Eisenstein series
E924212
Eisenstein series are special types of complex analytic functions on the upper half-plane (or more general symmetric spaces) that play a central role in the theory of modular and automorphic forms, connecting number theory, representation theory, and harmonic analysis.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Eisenstein series canonical | 8 |
| Eisenstein series E4 | 1 |
| Eisenstein series E6 | 1 |
| Siegel Eisenstein series | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11411969 — 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: Eisenstein series Context triple: [Representation Theory and Automorphic Functions, topic, Eisenstein series]
-
A.
Siegel modular form
A Siegel modular form is a type of complex analytic function defined on the Siegel upper half-space that generalizes classical modular forms to higher dimensions and plays a central role in number theory and algebraic geometry.
-
B.
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.
-
C.
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.
-
D.
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.
-
E.
Hecke theory
Hecke theory is a branch of number theory centered on Hecke operators and modular forms, providing powerful tools to study arithmetic properties of modular forms and related objects.
- 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: Eisenstein series Target entity description: Eisenstein series are special types of complex analytic functions on the upper half-plane (or more general symmetric spaces) that play a central role in the theory of modular and automorphic forms, connecting number theory, representation theory, and harmonic analysis.
-
A.
Siegel modular form
A Siegel modular form is a type of complex analytic function defined on the Siegel upper half-space that generalizes classical modular forms to higher dimensions and plays a central role in number theory and algebraic geometry.
-
B.
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.
-
C.
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.
-
D.
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.
-
E.
Hecke theory
Hecke theory is a branch of number theory centered on Hecke operators and modular forms, providing powerful tools to study arithmetic properties of modular forms and related objects.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
automorphic form
ⓘ
complex analytic function ⓘ mathematical object ⓘ modular form ⓘ |
| definedOn |
complex upper half-plane
ⓘ
symmetric space ⓘ |
| field |
harmonic analysis
ⓘ
number theory ⓘ representation theory ⓘ |
| hasProperty |
admits Fourier expansion
ⓘ
holomorphic in the upper half-plane for suitable parameters ⓘ invariant under modular group action ⓘ meromorphic continuation in complex parameter ⓘ satisfies functional equation ⓘ |
| introducedBy | Gotthold Eisenstein ⓘ |
| relatedTo |
Bernoulli numbers
ⓘ
Dedekind zeta function ⓘ
linked to:
Dedekind zeta functions
Dirichlet series ⓘ Fourier expansion ⓘ Hecke L-functions ⓘ
linked to:
L-functions
Hecke operators ⓘ Hilbert modular forms ⓘ
linked to:
Hilbert modular varieties
L-functions ⓘ Langlands Eisenstein series ⓘ Langlands program ⓘ Maass forms ⓘ Poisson summation formula ⓘ Rankin–Selberg method ⓘ Riemann zeta function ⓘ Selberg trace formula ⓘ Siegel modular forms ⓘ automorphic forms ⓘ constant term formula ⓘ constant term of automorphic forms ⓘ cusp forms ⓘ elliptic modular forms ⓘ functional equation of L-functions ⓘ induced representations ⓘ modular curves ⓘ modular forms ⓘ parabolic subgroups ⓘ principal series representations ⓘ spectral decomposition of automorphic forms ⓘ spectral theory of automorphic forms ⓘ theta functions ⓘ |
| usedFor |
construction of L-functions
ⓘ
explicit formulas in modular form theory ⓘ proofs of functional equations ⓘ spectral decomposition of L2 of arithmetic quotients ⓘ study of special values of L-functions ⓘ |
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: Eisenstein series Description of subject: Eisenstein series are special types of complex analytic functions on the upper half-plane (or more general symmetric spaces) that play a central role in the theory of modular and automorphic forms, connecting number theory, representation theory, and harmonic analysis.
Referenced by (11)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Eisenstein series
linked to: Eisenstein series
linked to: Eisenstein series