Hecke theory
E438309
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Hecke theory canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4410552 — 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: Hecke theory Context triple: [Ramanujan partition congruences, laterProvedUsing, Hecke theory]
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A.
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.
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B.
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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C.
Deligne–Lusztig theory
Deligne–Lusztig theory is a framework in algebraic geometry and representation theory that constructs and studies representations of finite groups of Lie type using varieties defined over finite fields.
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D.
Iwasawa theory
Iwasawa theory is a branch of number theory that studies the growth of arithmetic invariants in infinite towers of number fields, particularly using p-adic methods.
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E.
L-functions
L-functions are complex analytic functions, often arising from number theory and algebraic geometry, that encode deep arithmetic information and generalize the Riemann zeta function.
- 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: Hecke theory Target entity description: 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.
-
A.
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.
-
B.
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.
-
C.
Deligne–Lusztig theory
Deligne–Lusztig theory is a framework in algebraic geometry and representation theory that constructs and studies representations of finite groups of Lie type using varieties defined over finite fields.
-
D.
Iwasawa theory
Iwasawa theory is a branch of number theory that studies the growth of arithmetic invariants in infinite towers of number fields, particularly using p-adic methods.
-
E.
L-functions
L-functions are complex analytic functions, often arising from number theory and algebraic geometry, that encode deep arithmetic information and generalize the Riemann zeta function.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf | branch of number theory ⓘ |
| appliesTo |
Hilbert modular forms
ⓘ
Siegel modular forms ⓘ automorphic representations ⓘ modular forms of half-integral weight ⓘ modular forms of integral weight ⓘ |
| developedFrom |
theory of Dirichlet L-functions
ⓘ
theory of modular forms ⓘ |
| field | number theory ⓘ |
| focusesOn |
Hecke operators
ⓘ
modular forms ⓘ |
| historicalOrigin | work of Erich Hecke ⓘ |
| providesToolsFor |
classifying eigenforms
ⓘ
constructing L-functions ⓘ decomposing spaces of modular forms ⓘ relating modular forms to Galois representations ⓘ studying congruences between modular forms ⓘ |
| relatedTo |
Langlands program
ⓘ
algebraic geometry ⓘ arithmetic geometry ⓘ class field theory ⓘ
linked to:
global class field theory
modularity theorem ⓘ
linked to:
Taniyama–Shimura–Weil conjecture
representation theory ⓘ |
| studies |
Dirichlet series
ⓘ
Eisenstein series ⓘ Euler products ⓘ Fourier coefficients of modular forms ⓘ Galois representations attached to modular forms ⓘ Hecke algebras ⓘ
linked to:
Hecke algebra
Hecke eigenvalues ⓘ L-functions ⓘ arithmetic properties of modular forms ⓘ automorphic forms ⓘ cusp forms ⓘ eigenforms ⓘ local factors of L-functions ⓘ modular curves ⓘ newforms ⓘ oldforms ⓘ |
| usesConcept |
Grössencharacters
ⓘ
Hecke algebra action on modular forms ⓘ Hecke characters ⓘ Hecke eigenbasis ⓘ Satake parameters ⓘ adelic methods ⓘ commuting family of operators ⓘ multiplicity one theorem ⓘ representation theory of adelic groups ⓘ |
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: Hecke theory Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.