On the Brauer-Manin Obstruction Applied to Ramified Covers

TL;DR

This paper proposes a new obstruction, extending the Brauer-Manin approach to ramified covers, to explain failures of the local-global principle not accounted for by previous obstructions, with applications to Poonen's counterexample.

Contribution

It introduces a novel obstruction applying the Brauer-Manin method to ramified covers, enhancing the understanding of failures of the local-global principle.

Findings

The new obstruction explains Poonen's counterexample over totally imaginary number fields.

It extends the Brauer-Manin obstruction to ramified covers.

The approach offers a finer tool for analyzing rational points on varieties.

Abstract

The Brauer-Manin obstruction is used to explain the failure of the local-global principle for algebraic varieties. In 1999 Skorobogatov gave the first example of a variety that does not satisfy the local-global principle which is not explained by the Brauer-Manin obstruction. He did so by applying the Brauer-Manin obstruction to \'etale covers of the variety, and thus defining a finer obstruction. In 2008 Poonen gave the first example of failure of the local-global principle which cannot be explained for by Skorobogatov's \'etale-Brauer obstruction. However, Poonen's construction was not accompanied by a definition of a new finer obstruction. In this paper I shall present a possible definition for such an obstruction by allowing to apply the Brauer-Manin obstruction to some ramified covers as well, and show that this new obstruction can explain Poonen counterexample in the case of a totally imaginary number field.