Existence of rational points on smooth projective varieties

Details Existence of rational points on smooth projective varieties Existence of rational points on smooth projective varieties

2007-12-11

arxiv.org/abs/0712.1782v1

Mathematics

Number Theory

11 pages

Scientific paper

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 Chatelet surfaces such that exactly one of the surfaces fails to have a k-point.

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