Euler products for automorphic L-functions

E304347

Euler product L-function theory object mathematical concept

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.

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All labels observed (5)

Label	Occurrences
Euler product expansions	2
Euler products	2
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.
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
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 ⓘ local Langlands correspondence ⓘ linked to: Langlands program
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-function Euler products ⓘ linked to: Dirichlet L-functions 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 ⓘ Langlands L-functions ⓘ linked to: Euler products for automorphic L-functions Rankin–Selberg L-functions ⓘ linked to: 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