Editing Hecke operator (section)

== Mathematical description ==

Hecke operators can be realized in a number of contexts. The simplest meaning is combinatorial, namely as taking for a given integer {{math|''n''}} some function {{math|''f''(''&Lambda;'')}} defined on the [[lattice (group)|lattices]] of fixed rank to

:<math>\sum f(\Lambda')</math>

with the sum taken over all the {{math|Λ&prime;}} that are [[subgroup]]s of {{math|Λ}} of index {{math|''n''}}. For example, with {{math|1=''n=2''}} and two dimensions, there are three such {{math|Λ&prime;}}. [[Modular form]]s are particular kinds of functions of a lattice, subject to conditions making them [[analytic function]]s and [[homogeneous function|homogeneous]] with respect to [[Homothetic transformation|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 coset]]s in the [[modular group]]. In the contemporary [[adelic]] approach, this translates to double cosets with respect to some compact subgroups.

=== Explicit formula ===

Let {{math|''M''<sub>''m''</sub>}} be the set of {{math|2×2}} integral matrices with [[determinant]] {{math|''m''}} and {{math|1=''&Gamma;'' =  ''M''<sub>1</sub>}} be the full [[modular group]] {{math|''SL''(2, '''Z''')}}. Given a modular form {{math|''f''(''z'')}} of weight {{math|''k''}}, the {{math|''m''}}th Hecke operator acts by the formula

: <math> 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), </math>

where {{math|''z''}} is in the [[upper half-plane]] and the normalization constant {{math|''m''<sup>''k''&minus;1</sup>}} assures that the image of a form with integer Fourier coefficients has integer Fourier coefficients. This can be rewritten in the form

: <math> T_m f(z) = m^{k-1}\sum_{a,d>0, ad=m}\frac{1}{d^k}\sum_{b \pmod d} f\left(\frac{az+b}{d}\right), </math>

which leads to the formula for the Fourier coefficients of {{math|1=''T''<sub>''m''</sub>(''f''(''z'')) = Σ ''b''<sub>''n''</sub>''q''<sup>''n''</sup>}} in terms of the Fourier coefficients of {{math|1=''f''(''z'') = Σ&nbsp;''a''<sub>''n''</sub>''q''<sup>''n''</sup>}}:

: <math> b_n = \sum_{r>0, r|(m,n)}r^{k-1}a_{mn/r^2}.</math>

One can see from this explicit formula that Hecke operators with different indices commute and that if {{math|1=''a''<sub>0</sub> = 0}} then {{math|1=''b''<sub>0</sub> = 0}}, so the subspace {{math|''S''<sub>''k''</sub>}} of cusp forms of weight {{math|''k''}} is preserved by the Hecke operators. If a (non-zero) cusp form {{math|''f''}} is a [[Eigenform|simultaneous eigenform]] of all Hecke operators {{math|''T''<sub>''m''</sub>}} with eigenvalues {{math|''&lambda;''<sub>''m''</sub>}} then {{math|1=''a''<sub>''m''</sub> = ''&lambda;''<sub>''m''</sub>''a''<sub>1</sub>}} and {{math|''a''<sub>1</sub> ≠ 0}}. Hecke eigenforms are '''normalized''' so that {{math|1=''a''<sub>1</sub> = 1}}, then

: <math> T_m f = a_m f, \quad a_m a_n = \sum_{r>0, r|(m,n)}r^{k-1}a_{mn/r^2},\ m,n\geq 1. </math>

Thus for normalized cuspidal Hecke eigenforms of integer weight, their Fourier coefficients coincide with their Hecke eigenvalues.