Existence of rational points on smooth projective varieties

• Autores: Bjorn Poonen
• Localización: Journal of the European Mathematical Society, ISSN 1435-9855, Vol. 11, Nº 3, 2009, págs. 529-543
• Idioma: inglés
• DOI: 10.4171/jems/159
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• Resumen

  • 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.