Tate Conjecture
E680776
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.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Tate conjecture | 2 |
| Hasse–Weil conjecture | 1 |
| Tate Conjecture canonical | 1 |
| Tate Conjecture for divisors | 1 |
| Tate Conjecture for higher codimension cycles | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7678369 — 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: Tate Conjecture Context triple: [Hodge Conjecture, relatedTo, Tate Conjecture]
-
A.
Hodge Conjecture
The Hodge Conjecture is a major unsolved problem in algebraic geometry that predicts which cohomology classes on a non-singular projective complex variety arise from algebraic subvarieties.
-
B.
Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
-
C.
Weil conjectures
The Weil conjectures are a set of deep statements about the zeta functions of algebraic varieties over finite fields that guided the development of modern algebraic geometry and were ultimately proved using étale cohomology.
-
D.
Taniyama–Shimura–Weil conjecture
The Taniyama–Shimura–Weil conjecture, now the modularity theorem, asserts that every elliptic curve over the rational numbers is modular and played a central role in the proof of Fermat’s Last Theorem.
-
E.
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.
- 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: Tate Conjecture Target entity description: 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.
-
A.
Hodge Conjecture
The Hodge Conjecture is a major unsolved problem in algebraic geometry that predicts which cohomology classes on a non-singular projective complex variety arise from algebraic subvarieties.
-
B.
Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
-
C.
Weil conjectures
The Weil conjectures are a set of deep statements about the zeta functions of algebraic varieties over finite fields that guided the development of modern algebraic geometry and were ultimately proved using étale cohomology.
-
D.
Taniyama–Shimura–Weil conjecture
The Taniyama–Shimura–Weil conjecture, now the modularity theorem, asserts that every elliptic curve over the rational numbers is modular and played a central role in the proof of Fermat’s Last Theorem.
-
E.
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.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical conjecture
ⓘ
open problem in arithmetic geometry ⓘ |
| appliesTo | smooth projective varieties over finite fields ⓘ |
| assumes | prime l different from the characteristic of the finite field ⓘ |
| codimensionParameter | r ⓘ |
| cohomologyDegree | 2r ⓘ |
| concerns |
Galois representations
ⓘ
algebraic cycles ⓘ varieties over finite fields ⓘ étale cohomology ⓘ |
| domain | smooth projective variety over a finite field ⓘ |
| equates |
dimension of the space of Galois-invariant cohomology classes
ⓘ
rank of the group of algebraic cycles modulo numerical equivalence ⓘ |
| field |
algebraic geometry
ⓘ
arithmetic geometry ⓘ number theory ⓘ |
| formulatedBy | John Tate ⓘ |
| groupActing | absolute Galois group of the finite field ⓘ |
| hasConsequence |
control of algebraic cycles by Galois action
ⓘ
description of Néron–Severi group via Galois cohomology ⓘ |
| hasVariant |
Tate Conjecture for divisors
ⓘ
linked to:
Tate Conjecture
Tate Conjecture for higher codimension cycles ⓘ
linked to:
Tate Conjecture
|
| implies |
finiteness of the Néron–Severi group over finite fields
ⓘ
semisimplicity of certain Galois representations ⓘ |
| isAnalogOf |
Hodge Conjecture
ⓘ
Mumford–Tate Conjecture ⓘ |
| isCentralIn |
study of algebraic cycles over finite fields
ⓘ
theory of motives ⓘ |
| isRelatedTo |
Beilinson Conjectures
ⓘ
linked to:
Beilinson conjectures
Birch and Swinnerton-Dyer Conjecture ⓘ Weil Conjectures ⓘ
linked to:
Weil conjectures
|
| isSpecialCaseOf | standard conjectures on algebraic cycles ⓘ |
| knownFor | deep connection between geometry and arithmetic of varieties over finite fields ⓘ |
| knownToHoldFor |
K3 surfaces in some cases
ⓘ
abelian varieties over finite fields in many cases ⓘ divisors on abelian varieties over finite fields ⓘ |
| motivated | study of zeta functions of varieties over finite fields ⓘ |
| namedAfter | John Tate ⓘ |
| predicts | equality between algebraic cycles and Galois-invariant cohomology classes ⓘ |
| relates |
Galois-invariant subspace of cohomology
ⓘ
algebraic cycles of codimension r ⓘ l-adic étale cohomology ⓘ |
| status | open in general ⓘ |
| type | cohomological conjecture ⓘ |
| uses |
absolute Galois group of a finite field
ⓘ
l-adic cohomology ⓘ |
| yearProposed | 1963 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Tate Conjecture Description of subject: 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.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Tate Conjecture
linked to: Tate Conjecture
linked to: Tate Conjecture
linked to: Tate Conjecture
linked to: Tate Conjecture