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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.

MSC:
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)
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