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.
-
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
NERFINISHED
ⓘ
absolute Hodge cycles ⓘ mixed Hodge structures ⓘ motivic cohomology ⓘ |
| centralTo |
Beilinson’s conjectural description of L-values
ⓘ
arithmetic of motives ⓘ |
| codomain |
Deligne cohomology
NERFINISHED
ⓘ
absolute Hodge cohomology NERFINISHED ⓘ |
| constructionUses |
Deligne complexes
NERFINISHED
ⓘ
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
NERFINISHED
ⓘ
Dirichlet regulator NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
Bloch–Kato conjectures NERFINISHED ⓘ Deligne’s conjectures on critical values of L-functions NERFINISHED ⓘ arithmetic of elliptic curves ⓘ arithmetic of modular forms ⓘ formulas for special values of motivic L-functions ⓘ study of motives ⓘ |
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
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
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.
this entity surface form:
Beilinson regulators