A cohomological obstruction to the Hasse principle for homogeneous spaces (Q1298166)

Let \(G\) be a connected algebraic group defined over a number field \(k\), and let \(X\) be a homogeneous space of \(G\) defined over \(k\), whose stabiliser is either connected or Abelian. A few years ago, the author proved [see \textit{M. Borovoi}, J. Reine Angew. Math. 473, 181-194 (1996; Zbl 0844.14020)] that the Brauer-Manin obstruction is the only obstruction to the Hasse principle for \(X\). In this paper, the Brauer-Manin obstruction is computed in terms of the Galois hypercohomology of \(G\) with coefficients in the complex of Abelian \(k\)-groups.