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Descent obstruction is equivalent to étale Brauer-Manin obstruction. (English) Zbl 1180.14017
The author completes the proof of the result stated in the title.
More precisely, for a smooth projective geometrically integral \(k\)-variety \(X\) defined over a number field \(k\) and a torsor \(f: Y\to X\) under a linear \(k\)-group \(G\) denote \[ X(\mathbb A_k)^f = \bigcup_{[\sigma ]\in H^1(k,G)} f^{\sigma }(Y^{\sigma }(\mathbb A_k)), \] where \(f^{\sigma}: Y^{\sigma }\to X\) is the twist of \(f\) by the 1-cocycle \(\sigma\) and \(\mathbb A_k\) stands for the ring of adèles of \(k\). Then the descent obstruction \(X(\mathbb A_k)^{\text{{desc}}}\) is defined as the intersection of \(X(\mathbb A_k)^f\) where \(f\) ranges over all torsors under all linear \(k\)-groups.
The author’s goal is to compare \(X(\mathbb A_k)^{\text{{desc}}}\) with the étale Brauer–Manin obstruction, defined as \[ X(\mathbb A_k)^{\text{{ét, Br}}} = \bigcap_f\bigcup_{[\sigma ]\in H^1(k,F)} f^{\sigma }\left(Y^{\sigma }(\mathbb A_k)^{\text{{Br}}}\right), \] where \(f\) ranges over all torsors under all finite \(k\)-groups.
The question is motivated by the fact that the étale Brauer–Manin obstruction explains the counter-example to the Hasse principle constructed by the author [Invent. Math. 135, No. 2, 399–424 (1999; Zbl 0951.14013)] but does not explain the counter-example constructed by B. Poonen [Insufficiency of the Brauer–Manin obstruction applied to étale covers, to appear in Ann. Math., cf. arXiv:0806.1312]. In the above cited paper Poonen asked whether in his counter-example the descent obstruction is empty or not, and, more generally, whether one always has the inclusion \[ X(\mathbb A_k)^{\text{ét, Br}} \subset X(\mathbb A_k)^{\text{desc}}. \] The latter inclusion was proved by C. Demarche [Algebra and Number Theory 3, No. 2, 237–254 (2009; Zbl 1247.11090)]. In the paper under review the opposite inclusion is proved. The key result is the following formula for the descent obstruction (Theorem 1.1): if \(f: Y\to X\) is a torsor under a finite \(k\)-group scheme \(F\), then \[ X(\mathbb A_k)^{\text{{desc}}} = \bigcup_{[\sigma ]\in H^1(k,F)} f^{\sigma }\left(Y^{\sigma }(\mathbb A_k)^{\text{{desc}}}\right). \] (Note that a similar property does not hold for the Brauer–Manin set \(X(\mathbb A_k)^{\text{{desc}}}\): this is what happens in the above mentioned counter-example to the Hasse principle due to the author.)
The proof of Theorem 1.1 is based on a method of M. Stoll [Algebra and Number Theory 1, No. 4, 349–391 (2007; Zbl 1167.11024)] where a similar formula has been established for another variant of the descent obstruction, in which the intersection is taken over \(f\) ranging over all torsors under all finite \(k\)-groups.
As an application, the author presents some results for surfaces of Kodaira dimension zero. In particular, he proves the following conditional statement (Corollary 3.3): if the Brauer–Manin obstruction is the only obstruction to weak approximation on \(K3\) surfaces, then the descent obstruction is the only obstruction to weak approximation on Enriques surfaces.

14G05 Rational points
14G25 Global ground fields in algebraic geometry
14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations
14J28 \(K3\) surfaces and Enriques surfaces
14L30 Group actions on varieties or schemes (quotients)
Full Text: DOI
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