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

## TL;DR

This paper constructs simply connected fourfolds over global function fields where the Brauer-Manin obstruction fails, using a new Mordell-type theorem for orbifold structures, advancing understanding of rational points and obstructions in positive characteristic.

## Contribution

It provides the first examples of simply connected fourfolds over global fields where the Brauer-Manin principle fails, without relying on major conjectures, via a novel orbifold Mordell theorem.

## Key findings

- Constructed simply connected fourfolds with Brauer-Manin failure over global fields.
- Established a Mordell-type theorem for orbifolds in positive characteristic.
- First example of simply connected surface of general type with non-Zariski dense rational points.

## Abstract

Almost one decade ago, Poonen constructed the first examples of algebraic varieties over global fields for which Skorobogatov's etale 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.