Arthur trace formula
E865100
The Arthur trace formula is a far-reaching generalization of the Selberg trace formula that provides a powerful analytic tool for studying automorphic representations and establishing instances of the Langlands program.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Arthur trace formula canonical | 2 |
| the Arthur trace formula | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10462026 — 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: Arthur trace formula Context triple: [Selberg trace formula, inspired, Arthur trace formula]
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A.
Selberg trace formula
The Selberg trace formula is a fundamental result in analytic number theory and spectral theory that relates lengths of closed geodesics on a Riemannian manifold to the spectrum of its Laplace operator, serving as a non-abelian analogue of the Poisson summation formula.
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B.
Gutzwiller trace formula
The Gutzwiller trace formula is a semiclassical tool in quantum chaos that links the quantum energy spectrum of a system to the properties of its classical periodic orbits.
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C.
Hecke operators
Hecke operators are algebraic operators acting on modular forms that play a central role in number theory, particularly in understanding congruences, L-functions, and the arithmetic of modular forms.
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D.
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].
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E.
Euler products for automorphic L-functions
Euler products for automorphic L-functions are infinite product expansions attached to automorphic representations that encode deep arithmetic information and generalize the classical Euler product of the Riemann zeta function to a broad class of L-functions in the Langlands program.
- 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: Arthur trace formula Target entity description: The Arthur trace formula is a far-reaching generalization of the Selberg trace formula that provides a powerful analytic tool for studying automorphic representations and establishing instances of the Langlands program.
-
A.
Selberg trace formula
The Selberg trace formula is a fundamental result in analytic number theory and spectral theory that relates lengths of closed geodesics on a Riemannian manifold to the spectrum of its Laplace operator, serving as a non-abelian analogue of the Poisson summation formula.
-
B.
Gutzwiller trace formula
The Gutzwiller trace formula is a semiclassical tool in quantum chaos that links the quantum energy spectrum of a system to the properties of its classical periodic orbits.
-
C.
Hecke operators
Hecke operators are algebraic operators acting on modular forms that play a central role in number theory, particularly in understanding congruences, L-functions, and the arithmetic of modular forms.
-
D.
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].
-
E.
Euler products for automorphic L-functions
Euler products for automorphic L-functions are infinite product expansions attached to automorphic representations that encode deep arithmetic information and generalize the classical Euler product of the Riemann zeta function to a broad class of L-functions in the Langlands program.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
analytic tool in number theory
ⓘ
generalization of the Selberg trace formula ⓘ trace formula ⓘ |
| appliesTo |
automorphic representations
ⓘ
reductive groups over global fields ⓘ |
| context |
adelic groups
ⓘ
reductive algebraic groups over number fields ⓘ |
| developedBy | James Arthur NERFINISHED ⓘ |
| field |
Langlands program
NERFINISHED
ⓘ
automorphic forms ⓘ harmonic analysis on reductive groups ⓘ number theory ⓘ representation theory ⓘ |
| generalizes | Selberg trace formula NERFINISHED ⓘ |
| hasAspect |
geometric side
ⓘ
spectral side ⓘ |
| hasComponent |
global trace formula
NERFINISHED
ⓘ
local trace formula ⓘ |
| hasGoal |
comparison of automorphic spectra for different groups
ⓘ
stabilization for use in endoscopic classification ⓘ |
| involves |
Arthur parameters
ⓘ
Eisenstein series NERFINISHED ⓘ Levi subgroups ⓘ characters of automorphic representations ⓘ continuous spectrum ⓘ discrete spectrum ⓘ endoscopy ⓘ intertwining operators ⓘ orbital integrals ⓘ parabolic subgroups ⓘ stabilization ⓘ truncation operators ⓘ weighted orbital integrals ⓘ |
| namedAfter | James Arthur NERFINISHED ⓘ |
| relatedTo |
Arthur–Selberg trace formula
NERFINISHED
ⓘ
Langlands functoriality NERFINISHED ⓘ automorphic L-functions ⓘ endoscopic transfer ⓘ stable trace formula ⓘ |
| timePeriod | late 20th century ⓘ |
| usedFor |
classification of automorphic representations
ⓘ
comparison of trace formulas ⓘ establishing instances of functoriality ⓘ establishing instances of the Langlands correspondence ⓘ spectral decomposition of automorphic forms ⓘ stabilization of trace formulas ⓘ studying automorphic representations ⓘ |
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: Arthur trace formula Description of subject: The Arthur trace formula is a far-reaching generalization of the Selberg trace formula that provides a powerful analytic tool for studying automorphic representations and establishing instances of the Langlands program.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
the Arthur trace formula