Geodesic
Existence of rational points on smooth projective varieties
Journal of the European Mathematical Society, Tome 11 (2009) no. 3, pp. 529-543

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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.
DOI : 10.4171/jems/159
Classification : 14-XX, 11-XX, 00-XX
Keywords: Brauer–Manin obstruction, Hasse principle, Châtelet surface, conic bundle, rational points 
Bjorn Poonen. Existence of rational points on smooth projective varieties. Journal of the European Mathematical Society, Tome 11 (2009) no. 3, pp. 529-543. doi: 10.4171/jems/159
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     author = {Bjorn Poonen},
     title = {Existence of rational points on smooth projective varieties},
     journal = {Journal of the European Mathematical Society},
     pages = {529--543},
     year = {2009},
     volume = {11},
     number = {3},
     doi = {10.4171/jems/159},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/159/}
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