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Concept
Hecke operator
Summary
In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by , is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic representations.
History
used Hecke operators on modular forms in a paper on the special cusp form of Ramanujan, ahead of the general theory given by . Mordell proved that the Ramanujan tau function, expressing the coefficients of the Ramanujan form,
: \Delta(z)=q\left(\prod_{n=1}^{\infty}(1-q^n)\right)^{24}=
\sum_{n=1}^{\infty} \tau(n)q^n, \quad q=e^{2\pi iz},
is a multiplicative function:
: \tau(mn)=\tau(m)\tau(n) \quad \text{ for } (m,n)=1.
The idea goes back to earlier work of Adolf Hurwitz, who treated algebraic correspondences between modular curves which realise some individual Hecke operators.
Mathematical description
Hecke operators can be realized in a number of contexts. Th
Official source
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