Insufficiency of the Brauer-Manin obstruction applied to étale covers. (English) Zbl 1284.11096

Poonen, Bjorn

Ann. Math. (2) 171, No. 3, 2157-2169 (2010).

Summary: Let \(k\) be any global field of characteristic not 2. We construct a \(k\)-variety \(X\) such that \(X(k)\) is empty, but for which the emptiness cannot be explained by the Brauer-Manin obstruction or even by the Brauer-Manin obstruction applied to finite étale covers.

MSC:

11G35	Varieties over global fields
14G05	Rational points
14G25	Global ground fields in algebraic geometry
14J20	Arithmetic ground fields for surfaces or higher-dimensional varieties

Keywords:

rational points; Hasse principle; Brauer-Manin obstruction; descent obstruction; Châtelet surface; conic bundle

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