Higher dimensional analogues of Ch\^atelet surfaces

Abstract

We discuss the geometry and arithmetic of higher-dimensional analogues of Ch\^atelet 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 \'etale-Brauer obstruction to the Hasse principle.