Brauer–Manin obstruction
E753151
arithmetic-geometric obstruction
concept in arithmetic geometry
obstruction to the Hasse principle
obstruction to weak approximation
The Brauer–Manin obstruction is an arithmetic-geometric mechanism using the Brauer group and adelic points to explain failures of the Hasse principle and weak approximation for rational points on varieties.
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
arithmetic-geometric obstruction
ⓘ
concept in arithmetic geometry ⓘ obstruction to the Hasse principle ⓘ obstruction to weak approximation ⓘ |
| appliesTo | rational points on varieties ⓘ |
| arisesFrom | pairing between Brauer group and adelic points ⓘ |
| canBeNontrivialFor |
Châtelet surfaces
NERFINISHED
ⓘ
K3 surfaces NERFINISHED ⓘ diagonal cubic surfaces ⓘ |
| canExplain |
absence of rational points despite local solvability
ⓘ
failure of weak approximation despite Zariski density of rational points ⓘ |
| compares |
adelic points
ⓘ
rational points ⓘ |
| definedOn | varieties over number fields ⓘ |
| definedUsing |
Brauer group of a variety
NERFINISHED
ⓘ
adelic points of a variety ⓘ |
| explains |
failures of the Hasse principle
ⓘ
failures of weak approximation ⓘ |
| formalizedAs | Brauer–Manin set NERFINISHED ⓘ |
| generalizes | class field theoretic obstructions ⓘ |
| givesConditionFor |
density of rational points in adelic points
ⓘ
existence of rational points ⓘ |
| hasVariant |
descent obstruction
ⓘ
étale Brauer–Manin obstruction ⓘ |
| introducedBy | Yuri Manin NERFINISHED ⓘ |
| isCentralIn |
the study of counterexamples to the Hasse principle
ⓘ
the study of rational points via cohomological methods ⓘ |
| isToolFor |
studying rational points on curves
ⓘ
studying rational points on higher-dimensional varieties ⓘ studying rational points on surfaces ⓘ |
| motivated | development of refined obstructions in arithmetic geometry ⓘ |
| namedAfter |
Richard Brauer
NERFINISHED
ⓘ
Yuri Manin NERFINISHED ⓘ |
| relatedTo |
Brauer group of a field
ⓘ
Diophantine equations ⓘ Hasse principle NERFINISHED ⓘ local-global principles ⓘ rational points ⓘ weak approximation ⓘ |
| studiedIn |
algebraic geometry
ⓘ
number theory ⓘ |
| uses |
Brauer group
NERFINISHED
ⓘ
adelic points ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.