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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Brauer–Manin obstruction canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T8733360 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Brauer–Manin obstruction Context triple: [Hasse principle, relatedConcept, Brauer–Manin obstruction]
-
A.
Hasse principle
The Hasse principle is a concept in number theory stating that a Diophantine equation has a rational solution if and only if it has solutions in all completions of the rationals, such as the real numbers and p-adic numbers.
-
B.
Standard Conjectures on Algebraic Cycles
The Standard Conjectures on Algebraic Cycles are a set of deep, still unproven hypotheses in algebraic geometry that aim to provide a foundational theory of algebraic cycles and their cohomological properties, underpinning much of the modern theory of motives.
-
C.
Bombieri–Lang conjecture
The Bombieri–Lang conjecture is a major unsolved conjecture in number theory and arithmetic geometry predicting that varieties of general type over number fields have only finitely many rational points.
-
D.
Hodge–Riemann bilinear relations
The Hodge–Riemann bilinear relations are fundamental positivity and orthogonality conditions on the intersection form in Hodge theory that underpin results such as the hard Lefschetz theorem and the Hodge index theorem.
-
E.
Faltings' theorem
Faltings' theorem is a landmark result in arithmetic geometry that proves every algebraic curve of genus greater than one over a number field has only finitely many rational points.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Brauer–Manin obstruction Target entity description: 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.
-
A.
Hasse principle
The Hasse principle is a concept in number theory stating that a Diophantine equation has a rational solution if and only if it has solutions in all completions of the rationals, such as the real numbers and p-adic numbers.
-
B.
Standard Conjectures on Algebraic Cycles
The Standard Conjectures on Algebraic Cycles are a set of deep, still unproven hypotheses in algebraic geometry that aim to provide a foundational theory of algebraic cycles and their cohomological properties, underpinning much of the modern theory of motives.
-
C.
Bombieri–Lang conjecture
The Bombieri–Lang conjecture is a major unsolved conjecture in number theory and arithmetic geometry predicting that varieties of general type over number fields have only finitely many rational points.
-
D.
Hodge–Riemann bilinear relations
The Hodge–Riemann bilinear relations are fundamental positivity and orthogonality conditions on the intersection form in Hodge theory that underpin results such as the hard Lefschetz theorem and the Hodge index theorem.
-
E.
Faltings' theorem
Faltings' theorem is a landmark result in arithmetic geometry that proves every algebraic curve of genus greater than one over a number field has only finitely many rational points.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
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Input
Subject: Brauer–Manin obstruction Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.