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Non-abelian cohomology and rational points. (English) Zbl 1019.14012
The main object of the paper under review is a (right) torsor $$f: Y\to X$$ under an algebraic group $$G$$ (possibly non-abelian and disconnected). For a cocycle $$\sigma\in Z^1(k,G)$$ one can define the twisted torsor $$f^{\sigma }: Y^{\sigma }\to X$$ under the inner form $$G^{\sigma }$$. If $$k$$ is a number field, the authors define $X(A_k)^f=\bigcup_{[\sigma ]\in H^1(k,G)}f^{\sigma }(Y^{\sigma }(A_k))$ (here $$A_k$$ is the ring of adèles of $$k$$). Since $$X(k)\subset X(A_k)^f\subset X(A_k)$$, the emptiness of $$X(A_k)^f$$ is an obstruction to the existence of a $$k$$-point on $$X$$. This obstruction is called the descent obstruction defined by the torsor $$f: Y\to X$$. This new obstruction turns out to be a refinement of the classical Manin obstruction and explains a counter-example to the Hasse principle not accounted for by the Manin obstruction [A. N. Skorobogatov, Invent. Math. 135, 399-424 (1999; Zbl 0951.14013)]. It also explains “transcendental” Manin obstructions [D. Harari, Sém. Théorie des Nombres de Paris 1993-1994, Lond. Math. Soc. Lect. Note Ser. 235, 75-87 (1996; Zbl 0926.14009)]. Along similar lines, the authors define the descent obstruction to weak approximation which, in particular, explains counter-examples related to non-abelian fundamental groups [D. Harari, Ann. Sci. Éc. Norm. Sup. (4) 33, 467-484 (2000; Zbl 1073.14522)].

##### MSC:
 14G05 Rational points 14M17 Homogeneous spaces and generalizations 11G35 Varieties over global fields 14G25 Global ground fields in algebraic geometry 14F22 Brauer groups of schemes
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