Higher dimensional analogues of Châtelet surfaces

Abstract

We discuss the geometry and arithmetic of higher-dimensional analogues of Châtelet surfaces; namely, we describe the structure of their Brauer and Picard groups and show that they can violate the Hasse principle. In addition, we use these varieties to give straightforward generalizations of two recent results of Poonen. Specifically, we prove that, assuming Schinzel's hypothesis, the non-m^{th} powers of a number field are diophantine. Also, given a global field k such that Char(k) = p or k contains the p^{th} roots of unity, we construct a (p+1)-fold that has no k-points and no étale-Brauer obstruction to the Hasse principle.

LaTeX, 10 pages. Spurious extra copy of source file removed

References in corpus (4)

• Zéro-cycles sur les fibrations au-dessus d'une courbe de genre quelconque
• Higher dimensional analogues of Châtelet surfaces
• Failure of the Hasse principle for Chatelet surfaces in characteristic 2
• Principe local-global pour les zéro-cycles sur certaines fibrations au-dessus d'une courbe : I

Cited by in corpus (6)

• On the fibration method for zero-cycles and rational points
• Higher dimensional analogues of Châtelet surfaces
• Principes locaux-globaux pour certaines fibrations en torseurs sous un tore
• Rational points on fibrations with few non-split fibres
• Le complémentaire des puissances $n$-ièmes dans un corps de nombres est un ensemble diophantien
• Vertical unramified Brauer groups of Galois normic bundles