The elementary obstruction and homogeneous spaces (Q2472321)

The main object of the paper under review is the elementary obstruction \(ob(X)\) for the existence of a (smooth) rational point on a geometrically integral variety \(X\) defined over a field \(k\) of characteristic zero. This obstruction, first introduced and studied by \textit{J.-L.~Colliot-Thélène and J.-J.~Sansuc} [Duke Math. J. 54, 375--492 (1987; Zbl 0659.14028)], is defined as the class in \({\text{Ext}}^1_{\mathfrak g}(\bar k(X)^*/\bar k^*, \bar k^*)\) of the extension of \(\mathfrak g\)-modules \(1\to \bar k^*\to \bar k(X)^* \to \bar k(X)^*/\bar k^*\to 1\) (here \(\mathfrak g\) stands for the absolute Galois group of \(k\)). The authors' goal is to determine for which fields \(k\) and \(k\)-varieties \(X\) this obstruction is the only one. This turns out to be true for any homogeneous space \(X\) of a connected algebraic group \(G\) (with connected geometric stabilizer) in the following cases: \(k\) is a \(p\)-adic field; or (provided \(G\) is linear) a ``good'' field of cohomological dimension 2; or a number field (provided \(G\) is linear and \(X\) has points in the real completions of \(k\)); or a totally imaginary number field (assuming finiteness of the Tate--Shafarevich group of the maximal abelian variety quotient of \(G\)). They show that some of these results are sharp by exhibiting a striking example of a \(G\)-torsor \(X\) defined over \(\mathbb Q\) with vanishing elementary obstruction which has points everywhere locally but has no \(\mathbb Q\)-points (in their construction \(G\) is an extension of an elliptic curve by \(SL_1(D)\) where \(D\) denotes the Hamilton quaternions).