**Graduate Studies in Mathematics**

Volume: 186;
2017;
337 pp;
Hardcover

MSC: Primary 14;
Secondary 11

Print ISBN: 978-1-4704-3773-2

Product Code: GSM/186

List Price: $83.00

AMS Member Price: $66.40

MAA Member Price: $74.70

**Electronic ISBN: 978-1-4704-4315-3
Product Code: GSM/186.E**

List Price: $83.00

AMS Member Price: $66.40

MAA Member Price: $74.70

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#### Supplemental Materials

# Rational Points on Varieties

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*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.

#### Readership

Graduate students and researchers interested in arithmetic geometry.

#### Reviews & Endorsements

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

#### Table of Contents

# Table of Contents

## Rational Points on Varieties

- Cover Cover11
- Title page iii4
- Contents v6
- Preface ix10
- Chapter 1. Fields 118
- Chapter 2. Varieties over arbitrary fields 3148
- Chapter 3. Properties of morphisms 5774
- Chapter 4. Faithfully flat descent 101118
- Chapter 5. Algebraic groups 119136
- 5.1. Group schemes 119136
- 5.2. Fppf group schemes over a field 125142
- 5.3. Affine algebraic groups 130147
- 5.4. Unipotent groups 131148
- 5.5. Tori 134151
- 5.6. Semisimple and reductive algebraic groups 136153
- 5.7. Abelian varieties 142159
- 5.8. Finite étale group schemes 150167
- 5.9. Classification of smooth algebraic groups 151168
- 5.10. Approximation theorems 154171
- 5.11. Inner twists 155172
- 5.12. Torsors 156173
- Exercises 164181

- Chapter 6. Étale and fppf cohomology 169186
- 6.1. The reasons for étale cohomology 169186
- 6.2. Grothendieck topologies 171188
- 6.3. Presheaves and sheaves 173190
- 6.4. Cohomology 178195
- 6.5. Torsors over an arbitrary base 182199
- 6.6. Brauer groups 190207
- 6.7. Spectral sequences 196213
- 6.8. Residue homomorphisms 200217
- 6.9. Examples of Brauer groups 203220
- Exercises 207224

- Chapter 7. The Weil conjectures 209226
- Chapter 8. Cohomological obstructions to rational points 231248
- Chapter 9. Surfaces 261278
- Appendix A. Universes 295312
- Appendix B. Other kinds of fields 299316
- Appendix C. Properties under base extension 303320
- Bibliography 311328
- Index 331348
- Back Cover Back Cover1358