
Author :
Publisher : American Mathematical Soc.
Total Pages : 280
Release :
ISBN-10 : 9781470418823
ISBN-13 : 1470418827
Rating : 4/5 (23 Downloads)

Synopsis Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties by : Jorg 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.

Equidistribution in Number Theory, An Introduction

Equidistribution in Number Theory, An Introduction
Author :
Publisher : Springer Science & Business Media
Total Pages : 356
Release :
ISBN-10 : 9781402054044
ISBN-13 : 1402054041
Rating : 4/5 (44 Downloads)

Synopsis Equidistribution in Number Theory, An Introduction by : Andrew Granville

This set of lectures provides a structured introduction to the concept of equidistribution in number theory. This concept is of growing importance in many areas, including cryptography, zeros of L-functions, Heegner points, prime number theory, the theory of quadratic forms, and the arithmetic aspects of quantum chaos. The volume brings together leading researchers from a range of fields who reveal fascinating links between seemingly disparate areas.

Rational Points on Varieties

Rational Points on Varieties
Author :
Publisher : American Mathematical Society
Total Pages : 357
Release :
ISBN-10 : 9781470474584
ISBN-13 : 1470474581
Rating : 4/5 (84 Downloads)

Synopsis Rational Points on Varieties by : Bjorn Poonen

This book is motivated by the problem of determining the set of rational points on a variety, but its true goal is to equip readers with a broad range of tools essential for current research in algebraic geometry and number theory. The book is unconventional in that it provides concise accounts of many topics instead of a comprehensive account of just one—this is intentionally designed to bring readers up to speed rapidly. Among the topics included are Brauer groups, faithfully flat descent, algebraic groups, torsors, étale and fppf cohomology, the Weil conjectures, and the Brauer-Manin and descent obstructions. A final chapter applies all these to study the arithmetic of surfaces. The down-to-earth explanations and the over 100 exercises make the book suitable for use as a graduate-level textbook, but even experts will appreciate having a single source covering many aspects of geometry over an unrestricted ground field and containing some material that cannot be found elsewhere. The origins of arithmetic (or Diophantine) geometry can be traced back to antiquity, and it remains a lively and wide research domain up to our days. The book by Bjorn Poonen, a leading expert in the field, opens doors to this vast field for many readers with different experiences and backgrounds. It leads through various algebraic geometric constructions towards its central subject: obstructions to existence of rational points. —Yuri Manin, Max-Planck-Institute, Bonn It is clear that my mathematical life would have been very different if a book like this had been around at the time I was a student. —Hendrik Lenstra, University Leiden Understanding rational points on arbitrary algebraic varieties is the ultimate challenge. We have conjectures but few results. Poonen's book, with its mixture of basic constructions and openings into current research, will attract new generations to the Queen of Mathematics. —Jean-Louis Colliot-Thélène, Université Paris-Sud A beautiful subject, handled by a master. —Joseph Silverman, Brown University

Cox Rings

Cox Rings
Author :
Publisher : Cambridge University Press
Total Pages : 539
Release :
ISBN-10 : 9781107024625
ISBN-13 : 1107024625
Rating : 4/5 (25 Downloads)

Synopsis Cox Rings by : Ivan Arzhantsev

This book provides a largely self-contained introduction to Cox rings and their applications in algebraic and arithmetic geometry.