Existence of rational points on smooth projective varieties

TL;DR

The paper demonstrates that the existence of an algorithm for determining rational points on smooth projective varieties over a number field implies the existence of algorithms for more general varieties and for computing rational points when finite.

Contribution

It establishes a reduction showing that deciding rational points on smooth projective varieties is as hard as for all varieties, linking decision problems to explicit point computation.

Findings

If an algorithm exists for smooth projective varieties, then algorithms exist for all varieties.

Constructs a family of Chatelet surfaces with exactly one failure to have a rational point.

Shows the equivalence of decision problems and explicit computation of rational points.

Abstract

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.