Descent obstruction is equivalent to étale Brauer-Manin obstruction
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 \textit{B.~Poonen} [Insufficiency of the Brauer--Manin obstruction applied to étale covers, to appear in Ann. Math., cf. \url{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 \textit{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 \textit{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 \textit{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.
scientific article; zbMATH DE number 5565277