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Torsors and rational points. (English) Zbl 0972.14015
Cambridge Tracts in Mathematics. 144. Cambridge: Cambridge University Press. viii, 187 p. (2001).
Contents: 1. Introduction;
Part one: Torsors:
2. Torsors: general theory; 3. Examples of torsors; 4. Abelian torsors;
Part two: Descent and Manin obstruction:
5. Obstructions over number fields; 6. Abelian descent and Manin obstruction; 7. Abelian descent on conic bundle surfaces; 8. Non-abelian descent on bielliptic surfaces; 9. Homogenous spaces and non-abelian cohomology; References; Index.
Let $$X$$ be a smooth projective variety defined over a number field $$k$$. One of the basic questions about rational points on $$X$$ is whether the so-called Hasse principle holds, i.e. whether the existence of points over each completion $$k_v$$ of $$k$$ (or equivalently, the nonvoidness of the set $$X({\mathbb{A}}_k)\neq 0$$ for $${\mathbb{A}}_k$$ the adele ring of $$k$$) implies $$X(k)\neq\emptyset$$. It has arisen already in the classical case of curves of genus 1 that this question can be attacked by studying torsors, or principal homogeneous spaces over $$X$$, for the following reason. Suppose $$f: Y\to X$$ is a torsor under a (linear algebraic) group $$G$$. Then of course any $$k$$-point of $$Y$$ projects onto a $$k$$-point of $$X$$, but even if the fibre $$Z=Y_P$$ over a point $$P\in X(k)$$ contains no $$k$$-point, there is a well-known twisting operation producing a torsor $$Y_Z\to X$$ under a “twisted” group $$G_Z$$ (equal to $$G$$ if $$G$$ is abelian) whose fibre over $$P$$ already contains one. This constitutes a link between the behaviour of rational points on $$X$$ and those on the $$Y_Z$$; in particular, the absence of rational, or even adelic points on the $$Y_Z$$ constitutes an obstruction to the Hasse principle on $$X$$, and in many cases this can be checked by a finite amount of computation.
In the already mentioned case of genus 1 curves $$G$$ is finite abelian, the torsors are the so-called $$n$$-coverings and one gets the classical theory of descent on elliptic curves.
By examining F. Châtelet’s classical work on the arithmetic of the surfaces named after him, J.-L. Colliot-Thélène and J.-J. Sansuc discovered towards the end of the 1970’s that in the case of rational varieties an equally fruitful theory can be developed by studying torsors under tori [cf. Duke Math. J. 54, 375-492 (1987; Zbl 0659.14028)]. Their descent theory was powerful enough to settle the question of the uniqueness of the so-called Manin obstruction in many cases. This obstruction is defined as follows: Define a pairing $$X({\mathbb{A}}_k)\times \text{Br}(X)\to {\mathbb{Q}}/{\mathbb{Z}}$$ (where $$\text{Br}(X)$$ is the cohomological Brauer group of $$X$$) by evaluating elements of $$\text{Br}(X)$$ at each component and then taking the sum of local invariants (which is known to be finite). By global class field theory, the diagonal image of $$X(k)$$ in $$X({\mathbb{A}}_k)$$ is then contained in the subset $$X({\mathbb{A}}_k)^{\text{Br}}$$ of adeles annihilated by the above pairing and thus the emptiness of $$X({\mathbb{A}}_k)^{\text{Br}}$$ is an obstruction to the Hasse principle if $$X({\mathbb{A}}_k)$$ itself is nonempty. The obstruction is said to be the only one if $$X({\mathbb{A}}_k)^{\text{Br}}=\emptyset$$ is equivalent to $$X(k)=\emptyset$$. In the case of a rational variety $$X$$, J.-L. Colliot-Thélène and J.-J. Sansuc were able to characterise the set $$X({\mathbb{A}}_k)^{\text{Br}}$$ as the set of adelic points coming from so-called universal torsors under the Néron-Severi torus of $$X$$, thereby reducing the uniqueness question of the Manin obstruction to the existence of universal torsors with adelic points, a question which is effectively solvable in many concrete situations.
Almost thirty years later, A. N. Skorobogator presented [Invent. Math. 135, 399-424 (1999; Zbl 0951.14013)] the first unconditional example of a variety (in fact a bielliptic surface) where the Manin obstruction does not suffice to explain the failure of the Hasse “principle; subsequently, D. Harari and the author developed [Non-abelian cohomology and rational points” (to appear)] a conceptual framework for explaining the new counter-example, relating it to torsors under non-abelian groups.
The book under review offers a clear and polished account of the most important results in the field, with emphasis on the contents of the three works cited above. The two opening chapters, which may be of independent interest, present a nice collection of basic facts and examples about torsors which can hardly be found together elsewhere. Then the main results of the theory of Colliot-Thélène and Sansuc are presented in an elegant and streamlined manner thanks to the insights offered by subsequent developments and also to the use of derived categories which simplifies several proofs. Afterwards, the general theory is applied to treat specific classes of varieties, such as smooth compactifications of tori, the first case successfully handled by Colliot-Thélène and Sansuc, as well as several types of conic bundle surfaces whose study has been pioneered by Swinnerton-Dyer. The book concludes by explaining some aspects of the theory of Harari and the author, with applications to the interpretation of the author’s counter-example mentioned above and of Borovoi’s work on the uniqueness of the Manin obstruction for homogeneous spaces under semisimple simply connected algebraic groups.
An attractive feature of the book is the healthy balance between the abstract and the concrete, in that the author does not refrain from using hard machinery for building up the conceptual framework but as a counterpoint works out several examples in detail. Thus, due to its reasonably self-contained nature and the careful choice of topics, the book provides an excellent account of the subject for the non-expert. As for the experts, they will probably learn less here since most of the material appeared in original sources in a rather similar form, but they will certainly appreciate disposing of a neat and handy reference and may gain stimulus for further applications of the descent method in diophantine geometry.

MSC:
 14G05 Rational points 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry 14G25 Global ground fields in algebraic geometry