Beilinson regulator
E876106
The Beilinson regulator is a deep arithmetic-geometric map connecting algebraic K-theory of varieties to their Deligne or absolute Hodge cohomology, playing a central role in conjectural formulas for special values of L-functions.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Beilinson regulator canonical | 1 |
| Beilinson regulators | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10617353 — 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 regulator Context triple: [Alexander Beilinson, knownFor, Beilinson regulator]
-
A.
Beilinson conjectures
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.
-
B.
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.
-
C.
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.
-
D.
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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E.
Arakelov theory
Arakelov theory is a framework in arithmetic geometry that extends intersection theory to arithmetic surfaces by incorporating both finite and infinite places, enabling analytic tools to study Diophantine problems.
- 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 regulator Target entity description: The Beilinson regulator is a deep arithmetic-geometric map connecting algebraic K-theory of varieties to their Deligne or absolute Hodge cohomology, playing a central role in conjectural formulas for special values of L-functions.
-
A.
Beilinson conjectures
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.
-
B.
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.
-
C.
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.
-
D.
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.
-
E.
Arakelov theory
Arakelov theory is a framework in arithmetic geometry that extends intersection theory to arithmetic surfaces by incorporating both finite and infinite places, enabling analytic tools to study Diophantine problems.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
arithmetic invariant
ⓘ
cohomological construction ⓘ regulator map ⓘ |
| appearsIn |
Beilinson’s 1980s papers on higher regulators
ⓘ
literature on special values of L-functions ⓘ |
| appliesTo |
motives over number fields
ⓘ
smooth algebraic varieties over number fields ⓘ |
| associatedWith |
Deligne–Beilinson cohomology
ⓘ
linked to:
Deligne cohomology
absolute Hodge cycles ⓘ mixed Hodge structures ⓘ motivic cohomology ⓘ |
| centralTo |
Beilinson’s conjectural description of L-values
ⓘ
arithmetic of motives ⓘ |
| codomain |
Deligne cohomology
ⓘ
absolute Hodge cohomology ⓘ |
| constructionUses |
Deligne complexes
ⓘ
linked to:
Deligne cohomology
differential forms with logarithmic singularities ⓘ higher Chow groups ⓘ |
| definedFor |
K-groups of motives
ⓘ
K-groups of varieties over number fields ⓘ |
| domain |
algebraic K-theory
ⓘ
higher K-groups ⓘ |
| field |
algebraic K-theory
ⓘ
arithmetic geometry ⓘ number theory ⓘ |
| generalizes |
Borel regulator
ⓘ
Dirichlet regulator ⓘ |
| hasRole |
bridge between K-theory and cohomology theories
ⓘ
defines regulator maps in motivic cohomology ⓘ provides periods in conjectural L-value formulas ⓘ |
| motivates |
development of motivic cohomology
ⓘ
study of absolute Hodge cohomology ⓘ |
| namedAfter | Alexander Beilinson ⓘ |
| relatedConcept |
higher regulators
ⓘ
periods of motives ⓘ polylogarithm motives ⓘ regulator map in algebraic K-theory ⓘ |
| relates |
algebraic K-theory to Hodge-theoretic invariants
ⓘ
arithmetic geometry to complex geometry ⓘ |
| targetProperty |
mixed Hodge structures with real structure
ⓘ
real Deligne cohomology groups ⓘ |
| usedIn |
Beilinson conjectures on special values of L-functions
ⓘ
linked to:
Beilinson conjectures
Bloch–Kato conjectures ⓘ
linked to:
Bloch–Kato conjecture
Deligne’s conjectures on critical values of L-functions ⓘ arithmetic of elliptic curves ⓘ arithmetic of modular forms ⓘ formulas for special values of motivic L-functions ⓘ study of motives ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Beilinson regulator Description of subject: The Beilinson regulator is a deep arithmetic-geometric map connecting algebraic K-theory of varieties to their Deligne or absolute Hodge cohomology, playing a central role in conjectural formulas for special values of L-functions.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Beilinson regulator