Rational Points on Varieties

358 Pages·2017·3.823 MB·English

by  Bjorn Poonen

GRADUATE STUDIES 186
IN MATHEMATICS
Rational
Points on
Varieties
Bjorn Poonen
American Mathematical Society
10.1090/gsm/186
Rational
Points on
Varieties

GRADUATE STUDIES 186
IN MATHEMATICS
Rational
Points on
Varieties
Bjorn Poonen
American Mathematical Society
Providence, Rhode Island
EDITORIAL COMMITTEE
Dan Abramovich
Daniel S. Freed (Chair)
Gigliola Staffilani
Jeff A. Viaclovsky
2010 Mathematics Subject Classification. Primary 14G05; Secondary 11G35.
For additional informationand updates on this book, visit
www.ams.org/bookpages/gsm-186
Library of Congress Cataloging-in-Publication Data
Names: Poonen,Bjorn,author.
Title: Rationalpointsonvarieties/BjornPoonen.
Description: Providence,RhodeIsland: AmericanMathematicalSociety,[2017]|Series: Gradu-
atestudiesinmathematics;volume186|Includesbibliographicalreferencesandindex.
Identifiers: LCCN 2017022803 | ISBN 9781470437732 (alk. paper) Subjects: LCSH: Rational
points (Geometry) | Algebraic varieties. | AMS: Algebraic geometry – Arithmetic problems.
Diophantinegeometry–Rationalpoints. msc|Numbertheory–Arithmeticalgebraicgeometry
(Diophantinegeometry)–Varietiesoverglobalfields. msc
Classification: LCCQA564.P652017|DDC516.3/5–dc23
LCrecordavailableathttps://lccn.loc.gov/2017022803
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Contents
Preface ix
§0.1. Prerequisites ix
§0.2. What kind of book this is ix
§0.3. The nominal goal x
§0.4. The true goal x
§0.5. The content x
§0.6. Anything new in this book? xii
§0.7. Standard notation xiii
§0.8. Acknowledgments xiv
Chapter 1. Fields 1
§1.1. Some fields arising in classical number theory 1
§1.2. C fields 3
r
§1.3. Galois theory 8
§1.4. Cohomological dimension 12
§1.5. Brauer groups of fields 16
Exercises 27
Chapter 2. Varieties over arbitrary fields 31
§2.1. Varieties 32
§2.2. Base extension 32
§2.3. Scheme-valued points 37
§2.4. Closed points 45
§2.5. Curves 47
v
vi Contents
§2.6. Rational points over special fields 49
Exercises 53
Chapter 3. Properties of morphisms 57
§3.1. Finiteness conditions 57
§3.2. Spreading out 60
§3.3. Flat morphisms 66
§3.4. Fppf and fpqc morphisms 68
§3.5. Smooth and étale morphisms 69
§3.6. Rational maps 92
§3.7. Frobenius morphisms 95
§3.8. Comparisons 97
Exercises 97
Chapter 4. Faithfully flat descent 101
§4.1. Motivation: Gluing sheaves 101
§4.2. Faithfully flat descent for quasi-coherent sheaves 103
§4.3. Faithfully flat descent for schemes 104
§4.4. Galois descent 106
§4.5. Twists 109
§4.6. Restriction of scalars 114
Exercises 116
Chapter 5. Algebraic groups 119
§5.1. Group schemes 119
§5.2. Fppf group schemes over a field 125
§5.3. Affine algebraic groups 130
§5.4. Unipotent groups 131
§5.5. Tori 134
§5.6. Semisimple and reductive algebraic groups 136
§5.7. Abelian varieties 142
§5.8. Finite étale group schemes 150
§5.9. Classification of smooth algebraic groups 151
§5.10. Approximation theorems 154
§5.11. Inner twists 155
§5.12. Torsors 156
Exercises 164
Contents vii
Chapter 6. Étale and fppf cohomology 169
§6.1. The reasons for étale cohomology 169
§6.2. Grothendieck topologies 171
§6.3. Presheaves and sheaves 173
§6.4. Cohomology 178
§6.5. Torsors over an arbitrary base 182
§6.6. Brauer groups 190
§6.7. Spectral sequences 196
§6.8. Residue homomorphisms 200
§6.9. Examples of Brauer groups 203
Exercises 207
Chapter 7. The Weil conjectures 209
§7.1. Statements 209
§7.2. The case of curves 210
§7.3. Zeta functions 211
§7.4. The Weil conjectures in terms of zeta functions 213
§7.5. Cohomological explanation 214
§7.6. Cycle class homomorphism 222
§7.7. Applications to varieties over global fields 226
Exercises 229
Chapter 8. Cohomological obstructions to rational points 231
§8.1. Obstructions from functors 231
§8.2. The Brauer–Manin obstruction 233
§8.3. An example of descent 240
§8.4. Descent 244
§8.5. Comparing the descent and Brauer–Manin obstructions 250
§8.6. Insufficiency of the obstructions 253
Exercises 258
Chapter 9. Surfaces 261
§9.1. Kodaira dimension 261
§9.2. Varieties that are close to being rational 263
§9.3. Classification of surfaces 271
§9.4. Del Pezzo surfaces 279
§9.5. Rational points on varieties of general type 290
viii Contents
Exercises 293
Appendix A. Universes 295
§A.1. Definition of universe 295
§A.2. The universe axiom 296
§A.3. Strongly inaccessible cardinals 296
§A.4. Universes and categories 297
§A.5. Avoiding universes 297
Exercises 298
Appendix B. Other kinds of fields 299
§B.1. Higher-dimensional local fields 299
§B.2. Formally real and real closed fields 299
§B.3. Henselian fields 300
§B.4. Hilbertian fields 301
§B.5. Pseudo-algebraically closed fields 302
Exercises 302
Appendix C. Properties under base extension 303
§C.1. Morphisms 303
§C.2. Varieties 308
§C.3. Algebraic groups 308
Bibliography 311
Index 331
Preface
0.1. Prerequisites
A person interested in reading this book should have the following back-
ground:
• Algebraic geometry (e.g., [Har77]: up to Chapter II, §8 as a mini-
mum, but familiarity with later chapters is also needed at times)—
this is not needed so much in our Chapter 1.
• Algebraic number theory (e.g., [Cas67,Frö67] or [Lan94, Part
One] or [Neu99, Chapters I and II]).
• Some group cohomology (e.g., [AW67] or [Mil13, Chapter 2]).
0.2. What kind of book this is
The literature on rational points is vast. To write a book on the subject, an
author must
1. write thousands of pages to cover all the topics comprehensively, or
2. focus on one aspect of the subject, or
3. writeanextended surveyservingasanintroduction tomanytopics,
with pointers to the literature for those who want to learn more
about any particular one.
Our approach is closest to 3, so as to bring newcomers quickly up to speed
while also providing more experienced researchers with directions for further
exploration.
This book originated as the lecture notes for a semester-long course,
taught during spring 2003 at the University of California, Berkeley, and fall
ix