Brauer–Manin Obstruction

A mechanism detecting failure of the Hasse principle for rational points on varieties over global fields: elements of the Brauer group of a variety pair with adelic points via local invariants, and a nontrivial global obstruction (nonzero sum of local invariants) can rule out the existence of rational points even when local points exist everywhere.

A mechanism using elements of the Brauer group of a variety to obstruct the existence of rational points even when local points exist everywhere; concretely, evaluation of Brauer classes at adelic points produces compatibility conditions whose failure explains counterexamples to the Hasse principle and refines weak approximation statements.