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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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