---
tag: XDWF0HW2QN
kind: concept
name: Brauer–Manin obstruction
msc: [14G05, 14F22, 11G35]
sources:
  - url: https://en.wikipedia.org/wiki/Manin_obstruction
same_as:
  wikidata: Q6749775
  wikipedia: 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|EMAR4X3W8D]].
It uses the [[Brauer group|BEW48TA9WZ]] of the variety and the reciprocity law of [[class field theory|BEKCEYHCHX]].
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.

## Explanations

### L1 human

To show an equation has no rational solution, one can check it has no real solution or no solution modulo powers of some prime.
Sometimes these local tests all pass and still there is no rational solution.
The Brauer–Manin obstruction is a finer test.
Each element of the Brauer group attaches to each local solution a number, and a genuine rational solution must make these numbers add up to zero.
If no family of local solutions passes this test, there is no rational point.