Selberg trace formula
E246698
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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Selberg trace formula canonical | 3 |
| compact case Selberg trace formula | 1 |
| non-compact case Selberg trace formula | 1 |
| the Selberg trace formula | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2252104 — 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: Selberg trace formula Context triple: [Atle Selberg, knownFor, Selberg trace formula]
-
A.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
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B.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
-
C.
Weyl law
The Weyl law is a fundamental result in spectral theory that describes the asymptotic distribution of eigenvalues of the Laplacian (or similar operators) in terms of the volume of the underlying domain or manifold.
-
D.
Multiplicative Number Theory
Multiplicative Number Theory is a branch of analytic number theory that studies arithmetic functions and prime number distributions through their multiplicative properties and associated Dirichlet series.
-
E.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
- 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: Selberg trace formula Target entity description: 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.
-
A.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
B.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
-
C.
Weyl law
The Weyl law is a fundamental result in spectral theory that describes the asymptotic distribution of eigenvalues of the Laplacian (or similar operators) in terms of the volume of the underlying domain or manifold.
-
D.
Multiplicative Number Theory
Multiplicative Number Theory is a branch of analytic number theory that studies arithmetic functions and prime number distributions through their multiplicative properties and associated Dirichlet series.
-
E.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in analytic number theory ⓘ result in spectral theory ⓘ trace formula ⓘ |
| appliesTo |
Riemannian manifolds of constant negative curvature
ⓘ
hyperbolic surfaces ⓘ locally symmetric spaces of rank one ⓘ quotients of the hyperbolic plane by Fuchsian groups ⓘ |
| author | Atle Selberg ⓘ |
| coreConcept |
Laplace operator
ⓘ
surface form:
Laplace–Beltrami operator
closed geodesics ⓘ eigenvalues of the Laplacian ⓘ geometric side ⓘ length spectrum ⓘ spectral side ⓘ |
| describedAs | non-abelian analogue of the Poisson summation formula ⓘ |
| field |
analytic number theory
ⓘ
automorphic forms ⓘ differential geometry ⓘ global analysis ⓘ representation theory ⓘ spectral theory ⓘ |
| firstDevelopedIn | mid-20th century ⓘ |
| generalizedBy | Arthur trace formula ⓘ |
| hasComponent |
elliptic contribution
ⓘ
hyperbolic contribution ⓘ identity contribution ⓘ parabolic contribution ⓘ |
| hasVersion |
Selberg trace formula
self-linksurface differs
ⓘ
surface form:
compact case Selberg trace formula
Selberg trace formula self-linksurface differs ⓘ
surface form:
non-compact case Selberg trace formula
|
| historicalPeriod | 20th century mathematics ⓘ |
| inspired | Arthur trace formula ⓘ |
| namedAfter | Atle Selberg ⓘ |
| relatedTo |
Poisson summation formula
ⓘ
Selberg zeta function ⓘ Weyl law ⓘ
surface form:
Weyl law for eigenvalues
prime geodesic theorem ⓘ |
| relates |
length spectrum of closed geodesics
ⓘ
spectrum of the Laplace operator ⓘ |
| requires |
harmonic analysis on Lie groups
ⓘ
spectral theory of self-adjoint operators ⓘ theory of unitary representations ⓘ |
| typicalSetting |
compact hyperbolic surfaces
ⓘ
finite-area hyperbolic surfaces with cusps ⓘ |
| usedFor |
investigating distribution of closed geodesics
ⓘ
proving results about automorphic L-functions ⓘ relating geometric invariants to spectral invariants ⓘ spectral decomposition of automorphic representations ⓘ studying eigenvalues of the Laplacian on Riemann surfaces ⓘ |
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: Selberg trace formula Description of subject: 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.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
compact case Selberg trace formula
this entity surface form:
non-compact case Selberg trace formula
this entity surface form:
the Selberg trace formula