Rational Points on Varieties (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 186)

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 Cover Title page Contents Preface 0.1. Prerequisites 0.2. What kind of book this is 0.3. The nominal goal 0.4. The true goal 0.5. The content 0.6. Anything new in this book? 0.7. Standard notation 0.8. Acknowledgments Chapter 1. Fields 1.1. Some fields arising in classical number theory 1.2. Cr fields 1.3. Galois theory 1.4. Cohomological dimension 1.5. Brauer groups of fields Exercises Chapter 2. Varieties over arbitrary fields 2.1. Varieties 2.2. Base extension 2.3. Scheme-valued points 2.4. Closed points 2.5. Curves 2.6. Rational points over special fields Exercises Chapter 3. Properties of morphisms 3.1. Finiteness conditions 3.2. Spreading out 3.3. Flat morphisms 3.4. Fppf and fpqc morphisms 3.5. Smooth and étale morphisms 3.6. Rational maps 3.7. Frobenius morphisms 3.8. Comparisons Exercises Chapter 4. Faithfully flat descent 4.1. Motivation: Gluing sheaves 4.2. Faithfully flat descent for quasi-coherent sheaves 4.3. Faithfully flat descent for schemes 4.4. Galois descent 4.5. Twists 4.6. Restriction of scalars Exercises Chapter 5. Algebraic groups 5.1. Group schemes 5.2. Fppf group schemes over a field 5.3. Affine algebraic groups 5.4. Unipotent groups 5.5. Tori 5.6. Semisimple and reductive algebraic groups 5.7. Abelian varieties 5.8. Finite étale group schemes 5.9. Classification of smooth algebraic groups 5.10. Approximation theorems 5.11. Inner twists 5.12. Torsors Exercises Chapter 6. Étale and fppf cohomology 6.1. The reasons for étale cohomology 6.2. Grothendieck topologies 6.3. Presheaves and sheaves 6.4. Cohomology 6.5. Torsors over an arbitrary base 6.6. Brauer groups 6.7. Spectral sequences 6.8. Residue homomorphisms 6.9. Examples of Brauer groups Exercises Chapter 7. The Weil conjectures 7.1. Statements 7.2. The case of curves 7.3. Zeta functions 7.4. The Weil conjectures in terms of zeta functions 7.5. Cohomological explanation 7.6. Cycle class homomorphism 7.7. Applications to varieties over global fields Exercises Chapter 8. Cohomological obstructions to rational points 8.1. Obstructions from functors 8.2. The Brauer–Manin obstruction 8.3. An example of descent 8.4. Descent 8.5. Comparing the descent and Brauer–Manin obstructions 8.6. Insufficiency of the obstructions Exercises Chapter 9. Surfaces 9.1. Kodaira dimension 9.2. Varieties that are close to being rational 9.3. Classification of surfaces 9.4. Del Pezzo surfaces 9.5. Rational points on varieties of general type Exercises Appendix A. Universes A.1. Definition of universe A.2. The universe axiom A.3. Strongly inaccessible cardinals A.4. Universes and categories A.5. Avoiding universes Exercises Appendix B. Other kinds of fields B.1. Higher-dimensional local fields B.2. Formally real and real closed fields B.3. Henselian fields B.4. Hilbertian fields B.5. Pseudo-algebraically closed fields Exercises Appendix C. Properties under base extension C.1. Morphisms C.2. Varieties C.3. Algebraic groups Bibliography Index Back Cover 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, etale 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. 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, etale 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