Rational points on varieties.

*(English)*Zbl 1387.14004
Graduate Studies in Mathematics 186. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-3773-2/hbk; 978-1-4704-4315-3/ebook). xv, 337 p. (2017).

Bjorn Poonen has written a textbook that takes readers to the frontiers of research in a very active field where the author himself is a prominent contributor. His main topic is the existence of rational points on algebraic varieties and local-global principles for them. These questions have a long history and the methods go back to such classics as Fermat’s infinite descent and Hasse’s local-global principle for quadratic forms. The modern development of the subject started with work of Manin in the early 1970’s. Manin introduced a cohomological obstruction that explained the failure of the Hasse principle (i.e. the existence of varieties over a number field with points over every completion but no rational point) in all cases that were known at the time – it was only in 1998 that Skorobogatov found the first example where the failure of the Hasse principle was unaccounted for by Manin’s obstruction.

These ideas are discussed in Chapter 8 of Poonen’s book where the Manin obstruction to the Hasse principle and to weak approximation is presented together with its later refinements. The most general of these is the descent obstruction of Harari and Skorobogatov which is a non-commutative extension of fundamental earlier work by Colliot-Thélène and Sansuc. We owe to the latter authors the crucial discovery of a link between Manin’s obstruction and the descent method in Diophantine geometry which until then had only been applied to finite Galois coverings. Poonen gives beautiful worked-out examples for both the Manin obstruction and the descent method, and presents in detail a counter-example to the Hasse principle that is not explained by the Manin obstruction nor, in fact, by the more general descent obstruction.

The core of Chapter 9 is devoted to del Pezzo surfaces that are a rich source of examples where the methods in the subject can be successfully applied. After a thorough survey of relevant algebro-geometric topics, Poonen discusses the validity of the Hasse principle and weak approximation for each possible degree.

The earlier chapters of the book are devoted to background material needed for the constructions of Chapter 8. Among the topics treated here we find descent theory, group schemes and torsors under them, Brauer groups and methods of étale cohomology. An important feature throughout these chapters is the attention paid to global fields of positive characteristic which are often left in the shade in favour of number fields. This is particularly noteworthy in the chapter on group schemes where recent work of Conrad, Gabber and Prasad gets its due share.

There is also a chapter devoted to basic scheme-theoretic constructions that are often used in arithmetic geometry, such as spreading out techniques. This part is very useful for non-experts as it gives a rigorous and detailed treatment of techniques that are usually labelled ‘standard arguments’ in research papers. More dispensable is the chapter surveying the Weil conjectures (proven by Grothendieck and Deligne) and the (still unproven) Tate conjecture. These subjects are a bit off-topic in the present book, and a reasonably thorough treatment would require a book of its own. Nevertheless, the reader is given a nice quick introduction.

Poonen’s exposition is atypical for an introductory textbook: as he himself writes, it is closer in style to an extended survey. Thus the reader should not expect complete and self-contained proofs for most of the results; their inclusion would have at least tripled the book’s current size. However, there is no handwaving either. Concepts are always introduced clearly and rigorously, followed by key examples and counterexamples. The more accessible statements are then given with proof, but the reader is often directed to some of the best available references in the literature instead. It should be emphasized that all concepts are presented in the way they occur in present-day research, even in cases where more elementary approaches would have been possible. This entails that the prerequisites for reading the book are rather on the high side: the reader should have a solid working knowledge of basic algebraic number theory and algebraic geometry including scheme theory, but some familiarity with more advanced topics such as Galois cohomology is also required. In return, the author always presents what is nowadays considered the ‘right point of view’, and he is never lost in arid technicalities. In so doing, he has done a great service to the community and his book will be much appreciated by those wishing to enter this fascinating field of research in arithmetic.

These ideas are discussed in Chapter 8 of Poonen’s book where the Manin obstruction to the Hasse principle and to weak approximation is presented together with its later refinements. The most general of these is the descent obstruction of Harari and Skorobogatov which is a non-commutative extension of fundamental earlier work by Colliot-Thélène and Sansuc. We owe to the latter authors the crucial discovery of a link between Manin’s obstruction and the descent method in Diophantine geometry which until then had only been applied to finite Galois coverings. Poonen gives beautiful worked-out examples for both the Manin obstruction and the descent method, and presents in detail a counter-example to the Hasse principle that is not explained by the Manin obstruction nor, in fact, by the more general descent obstruction.

The core of Chapter 9 is devoted to del Pezzo surfaces that are a rich source of examples where the methods in the subject can be successfully applied. After a thorough survey of relevant algebro-geometric topics, Poonen discusses the validity of the Hasse principle and weak approximation for each possible degree.

The earlier chapters of the book are devoted to background material needed for the constructions of Chapter 8. Among the topics treated here we find descent theory, group schemes and torsors under them, Brauer groups and methods of étale cohomology. An important feature throughout these chapters is the attention paid to global fields of positive characteristic which are often left in the shade in favour of number fields. This is particularly noteworthy in the chapter on group schemes where recent work of Conrad, Gabber and Prasad gets its due share.

There is also a chapter devoted to basic scheme-theoretic constructions that are often used in arithmetic geometry, such as spreading out techniques. This part is very useful for non-experts as it gives a rigorous and detailed treatment of techniques that are usually labelled ‘standard arguments’ in research papers. More dispensable is the chapter surveying the Weil conjectures (proven by Grothendieck and Deligne) and the (still unproven) Tate conjecture. These subjects are a bit off-topic in the present book, and a reasonably thorough treatment would require a book of its own. Nevertheless, the reader is given a nice quick introduction.

Poonen’s exposition is atypical for an introductory textbook: as he himself writes, it is closer in style to an extended survey. Thus the reader should not expect complete and self-contained proofs for most of the results; their inclusion would have at least tripled the book’s current size. However, there is no handwaving either. Concepts are always introduced clearly and rigorously, followed by key examples and counterexamples. The more accessible statements are then given with proof, but the reader is often directed to some of the best available references in the literature instead. It should be emphasized that all concepts are presented in the way they occur in present-day research, even in cases where more elementary approaches would have been possible. This entails that the prerequisites for reading the book are rather on the high side: the reader should have a solid working knowledge of basic algebraic number theory and algebraic geometry including scheme theory, but some familiarity with more advanced topics such as Galois cohomology is also required. In return, the author always presents what is nowadays considered the ‘right point of view’, and he is never lost in arid technicalities. In so doing, he has done a great service to the community and his book will be much appreciated by those wishing to enter this fascinating field of research in arithmetic.

Reviewer: Tamás Szamuely (Budapest)