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 (1)
| Label | Occurrences |
|---|---|
| Hecke operators canonical | 4 |
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]
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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
NERFINISHED
ⓘ
Deligne’s proof relating eigenvalues to Frobenius traces ⓘ Shimura correspondence NERFINISHED ⓘ modularity theorem NERFINISHED ⓘ |
| 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)
NERFINISHED
ⓘ
Hecke algebra of GL(n) NERFINISHED ⓘ global Hecke algebra ⓘ local Hecke algebras at primes ⓘ |
| centralRoleIn |
Langlands program
NERFINISHED
ⓘ
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 NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
Hecke operator T_p for prime p ⓘ U_p operator at prime p dividing the level ⓘ diamond operators ⓘ |
| namedAfter | Erich Hecke NERFINISHED ⓘ |
| relatedTo |
Euler factors of L-functions
ⓘ
Hecke characters ⓘ Satake parameters ⓘ newforms and oldforms decomposition ⓘ |
| usedToDefine |
Fourier coefficient multiplicativity of eigenforms
ⓘ
Hecke L-functions NERFINISHED ⓘ Hecke eigenforms NERFINISHED ⓘ Hecke eigenvalues NERFINISHED ⓘ |
| 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
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Instruction
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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 (4)
Full triples — surface form annotated when it differs from this entity's canonical label.