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

Abstract

A. Almost one decade ago, Poonen constructed the first examples of algebraic varieties over global fields for which Skorobogatov's étale Brauer-Manin 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 paper, we construct simply connected fourfolds over global fields of positive characteristic for which the Brauer-Manin 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 Campana's "geometric orbifolds" over function fields of positive characteristic. Along the way, we also construct the first example of simply connected surface of general type over a global field with a non-empty, but non-Zariski dense set of rational points.

Campana points of bounded height on vector group compactifications

“…In dimension 1, where both notions of Campana points coincide, the analogue of Mordell's conjecture for Campana points has been proved over function fields, first in characteristic 0 by Campana himself [Cam05], and only very recently in arbitrary characteristic [KPS19]. Over number fields, the only known result says that the abc conjecture implies Mordell's conjecture for Campana points; see [Sme17,Appendix] for a detailed argument.…”

Weak approximation and the Hilbert property for Campana points

“…Further, it naturally lends itself to number-theoretic problems involving m-full solutions of homogeneous equations. (Given m ∈ Z ≥2 , we say that n ∈ Z is m-full if p | n implies p m | n for all primes p.) Despite their arithmetic appeal, questions regarding the geometric abundance and distribution of Campana points remain largely unanswered outside of the case of curves ( [8,15], [25,Appendix]). In particular, the question of whether the Campana points of an orbifold are thin is open even for the most elementary orbifold structures on projective space.…”