Hecke operators

E438307

algebraic operator endomorphism of modular forms space linear operator

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.

Try in SPARQL Jump to: Surface forms Statements Referenced by

All labels observed (1)

Label	Occurrences
Hecke operators canonical	4

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 ⓘ

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Ramanujan partition congruences → relatedTo → Hecke operators ⓘ

Representation Theory and Automorphic Functions → topic → Hecke operators ⓘ

Euler products for automorphic L-functions → builtFrom → Hecke operators ⓘ

Ramanujan–Petersson conjecture → relatedTo → Hecke operators ⓘ