Английская Википедия:Hecke operator
In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by Шаблон:Harvs, 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
Шаблон:Harvs used Hecke operators on modular forms in a paper on the special cusp form of Ramanujan, ahead of the general theory given by Шаблон:Harvs. Mordell proved that the Ramanujan tau function, expressing the coefficients of the Ramanujan form,
- <math> \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}, </math>
is a multiplicative function:
- <math> \tau(mn)=\tau(m)\tau(n) \quad \text{ for } (m,n)=1. </math>
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. The simplest meaning is combinatorial, namely as taking for a given integer Шаблон:Math some function Шаблон:Math defined on the lattices of fixed rank to
- <math>\sum f(\Lambda')</math>
with the sum taken over all the Шаблон:Math that are subgroups of Шаблон:Math of index Шаблон:Math. For example, with Шаблон:Math and two dimensions, there are three such Шаблон:Math. 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 Шаблон:Math be the set of Шаблон:Math integral matrices with determinant Шаблон:Math and Шаблон:Math be the full modular group Шаблон:Math. Given a modular form Шаблон:Math of weight Шаблон:Math, the Шаблон:Mathth 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 Шаблон:Math is in the upper half-plane and the normalization constant Шаблон:Math 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 Шаблон:Math in terms of the Fourier coefficients of Шаблон:Math:
- <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 Шаблон:Math then Шаблон:Math, so the subspace Шаблон:Math of cusp forms of weight Шаблон:Math is preserved by the Hecke operators. If a (non-zero) cusp form Шаблон:Math is a simultaneous eigenform of all Hecke operators Шаблон:Math with eigenvalues Шаблон:Math then Шаблон:Math and Шаблон:Math. Hecke eigenforms are normalized so that Шаблон:Math, 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.
Hecke algebras
Шаблон:Main Algebras of Hecke operators are called "Hecke algebras", and are commutative rings. In the classical elliptic modular form theory, the Hecke operators Шаблон:Math with Шаблон:Math coprime to the level acting on the space of cusp forms of a given weight are self-adjoint with respect to the Petersson inner product. Therefore, the spectral theorem implies that there is a basis of modular forms that are eigenfunctions for these Hecke operators. Each of these basic forms possesses an Euler product. More precisely, its Mellin transform is the Dirichlet series that has Euler products with the local factor for each prime Шаблон:Math is the inverseШаблон:Clarify of the Hecke polynomial, a quadratic polynomial in Шаблон:Math. In the case treated by Mordell, the space of cusp forms of weight 12 with respect to the full modular group is one-dimensional. It follows that the Ramanujan form has an Euler product and establishes the multiplicativity of Шаблон:Math.
Other related mathematical rings are also called "Hecke algebras", although sometimes the link to Hecke operators is not entirely obvious. These algebras include certain quotients of the group algebras of braid groups. The presence of this commutative operator algebra plays a significant role in the harmonic analysis of modular forms and generalisations.
See also
- Eichler–Shimura congruence relation
- Hecke algebra
- Abstract algebra
- Wiles's proof of Fermat's Last Theorem
References
- Шаблон:Citation (See chapter 8.)
- Шаблон:Springer
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Jean-Pierre Serre, A course in arithmetic.
- Don Zagier, Elliptic Modular Forms and Their Applications, in The 1-2-3 of Modular Forms, Universitext, Springer, 2008 Шаблон:ISBN
