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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.
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,
is a multiplicative function:
The idea goes back to earlier work of Adolf Hurwitz, who treated algebraic correspondences between modular curves which realise some individual Hecke operators.
Hecke operators can be realized in a number of contexts. The simplest meaning is combinatorial, namely as taking for a given integer n some function f(Λ) defined on the lattices of fixed rank to
with the sum taken over all the Λ′ that are subgroups of Λ of index n. For example, with n=2 and two dimensions, there are three such Λ′. Modular forms are particular kinds of functions of a lattice, subject to conditions making them analytic functions and homogeneous with respect to homotheties, as well as moderate growth at infinity; these conditions are preserved by the summation, and so Hecke operators preserve the space of modular forms of a given weight.
Another way to express Hecke operators is by means of double cosets in the modular group. In the contemporary adelic approach, this translates to double cosets with respect to some compact subgroups.
Let Mm be the set of 2×2 integral matrices with determinant m and Γ = M1 be the full modular group SL(2, Z). Given a modular form f(z) of weight k, the mth Hecke operator acts by the formula
where z is in the upper half-plane and the normalization constant mk−1 assures that the image of a form with integer Fourier coefficients has integer Fourier coefficients.
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