Higher dimensional analogues of Ch\^atelet surfaces

TL;DR

This paper explores higher-dimensional analogues of Châtelet surfaces, analyzing their geometric and arithmetic properties, including Brauer and Picard groups, and demonstrates their ability to violate the Hasse principle, extending recent results in number theory.

Contribution

It introduces higher-dimensional analogues of Châtelet surfaces, detailing their structure and showing they can violate the Hasse principle, generalizing recent results of Poonen.

Findings

Higher-dimensional Châtelet analogues have complex Brauer and Picard groups.

These varieties can violate the Hasse principle without Brauer-Manin obstructions.

Results depend on hypotheses like Schinzel's conjecture and properties of global fields.

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.