Brauer–Manin obstruction

Summary

A variety over a number field can have points over every completion of the field and still have no rational point. The Brauer–Manin obstruction explains many such failures of the Hasse principle. It uses the Brauer group of the variety and the reciprocity law of class field theory. These cut out a subset of the adelic points that must contain all rational points. Manin introduced it in 1970, and it accounts for most known counterexamples to the Hasse principle.