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Rational Points on Varieties GRADUATE STUDIES IN MATHEMATICS 186 Rational Points on Varieties Bjorn Poonen American Mathematical Society 10.1090/gsm/186 Rational Points on Varieties GRADUATE STUDIES IN MATHEMATICS 186 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 information and updates on this book, visit www.ams.org/bookpages/gsm-186 Library of Congress Cataloging-in-Publication Data Names:Poonen,Bjorn,author. Title: Rational points on varieties / Bjorn Poonen. Description: Providence, Rhode Island : American Mathematical Society, [2017] | Series: Gradu- ate studies in mathematics ; volume 186 | Includes bibliographical references and index. Identifiers: LCCN 2017022803 | ISBN 9781470437732 (alk. paper) Subjects: LCSH: Rational points (Geometry) | Algebraic varieties. | AMS: Algebraic geometry – Arithmetic problems. Diophantine geometry – Rational points. msc | Number theory – Arithmetic algebraic geometry (Diophantine geometry) – Varieties over global fields. msc Classification: LCC QA564 .P65 2017 | DDC 516.3/5–dc23 LC record available at https://lccn.loc.gov/2017022803 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2017 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 222120191817 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. Cr fields 3 §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. write an extended survey serving as an introduction to many topics, 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 x Preface 2008 and fall 2013 at the Massachusetts Institute of Technology. But it has grown since then; probably now it is about 50% too large for a single semester, unless students are willing to read much of it outside of class. 0.3. The nominal goal Many techniques have been used to decide whether a variety over a number field has a rational point. Some generalize Fermat’s method of infinite de- scent, some use quadratic reciprocity, and others appear at first sight to be ad hoc. But over the past few decades, it was discovered that nearly all of these techniques could be understood as applications of just two cohomolog- ical obstructions, the étale-Brauer obstruction and the descent obstruction. Moreover, while this book was being written, it was proved that the étale- Brauer obstruction and the descent obstruction are equivalent! The topics in this book build up to an explanation of this “grand unified theory” of obstructions. 0.4. The true goal Our ulterior motive, however, is to introduce readers to techniques that they are likely to need while researching arithmetic geometry more broadly. Along the way, we mention open problems and applications that are interesting in their own right. 0.5. The content Chapter 1 introduces fields of special interest to arithmetic geometers, and it discusses properties and invariants (the Cr property, cohomological dimen- sion, and the Brauer group) that control the answers to some arithmetic questions about fields in general. Not all of Chapter 1 is needed in future chapters, but the Brauer group plays a key role later on (in Chapters 6 and 8). Chapter 2 discusses aspects of varieties with particular attention to the case of ground fields that are not algebraically closed. Ultimately, we aim to treat global fields of positive characteristic as well as number fields, so we do not require our ground field to be perfect. Among other topics, this chapter discusses properties of varieties under base extension (e.g., irreducible vs. geometrically irreducible), the functor of points of a scheme, closed points and their relation to field-valued points, and genus change of curves under field extension. A final section introduces the main questions about rational points that motivate the subject, such as the questions of whether the local- global principle and weak approximation hold.
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