Hecke operators
E438307
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Hecke operators canonical | 8 |
| Hecke operator | 1 |
| Hecke operator T_n | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4410545 — 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 operators Context triple: [Ramanujan partition congruences, relatedTo, Hecke operators]
-
A.
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.
-
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.
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.
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.
-
E.
Hilbert’s twelfth problem
Hilbert’s twelfth problem is one of David Hilbert’s famous list of 23 problems, asking for a general explicit class field theory that would generate all abelian extensions of a given number field using special values of analytic functions.
- 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 operators Target entity description: 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.
-
A.
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.
-
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.
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.
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.
-
E.
Hilbert’s twelfth problem
Hilbert’s twelfth problem is one of David Hilbert’s famous list of 23 problems, asking for a general explicit class field theory that would generate all abelian extensions of a given number field using special values of analytic functions.
- F. None of above. chosen
Statements (56)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic operator
ⓘ
endomorphism of modular forms space ⓘ linear operator ⓘ |
| actOn |
automorphic forms
ⓘ
cohomology of arithmetic groups ⓘ cusp forms ⓘ modular forms ⓘ modular symbols ⓘ spaces of modular forms of fixed weight and level ⓘ |
| appearIn |
Atkin–Lehner theory
ⓘ
Deligne’s proof relating eigenvalues to Frobenius traces ⓘ Shimura correspondence ⓘ modularity theorem ⓘ
linked to:
Taniyama–Shimura–Weil conjecture
|
| are |
commuting family of operators
ⓘ
diagonalizable over C on spaces of modular forms ⓘ normal operators with respect to Petersson inner product ⓘ |
| associatedWith |
Hecke algebra of GL(2)
ⓘ
linked to:
Hecke algebra
Hecke algebra of GL(n) ⓘ
linked to:
Hecke algebra
global Hecke algebra ⓘ local Hecke algebras at primes ⓘ |
| centralRoleIn |
Langlands program
ⓘ
arithmetic of modular forms ⓘ congruences between modular forms ⓘ theory of L-functions ⓘ theory of automorphic forms ⓘ theory of modular forms ⓘ |
| definedBy |
correspondences on modular curves
ⓘ
double coset operators ⓘ sums over divisors of integers ⓘ sums over lattices of given index ⓘ |
| definedOver | number theory ⓘ |
| generate | Hecke algebra ⓘ |
| haveProperty |
form a commutative algebra
ⓘ
mutually commute ⓘ preserve eigenform subspaces ⓘ preserve level up to Atkin–Lehner theory ⓘ preserve space of cusp forms ⓘ preserve weight of modular forms ⓘ self-adjoint with respect to Petersson inner product ⓘ |
| include |
Hecke operator T_n
ⓘ
linked to:
Hecke operators
Hecke operator T_p for prime p ⓘ U_p operator at prime p dividing the level ⓘ diamond operators ⓘ |
| namedAfter | Erich Hecke ⓘ |
| relatedTo |
Euler factors of L-functions
ⓘ
Hecke characters ⓘ Satake parameters ⓘ newforms and oldforms decomposition ⓘ |
| usedToDefine |
Fourier coefficient multiplicativity of eigenforms
ⓘ
Hecke L-functions ⓘ
linked to:
L-functions
Hecke eigenforms ⓘ Hecke eigenvalues ⓘ |
| usedToStudy |
Galois representations attached to modular forms
ⓘ
arithmetic of modular curves ⓘ congruences between modular forms modulo primes ⓘ modularity of elliptic curves ⓘ |
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 operators Description of subject: 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.
Referenced by (10)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Hecke operators
linked to: Hecke operators