Hasse principle and weak approximation on certain hypersurfaces (Q1814962)

The main object of the paper under review is an affine hypersurface \(V\) given by an equation of the form \( y^2-az^2=P(x_1,\dots ,x_{n-2}) \) where \(a\) is non-square and \(P\) is a polynomial of total degree at most \(4\). The ground field \(k\) is assumed to be a number field. The author is interested in the Hasse principle and weak approximation for a smooth projective model of \(V\) as above. To be more precise, the goal is to compute the Brauer-Manin obstruction which turned out to be the only one for this class of varieties [ cf. \textit{D. Harari}, Duke Math. J. 75, No. 1, 221-260 (1994; Zbl 0847.14001)]. The obtained results provide subclasses of varieties for which the Hasse principle and weak approximation hold as well as numerical counter-examples to both properties.