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  "body": "\n## Summary\n\nA variety over a number field can have points over every completion of the field and still have no rational point.\nThe Brauer–Manin obstruction explains many such failures of the [[Hasse principle|EMAR4X3W8D]].\nIt uses the [[Brauer group|BEW48TA9WZ]] of the variety and the reciprocity law of [[class field theory|BEKCEYHCHX]].\nThese cut out a subset of the adelic points that must contain all rational points.\nManin introduced it in 1970, and it accounts for most known counterexamples to the Hasse principle.\n\n## Explanations\n\n### L1 human\n\nTo show an equation has no rational solution, one can check it has no real solution or no solution modulo powers of some prime.\nSometimes these local tests all pass and still there is no rational solution.\nThe Brauer–Manin obstruction is a finer test.\nEach 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.\nIf no family of local solutions passes this test, there is no rational point.\n",
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