Fix a number field k. We prove that if there is an algorithm for deciding whether a smooth projective geometrically integral k-variety has a k-point, then there is an algorithm for deciding whether an arbitrary k-variety has a k-point and also an algorithm for computing X(k) for any k-variety X for which X(k) is finite. The proof involves the construction of a one-parameter algebraic family of Châtelet surfaces such that exactly one of the surfaces fails to have a k-point.
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Bjorn Poonen, Existence of rational points on smooth projective varieties. J. Eur. Math. Soc. 11 (2009), no. 3, pp. 529–543DOI 10.4171/JEMS/159