An error was encountered while trying to add the item to the cart. Please try again.
Copy To Clipboard
Successfully Copied!
Rational Points on Varieties

Bjorn Poonen Massachusetts Institute of Technology, Cambridge, MA
Available Formats:
Hardcover ISBN: 978-1-4704-3773-2
Product Code: GSM/186
List Price: $83.00 MAA Member Price:$74.70
AMS Member Price: $66.40 Electronic ISBN: 978-1-4704-4315-3 Product Code: GSM/186.E List Price:$83.00
MAA Member Price: $74.70 AMS Member Price:$66.40
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $124.50 MAA Member Price:$112.05
AMS Member Price: $99.60 Click above image for expanded view Rational Points on Varieties Bjorn Poonen Massachusetts Institute of Technology, Cambridge, MA Available Formats:  Hardcover ISBN: 978-1-4704-3773-2 Product Code: GSM/186  List Price:$83.00 MAA Member Price: $74.70 AMS Member Price:$66.40
 Electronic ISBN: 978-1-4704-4315-3 Product Code: GSM/186.E
 List Price: $83.00 MAA Member Price:$74.70 AMS Member Price: $66.40 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$124.50 MAA Member Price: $112.05 AMS Member Price:$99.60
• Book Details

Volume: 1862017; 337 pp
MSC: Primary 14; Secondary 11;

2023 Joseph L. Doob Prize Winner

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.

Graduate students and researchers interested in arithmetic geometry.

• Chapters
• Fields
• Varieties over arbitrary fields
• Properties of morphisms
• Faithfully flat descent
• Algebraic groups
• Étale and fppf cohomology
• The Weil conjecture
• Cohomological obstructions to rational points
• Surfaces
• Universes
• Other kinds of fields
• Properties under base extension

• Reviews

• The reviewer cannot emphasize enough how brilliant and necessary this book is. It will be a great reference text for researchers and essential reading for graduate students in arithmetic geometry for many years to come. I will certainly be recommending that all my Ph.D. students study it in great detail.

Daniel Loughran, Mathematical Reviews
• A monograph/textbook whose main goals are to introduce the interested reader to the methods and problems of arithmetic geometry and at the same time discuss open problems of interest for further research is therefore a most welcome addition to a classical subject..The choice of topics and the decisions on what to spell out and what to just barely sketch, with adequate pointers to the existing literature, make the book under review an excellent quick introduction and reference on this subject..The book is well structured, balancing explicit constructions, terse arguments, and precise references to the literature when needed.

Felipe Zaldivar, MAA Reviews
• 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
• Requests

Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 1862017; 337 pp
MSC: Primary 14; Secondary 11;

2023 Joseph L. Doob Prize Winner

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.

Graduate students and researchers interested in arithmetic geometry.

• Chapters
• Fields
• Varieties over arbitrary fields
• Properties of morphisms
• Faithfully flat descent
• Algebraic groups
• Étale and fppf cohomology
• The Weil conjecture
• Cohomological obstructions to rational points
• Surfaces
• Universes
• Other kinds of fields
• Properties under base extension
• The reviewer cannot emphasize enough how brilliant and necessary this book is. It will be a great reference text for researchers and essential reading for graduate students in arithmetic geometry for many years to come. I will certainly be recommending that all my Ph.D. students study it in great detail.

Daniel Loughran, Mathematical Reviews
• A monograph/textbook whose main goals are to introduce the interested reader to the methods and problems of arithmetic geometry and at the same time discuss open problems of interest for further research is therefore a most welcome addition to a classical subject..The choice of topics and the decisions on what to spell out and what to just barely sketch, with adequate pointers to the existing literature, make the book under review an excellent quick introduction and reference on this subject..The book is well structured, balancing explicit constructions, terse arguments, and precise references to the literature when needed.

Felipe Zaldivar, MAA Reviews
• 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
Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
You may be interested in...
Please select which format for which you are requesting permissions.