Euler products for automorphic L-functions
E304347
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.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Euler product expansions | 2 |
| Euler Products | 1 |
| Euler products | 1 |
| Euler products for automorphic L-functions canonical | 1 |
| Langlands L-functions | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2815444 — 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: Euler products for automorphic L-functions Context triple: [Euler product formula for the Riemann zeta function, generalizedBy, Euler products for automorphic L-functions]
-
A.
Dirichlet L-functions
Dirichlet L-functions are complex analytic functions built from Dirichlet characters that generalize the Riemann zeta function and play a central role in number theory, particularly in the study of primes in arithmetic progressions.
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B.
Representation Theory and Automorphic Functions
"Representation Theory and Automorphic Functions" is a seminal mathematical work by Israel Gelfand that develops the connections between representation theory of groups and the theory of automorphic forms, with deep applications in number theory and harmonic analysis.
-
C.
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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D.
Dedekind zeta functions
Dedekind zeta functions are number-theoretic functions attached to algebraic number fields that encode their arithmetic properties, such as the distribution of prime ideals and class numbers.
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E.
Selberg class
The Selberg class is a collection of Dirichlet series with specific analytic properties introduced to generalize and axiomatize L-functions in number theory.
- 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: Euler products for automorphic L-functions Target entity description: 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.
-
A.
Dirichlet L-functions
Dirichlet L-functions are complex analytic functions built from Dirichlet characters that generalize the Riemann zeta function and play a central role in number theory, particularly in the study of primes in arithmetic progressions.
-
B.
Representation Theory and Automorphic Functions
"Representation Theory and Automorphic Functions" is a seminal mathematical work by Israel Gelfand that develops the connections between representation theory of groups and the theory of automorphic forms, with deep applications in number theory and harmonic analysis.
-
C.
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.
-
D.
Dedekind zeta functions
Dedekind zeta functions are number-theoretic functions attached to algebraic number fields that encode their arithmetic properties, such as the distribution of prime ideals and class numbers.
-
E.
Selberg class
The Selberg class is a collection of Dirichlet series with specific analytic properties introduced to generalize and axiomatize L-functions in number theory.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
Euler product
ⓘ
L-function theory object ⓘ mathematical concept ⓘ |
| associatedTo |
automorphic representations
ⓘ
automorphic representations of reductive groups over global fields ⓘ cuspidal automorphic representations ⓘ |
| builtFrom |
Hecke operators
ⓘ
Satake isomorphism ⓘ Langlands program ⓘ
surface form:
local Langlands correspondence
|
| definedOver |
function fields
ⓘ
global fields ⓘ number fields ⓘ |
| encodes |
Hecke eigenvalues
ⓘ
Satake parameters ⓘ arithmetic information ⓘ local factors at primes ⓘ local-global compatibility ⓘ |
| field |
Langlands program
ⓘ
automorphic forms ⓘ number theory ⓘ |
| generalizes |
Dirichlet L-functions
ⓘ
surface form:
Dirichlet L-function Euler products
Euler product of the Riemann zeta function ⓘ |
| hasComponent |
archimedean local factors
ⓘ
local L-factors ⓘ non-archimedean local factors ⓘ |
| hasProperty |
absolute convergence in some right half-plane
ⓘ
local factors often given by characteristic polynomials of Frobenius elements ⓘ local factors rational in p^{-s} ⓘ meromorphic continuation expected to entire plane except possible poles ⓘ reflects unramified and ramified behavior at primes ⓘ |
| motivatedBy | classical Euler product for the Riemann zeta function ⓘ |
| relatedTo |
Artin L-functions
ⓘ
Euler products for automorphic L-functions self-linksurface differs ⓘ
surface form:
Langlands L-functions
L-functions ⓘ
surface form:
Rankin–Selberg L-functions
automorphic L-functions ⓘ functoriality in the Langlands program ⓘ standard L-functions of GL(n) ⓘ symmetric power L-functions ⓘ |
| satisfies |
Euler product factorization over all places
ⓘ
analytic continuation conjecturally ⓘ functional equation conjecturally ⓘ multiplicativity of local factors ⓘ |
| studiedIn |
algebraic number theory
ⓘ
arithmetic geometry ⓘ automorphic representation theory ⓘ |
| usedFor |
modularity and reciprocity questions
ⓘ
non-vanishing results for L-functions ⓘ relating automorphic representations and Galois representations ⓘ studying distribution of primes ⓘ subconvexity problems ⓘ |
How these facts were elicited
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Input
Subject: Euler products for automorphic L-functions Description of subject: 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.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
Euler product formula for the Riemann zeta function
→
generalizedBy
→
Euler products for automorphic L-functions
ⓘ
this entity surface form:
Euler products
this entity surface form:
Euler product expansions
this entity surface form:
Euler Products
Euler’s method of rearranging absolutely convergent series
→
relatedTo
→
Euler products for automorphic L-functions
ⓘ
this entity surface form:
Euler product expansions
Euler products for automorphic L-functions
→
relatedTo
→
Euler products for automorphic L-functions
self-linksurface differs
ⓘ
this entity surface form:
Langlands L-functions