Geodesic
Existence of rational points on smooth projective varieties
Journal of the European Mathematical Society, Tome 11 (2009) no. 3, pp. 529-543 Cet article a éte moissonné depuis la source EMS Press

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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 
@article{JEMS_2009_11_3_a3,
     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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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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