Beilinson conjectures
E685698
Beilinson conjectures are a set of deep conjectures in arithmetic geometry that relate special values of L-functions to algebraic K-theory and motivic cohomology, generalizing phenomena seen in cases like the Birch and Swinnerton-Dyer conjecture.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Beilinson conjectures canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T7743559 — 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: Beilinson conjectures Context triple: [Birch and Swinnerton-Dyer Conjecture, relatedTo, Beilinson conjectures]
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A.
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.
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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.
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.
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D.
Deligne cohomology
Deligne cohomology is a refined cohomology theory in algebraic geometry that combines singular cohomology and differential forms to capture both topological and arithmetic information about complex algebraic varieties.
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E.
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.
- 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: Beilinson conjectures Target entity description: Beilinson conjectures are a set of deep conjectures in arithmetic geometry that relate special values of L-functions to algebraic K-theory and motivic cohomology, generalizing phenomena seen in cases like the Birch and Swinnerton-Dyer conjecture.
-
A.
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.
-
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.
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.
Deligne cohomology
Deligne cohomology is a refined cohomology theory in algebraic geometry that combines singular cohomology and differential forms to capture both topological and arithmetic information about complex algebraic varieties.
-
E.
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.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
conjectural framework in arithmetic geometry
ⓘ
mathematical conjecture ⓘ |
| appliesTo |
L-functions of algebraic varieties
ⓘ
L-functions of motives ⓘ L-functions of number fields ⓘ |
| concerns |
critical values of L-functions
ⓘ
non-critical values of L-functions at integers ⓘ rational structures on cohomology ⓘ special values at negative integers ⓘ special values at positive integers ⓘ |
| field |
algebraic K-theory
ⓘ
arithmetic geometry ⓘ motivic cohomology ⓘ number theory ⓘ |
| formulatedBy | Alexander Beilinson NERFINISHED ⓘ |
| formulatedIn | 1980s ⓘ |
| generalizes |
Birch and Swinnerton-Dyer conjecture
NERFINISHED
ⓘ
Dirichlet’s class number formula NERFINISHED ⓘ class number formula ⓘ |
| hasPart |
Beilinson conjecture on motivic cohomology
NERFINISHED
ⓘ
Beilinson conjecture on regulators NERFINISHED ⓘ Beilinson conjecture on special values of L-functions NERFINISHED ⓘ |
| implies | parts of the Bloch–Kato conjecture in certain cases ⓘ |
| influenced |
development of motivic cohomology
ⓘ
formulation of the Bloch–Kato Tamagawa number conjecture ⓘ |
| mainTheme |
relation between special values of L-functions and algebraic K-theory
ⓘ
relation between special values of L-functions and motivic cohomology ⓘ |
| namedAfter | Alexander Beilinson NERFINISHED ⓘ |
| predicts |
leading Taylor coefficients of L-functions at integers
ⓘ
order of vanishing of L-functions at integers ⓘ rationality properties of normalized L-values ⓘ relations between special L-values and determinants of regulator pairings ⓘ |
| relatedTo |
Bloch–Kato conjecture
NERFINISHED
ⓘ
Bloch’s conjecture on Chow groups NERFINISHED ⓘ Deligne conjecture on critical values of L-functions ⓘ Equivariant Tamagawa number conjecture NERFINISHED ⓘ Lichtenbaum conjectures ⓘ Tamagawa number conjecture NERFINISHED ⓘ |
| relatesConcept |
Deligne cohomology
NERFINISHED
ⓘ
absolute Hodge cohomology ⓘ algebraic K-groups ⓘ higher regulators ⓘ motivic cohomology groups ⓘ regulator maps ⓘ |
| status | open ⓘ |
| typicalDomain |
motives over number fields
ⓘ
smooth projective varieties over number fields ⓘ |
| usesConcept |
Chow groups
ⓘ
Ext-groups in the category of mixed motives ⓘ higher Chow groups ⓘ |
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Subject: Beilinson conjectures Description of subject: Beilinson conjectures are a set of deep conjectures in arithmetic geometry that relate special values of L-functions to algebraic K-theory and motivic cohomology, generalizing phenomena seen in cases like the Birch and Swinnerton-Dyer conjecture.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.