Rational Points on Varieties

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. ?ᵣ 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