Rational Points On Algebraic Varieties

Author : Bjorn Poonen
Publisher : American Mathematical Society
Page : 357 pages
File Size : 48,9 Mb
Release : 2023-08-10
Category : Mathematics
ISBN : 9781470474584

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Rational Points on Varieties by Bjorn Poonen Pdf

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

Author : Emmanuel Peyre,Yuri Tschinkel
Publisher : Birkhauser
Page : 446 pages
File Size : 42,7 Mb
Release : 2001
Category : Mathematics
ISBN : 0817666125

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Rational Points on Algebraic Varieties by Emmanuel Peyre,Yuri Tschinkel Pdf

Author : Bjorn Poonen
Publisher : American Mathematical Soc.
Page : 337 pages
File Size : 53,7 Mb
Release : 2017-12-13
Category : Algebraic varieties
ISBN : 9781470437732

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Rational Points on Varieties by Bjorn Poonen Pdf

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.

Author : Emmanuel Peyre,Yuri Tschinkel
Publisher : Birkhäuser
Page : 455 pages
File Size : 53,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034883689

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Rational Points on Algebraic Varieties by Emmanuel Peyre,Yuri Tschinkel Pdf

This book is devoted to the study of rational and integral points on higher-dimensional algebraic varieties. It contains carefully selected research papers addressing the arithmetic geometry of varieties which are not of general type, with an emphasis on how rational points are distributed with respect to the classical, Zariski and adelic topologies. The present volume gives a glimpse of the state of the art of this rapidly expanding domain in arithmetic geometry. The techniques involve explicit geometric constructions, ideas from the minimal model program in algebraic geometry as well as analytic number theory and harmonic analysis on adelic groups.

Author : Károly Jr. Böröczky,János Kollár,Szamuely Tamas
Publisher : Springer Science & Business Media
Page : 307 pages
File Size : 45,9 Mb
Release : 2013-12-11
Category : Mathematics
ISBN : 9783662051238

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Higher Dimensional Varieties and Rational Points by Károly Jr. Böröczky,János Kollár,Szamuely Tamas Pdf

Exploring the connections between arithmetic and geometric properties of algebraic varieties has been the object of much fruitful study for a long time, especially in the case of curves. The aim of the Summer School and Conference on "Higher Dimensional Varieties and Rational Points" held in Budapest, Hungary during September 2001 was to bring together students and experts from the arithmetic and geometric sides of algebraic geometry in order to get a better understanding of the current problems, interactions and advances in higher dimension. The lecture series and conference lectures assembled in this volume give a comprehensive introduction to students and researchers in algebraic geometry and in related fields to the main ideas of this rapidly developing area.

Author : Jorg Jahnel
Publisher : American Mathematical Soc.
Page : 280 pages
File Size : 47,9 Mb
Release : 2014-12-02
Category : Mathematics
ISBN : 9781470418823

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Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties by Jorg Jahnel Pdf

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.

Author : Gavril Farkas,Gerard van der Geer,Mingmin Shen,Lenny Taelman
Publisher : Springer Nature
Page : 433 pages
File Size : 55,7 Mb
Release : 2021-10-19
Category : Mathematics
ISBN : 9783030754211

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Rationality of Varieties by Gavril Farkas,Gerard van der Geer,Mingmin Shen,Lenny Taelman Pdf

This book provides an overview of the latest progress on rationality questions in algebraic geometry. It discusses new developments such as universal triviality of the Chow group of zero cycles, various aspects of stable birationality, cubic and Fano fourfolds, rationality of moduli spaces and birational invariants of group actions on varieties, contributed by the foremost experts in their fields. The question of whether an algebraic variety can be parametrized by rational functions of as many variables as its dimension has a long history and played an important role in the history of algebraic geometry. Recent developments in algebraic geometry have made this question again a focal point of research and formed the impetus to organize a conference in the series of conferences on the island of Schiermonnikoog. The book follows in the tradition of earlier volumes, which originated from conferences on the islands Texel and Schiermonnikoog.

