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The reciprocity obstruction to the existence of rational points for certain varieties on the function field of a real curve. (L’obstruction de réciprocité à l’existence de points rationnels pour certaines variétés sur le corps des fonctions d’une courbe réelle.) (English) Zbl 0934.14012

Let \(K\) be the function field of an integral, smooth, projective curve \(C\), defined over a real closed field \(R\). For any closed point \(P\) of \(C\), let \(K_P\) denote the corresponding completion. A family \({\mathcal F}\) of integral, proper, smooth \(K\)-varieties is said to satisfy the Hasse principle if any variety \(Y\) belonging to \({\mathcal F}\) and such that \(\prod_P X(K_P)\) is not empty has a \(K\)-point. In this article, one defines a “reciprocity” obstruction to the Hasse principle and one gives a geometrical interpretation of this obstruction. Using this interpretation, some deep results of Scheiderer, the descente theory of Colliot-Thélène and Sansuc and the Stone-Weierstraß approximation theorem, one shows that for conic bundles over the projective line, the reciprocity obstruction to the Hasse principle is the only one when \(R\) is archimedean. This last restriction is crucial; for any non-archimedean \(R\), a counterexample is given.

MSC:

14G05 Rational points
14H05 Algebraic functions and function fields in algebraic geometry
11R58 Arithmetic theory of algebraic function fields
14P99 Real algebraic and real-analytic geometry
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