In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by Erich Hecke (1937a,1937b), 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.
Mathematical description Hecke operators can be realized in a number of contexts. The simplest meaning is combinatorial, namely as taking for a given integer n {\displaystyle n} some function f ( Λ ) {\textstyle f(\Lambda )} defined on the lattices of fixed rank to
∑ f ( Λ ′ ) {\displaystyle \sum f(\Lambda ')}
with the sum taken over all the Λ ′ {\textstyle \Lambda '} that are subgroups of Λ {\textstyle \Lambda } of index n {\displaystyle n} . For example, with n = 2 {\textstyle n=2} and two dimensions, there are three such Λ ′ {\textstyle \Lambda '} . 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.
Explicit formula Let M m {\textstyle M_{m}} be the set of 2 × 2 {\textstyle 2\times 2} integral matrices with determinant m {\displaystyle m} and Γ = M 1 {\textstyle \Gamma =M_{1}} be the full modular group SL 2 ( Z ) {\textstyle {\text{SL}}_{2}(\mathbb {Z} )} . Given a modular form f ( z ) {\textstyle f(z)} of weight k {\displaystyle k} , the m {\displaystyle m} th Hecke operator acts by the formula
T m f ( z ) = m k − 1 ∑ ( a b c d ) ∈ Γ ∖ M m ( c z + d ) − k f ( a z + b c z + d ) , {\displaystyle T_{m}f(z)=m^{k-1}\sum _{\left({\begin{smallmatrix}a&b\\c&d\end{smallmatrix}}\right)\in \Gamma \backslash M_{m}}(cz+d)^{-k}f\left({\frac {az+b}{cz+d}}\right),}
where z {\textstyle z} is in the upper half-plane and the normalization constant m k − 1 {\textstyle m^{k-1}} assures that the image of a form with integer Fourier coefficients has integer Fourier coefficients. This can be rewritten in the form
T m f ( z ) = m k − 1 ∑ a d = m a , d > 0 1 d k ∑ b ( mod d ) f ( a z + b d ) , {\displaystyle T_{m}f(z)=m^{k-1}\sum _{\begin{smallmatrix}ad=m\\a,d>0\end{smallmatrix}}{\frac {1}{d^{k}}}\sum _{b{\pmod {d}}}f\left({\frac {az+b}{d}}\right),}
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