On the Hasse principle for homogeneous spaces with finite stabilizers (Q1383415)

Let \(X\) be an algebraic variety over a number field \(k\). The variety \(X\) is called a counter example to the Hasse principle if \(X\) has a \(k_v\)-point for any completion \(k_v\) of \(k\) but has no \(k\)-points. \textit{Yu. I. Manin} [in: Actes Congr. Int. Math., Nice 1970, No. 1, 401-411 (1971; Zbl 0239.14010) and in his book: ``Cubic Forms: Algebra, Geometry, Arithmetic'' (1972; Zbl 0255.14002; see also the English translation thereof: 1974; Zbl 0277.14014)] proposed a general method of obtaining obstructions to the Hasse principle with the use of the Brauer group of \(X\). Manin's method gives an obstruction for \(X\) to have a \(k\)-point. The authors of this paper give another counter example to the Hasse principle.