Journal of the European Mathematical Society


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Volume 11, Issue 3, 2009, pp. 529–543
DOI: 10.4171/JEMS/159

Published online: 2009-06-30

Existence of rational points on smooth projective varieties

Bjorn Poonen[1]

(1) Massachusetts Institute of Technology, Cambridge, United States

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.

Keywords: Brauer–Manin obstruction, Hasse principle, Châtelet surface, conic bundle, rational points

Poonen Bjorn: Existence of rational points on smooth projective varieties. J. Eur. Math. Soc. 11 (2009), 529-543. doi: 10.4171/JEMS/159