Local solubility in generalised Châtelet varieties

Abstract

We obtain asymptotic formulas for averages of general multivariate arithmetic functions evaluated at polynomial arguments using recent work of Rydin Myerson and Rome-Yamagishi. We give several applications of our results in analytic number theory and arithmetic geometry. For example, we improve on the number of variables needed to prove the Hasse principle for certain polynomial systems, and we count the number of fibers with a rational point in families of high-dimensional Châtelet varieties, allowing for arbitrarily large subordinate Brauer groups.