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The elementary obstruction and homogeneous spaces. (English) Zbl 1135.14013

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 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).

MSC:

14G05 Rational points
11E72 Galois cohomology of linear algebraic groups
14F22 Brauer groups of schemes
14K15 Arithmetic ground fields for abelian varieties
14M17 Homogeneous spaces and generalizations
20G99 Linear algebraic groups and related topics
11G99 Arithmetic algebraic geometry (Diophantine geometry)
12G05 Galois cohomology

Citations:

Zbl 0659.14028
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References:

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