Author : Carlo Gasbarri,Steven Lu,Mike Roth,Yuri Tschinkel
Publisher : American Mathematical Soc.
Page : 165 pages
File Size : 49,7 Mb
Release : 2015-12-22
Category : Algebraic geometry -- Arithmetic problems. Diophantine geometry -- Arithmetic problems. Diophantine geometry
ISBN : 9781470414580

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Rational Points, Rational Curves, and Entire Holomorphic Curves on Projective Varieties by Carlo Gasbarri,Steven Lu,Mike Roth,Yuri Tschinkel Pdf

This volume contains papers from the Short Thematic Program on Rational Points, Rational Curves, and Entire Holomorphic Curves and Algebraic Varieties, held from June 3-28, 2013, at the Centre de Recherches Mathématiques, Université de Montréal, Québec, Canada. The program was dedicated to the study of subtle interconnections between geometric and arithmetic properties of higher-dimensional algebraic varieties. The main areas of the program were, among others, proving density of rational points in Zariski or analytic topology on special varieties, understanding global geometric properties of rationally connected varieties, as well as connections between geometry and algebraic dynamics exploring new geometric techniques in Diophantine approximation. This book is co-published with the Centre de Recherches Mathématiques.

Author : Bjorn Poonen,Yuri Tschinkel
Publisher : Springer Science & Business Media
Page : 292 pages
File Size : 47,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9780817681708

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Arithmetic of Higher-Dimensional Algebraic Varieties by Bjorn Poonen,Yuri Tschinkel Pdf

This text offers a collection of survey and research papers by leading specialists in the field documenting the current understanding of higher dimensional varieties. Recently, it has become clear that ideas from many branches of mathematics can be successfully employed in the study of rational and integral points. This book will be very valuable for researchers from these various fields who have an interest in arithmetic applications, specialists in arithmetic geometry itself, and graduate students wishing to pursue research in this area.

Author : Jakob Stix
Publisher : Springer
Page : 257 pages
File Size : 48,7 Mb
Release : 2012-10-19
Category : Mathematics
ISBN : 9783642306747

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Rational Points and Arithmetic of Fundamental Groups by Jakob Stix Pdf

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.

Author : K. Böröczky,János Kollár,Tamás Szamuely
Publisher : Unknown
Page : 310 pages
File Size : 46,6 Mb
Release : 2003
Category : Algebraic varieties
ISBN : 9639453021

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Higher Dimensional Varieties and Rational Points by K. Böröczky,János Kollár,Tamás Szamuely Pdf

Author : Gerd Faltings,Gisbert Wüstholz
Publisher : Springer Science & Business Media
Page : 322 pages
File Size : 46,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783322803405

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Rational Points by Gerd Faltings,Gisbert Wüstholz Pdf

This book consists of the notes from the seminar Bonn/ Wuppertal 1983/ 84 on Arithmetic Geometry. It contains a proof for the Mordell conjecture and may be useful as an introduction to Arakelov's point of view in diophantine geometry. The third edition includes an appendix in which a detailed survey on the spectacular recent developments in arithmetic algebraic geometry is given. These beautiful new results have their roots in the material covered by this book.

Author : Robin Hartshorne
Publisher : Springer
Page : 271 pages
File Size : 44,5 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540363453

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Ample Subvarieties of Algebraic Varieties by Robin Hartshorne Pdf

Author : Carlo Gasbarri,Steven Lu,Mike Roth,Yuri Tschinkel
Publisher : Unknown
Page : 165 pages
File Size : 52,7 Mb
Release : 2015
Category : Algebraic varieties
ISBN : 1470428415

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Rational Points, Rational Curves, and Entire Holomorphic Curves on Projective Varieties by Carlo Gasbarri,Steven Lu,Mike Roth,Yuri Tschinkel Pdf

Author : Pietro Corvaja
Publisher : Springer
Page : 75 pages
File Size : 49,5 Mb
Release : 2016-11-23
Category : Mathematics
ISBN : 9789811026485

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Integral Points on Algebraic Varieties by Pietro Corvaja Pdf

This book is intended to be an introduction to Diophantine geometry. The central theme of the book is to investigate the distribution of integral points on algebraic varieties. This text rapidly introduces problems in Diophantine geometry, especially those involving integral points, assuming a geometrical perspective. It presents recent results not available in textbooks and also new viewpoints on classical material. In some instances, proofs have been replaced by a detailed analysis of particular cases, referring to the quoted papers for complete proofs. A central role is played by Siegel’s finiteness theorem for integral points on curves. The book ends with the analysis of integral points on surfaces.