2011/12/01 by Anthony Várilly-Alvarado, Anthony Várilly‐Alvarado, Bianca Viray · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Analytic Number Theory Research
paper · doi:10.1112/blms/bdr075
We discuss the geometry and arithmetic of higher-dimensional analogs 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-mth powers of a number field are diophantine. Also, given a global field k such that Char(k)=p or k contains the pth roots of unity, we construct a (p+1)-fold that has no k-points and no étale-Brauer obstruction to the Hasse principle.