Failure of the Brauer–Manin principle for a simply connected fourfold over a global function field, via orbifold Mordell

Abstract

Almost one decade ago, Poonen constructed the first examples of algebraic varieties over global fields for which Skorobogatovs étale BrauerManin obstruction does not explain the failure of the Hasse principle. By now, several constructions are known, but they all share common geometric features such as large fundamental groups.

In this article, we construct simply connected fourfolds over global fields of positive characteristic for which the BrauerManin machinery fails. Contrary to earlier work in this direction, our construction does not rely on major conjectures. Instead, we establish a new Diophantine result of independent interest: a Mordell-type theorem for Campanas geometric orbifolds over function fields of positive characteristic. Along the way, we also construct the first example of a simply connected surface of general type over a global field with a nonempty, but non-Zariski-dense set of rational points.