Poitou–Tate duality
E883485
Poitou–Tate duality is a fundamental result in Galois cohomology that establishes deep duality relationships between global and local cohomology groups of number fields.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Poitou–Tate duality canonical | 6 |
| Poitou–Tate duality in étale cohomology | 1 |
| Tate duality | 1 |
| local Tate duality | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10732942 — 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: Poitou–Tate duality Context triple: [Cohomologie Galoisienne, topic, Poitou–Tate duality]
-
A.
Cassels–Tate pairing
The Cassels–Tate pairing is a bilinear pairing on the Tate–Shafarevich group of an abelian variety over a number field that plays a central role in arithmetic geometry and the study of rational points.
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B.
Tate Conjecture
The Tate Conjecture is a major open problem in arithmetic geometry that predicts a deep connection between algebraic cycles on varieties over finite fields and their Galois-invariant étale cohomology classes.
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C.
Sato–Tate distribution (for families of elliptic curves)
The Sato–Tate distribution (for families of elliptic curves) is a probabilistic law describing how the normalized Frobenius traces (or equivalently, the angles in the Hasse bound) of elliptic curves are distributed, typically following a specific sine-squared measure on the interval [0, π].
-
D.
Pontryagin duality
Pontryagin duality is a fundamental theorem in harmonic analysis and topological group theory that establishes a duality between locally compact abelian groups and their groups of continuous characters.
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E.
Deligne–Lusztig theory
Deligne–Lusztig theory is a framework in algebraic geometry and representation theory that constructs and studies representations of finite groups of Lie type using varieties defined over finite fields.
- 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: Poitou–Tate duality Target entity description: Poitou–Tate duality is a fundamental result in Galois cohomology that establishes deep duality relationships between global and local cohomology groups of number fields.
-
A.
Cassels–Tate pairing
The Cassels–Tate pairing is a bilinear pairing on the Tate–Shafarevich group of an abelian variety over a number field that plays a central role in arithmetic geometry and the study of rational points.
-
B.
Tate Conjecture
The Tate Conjecture is a major open problem in arithmetic geometry that predicts a deep connection between algebraic cycles on varieties over finite fields and their Galois-invariant étale cohomology classes.
-
C.
Sato–Tate distribution (for families of elliptic curves)
The Sato–Tate distribution (for families of elliptic curves) is a probabilistic law describing how the normalized Frobenius traces (or equivalently, the angles in the Hasse bound) of elliptic curves are distributed, typically following a specific sine-squared measure on the interval [0, π].
-
D.
Pontryagin duality
Pontryagin duality is a fundamental theorem in harmonic analysis and topological group theory that establishes a duality between locally compact abelian groups and their groups of continuous characters.
-
E.
Deligne–Lusztig theory
Deligne–Lusztig theory is a framework in algebraic geometry and representation theory that constructs and studies representations of finite groups of Lie type using varieties defined over finite fields.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
duality theorem
ⓘ
result in Galois cohomology ⓘ |
| appliesTo |
absolute Galois groups of number fields
ⓘ
global fields ⓘ number fields ⓘ |
| assumes | finite Galois module with continuous action ⓘ |
| concerns |
cohomology in degrees 0, 1, and 2
ⓘ
cohomology of Galois groups of number fields ⓘ |
| context | cohomological dimension 2 of global Galois groups ⓘ |
| describes | compatibility of global and local pairings ⓘ |
| establishes | duality between global and local cohomology groups ⓘ |
| expressedAs | nine-term exact sequence in Galois cohomology ⓘ |
| field |
Galois cohomology
ⓘ
algebraic number theory ⓘ |
| framework | cohomology of profinite groups ⓘ |
| generalizes | Tate local duality to global fields ⓘ |
| hasGeneralization |
Poitou–Tate duality for p-adic representations
ⓘ
Poitou–Tate duality in étale cohomology ⓘ
linked to:
Poitou–Tate duality
|
| implies |
duality for Selmer and Tate–Shafarevich groups
ⓘ
finiteness properties of Galois cohomology groups ⓘ |
| involves |
Pontryagin duality
ⓘ
Tate–Shafarevich groups ⓘ
linked to:
Tate–Shafarevich group
cohomology groups with restricted ramification ⓘ cohomology with compact support ⓘ discrete Galois modules ⓘ finite Galois modules ⓘ |
| namedAfter |
Claude Poitou
ⓘ
John Tate ⓘ |
| provides |
exact sequences relating global and local cohomology
ⓘ
orthogonality relations for local conditions ⓘ perfect pairings between cohomology groups ⓘ |
| relatedTo |
Artin–Verdier duality
ⓘ
linked to:
Verdier duality
Grothendieck duality ⓘ local Tate duality ⓘ |
| relates |
cohomology groups H^i(G_K,M) and H^{3-i}(G_K,M^∨(1))
ⓘ
global Galois cohomology ⓘ local Galois cohomology ⓘ |
| requires | choice of a finite set of primes containing archimedean places ⓘ |
| typicalDomain | finite sets of places of a number field ⓘ |
| usedIn |
Bloch–Kato conjectures
ⓘ
linked to:
Bloch–Kato conjecture
Galois deformation theory ⓘ Iwasawa theory ⓘ arithmetic of elliptic curves ⓘ class field theory ⓘ modularity lifting theorems ⓘ study of Selmer groups ⓘ |
| uses |
Tate local duality
ⓘ
linked to:
Tate cohomology
|
How these facts were elicited
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Input
Subject: Poitou–Tate duality Description of subject: Poitou–Tate duality is a fundamental result in Galois cohomology that establishes deep duality relationships between global and local cohomology groups of number fields.
Referenced by (9)
Full triples — surface form annotated when it differs from this entity's canonical label.