Pathologies of the Brauer-Manin obstruction

TL;DR

This paper constructs explicit counterexamples to the Hasse principle over real quadratic fields, demonstrating limitations of the étale Brauer-Manin obstruction in explaining rational points on certain algebraic varieties.

Contribution

It provides new explicit examples of varieties where the Brauer-Manin obstruction fails to account for the absence of rational points, expanding understanding of its limitations.

Findings

Counterexamples to the Hasse principle not explained by the étale Brauer-Manin obstruction

Construction of conic bundles over elliptic curves with these properties

Examples of threefolds as families of quadrics with similar behavior

Abstract

We construct a conic bundle over an elliptic curve over a real quadratic field that is a counterexample to the Hasse principle not explained by the \'etale Brauer-Manin obstruction. We also give simple examples of threefolds with the same property that are families of 2-dimensional quadrics, and discuss some other examples and general properties of the Brauer-Manin obstruction.