Failure of the Hasse principle for Chatelet surfaces in characteristic 2

TL;DR

This paper constructs a Chatelet surface over a global field of characteristic 2 that fails the Hasse principle due to a Brauer-Manin obstruction, extending previous results to characteristic 2.

Contribution

It extends the known failure of the Hasse principle via Brauer-Manin obstruction to characteristic 2, previously established in other characteristics.

Findings

Constructed a Chatelet surface in characteristic 2 failing the Hasse principle.

Demonstrated the failure is due to a Brauer-Manin obstruction.

Extended Poonen's result to characteristic 2.

Abstract

Given any global field k of characteristic 2, we construct a Chatelet surface over k which fails to satisfy the Hasse principle. This failure is due to a Brauer-Manin obstruction. This construction extends a result of Poonen to characteristic 2, thereby showing that the etale-Brauer obstruction is insufficient to explain all failures of the Hasse principle over a global field of any characteristic.