# 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