Editing Hecke operator

{{Short description|Linear operator acting on modular forms}}
In [[mathematics]], in particular in the theory of [[modular form]]s, a '''Hecke operator''', studied by {{harvs|txt|authorlink=Erich Hecke|last=Hecke|first=Erich|year=1937a,1937b}}, is a certain kind of "averaging" operator that plays a significant role in the structure of [[vector space]]s of modular forms and more general [[automorphic representation]]s.

== History ==

{{harvs|txt|last=Mordell|authorlink=Louis J. Mordell|year=1917}} used Hecke operators on modular forms in a paper on the special [[cusp form]] of [[Ramanujan]], ahead of the general theory given by {{harvs|txt|authorlink=Erich Hecke|last=Hecke|year=1937a,1937b}}. 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 correspondence]]s between [[modular curve]]s 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 {{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.

== Hecke algebras ==
{{Main|Hecke algebra}}
Algebras of Hecke operators are called "Hecke algebras", and are [[commutative ring]]s. In the classical [[elliptic modular form]] theory,  the Hecke operators {{math|''T''<sub>''n''</sub>}} with {{math|''n''}} coprime to the level acting on the space of cusp forms of a given weight are [[self-adjoint operator|self-adjoint]] with respect to the [[Petersson inner product]]. Therefore, the [[spectral theorem]] implies that there is a basis of modular forms that are [[eigenfunction]]s 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 product]]s with the local factor for each prime {{math|''p''}} is the inverse{{clarify|reason=Syntax error|date=April 2014}} of the '''Hecke polynomial''', a quadratic polynomial in {{math|''p''<sup>&minus;''s''</sup>}}. 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 {{math|''&tau;''(''n'')}}.

[[Hecke algebra (disambiguation)|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 ring|group algebra]]s of [[braid group]]s. 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 ==

* {{Citation | last1=Apostol | first1=Tom M. | author1-link=Tom M. Apostol | title=Modular functions and Dirichlet series in number theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | isbn=978-0-387-97127-8 | year=1990 | url-access=registration | url=https://archive.org/details/modularfunctions0000apos }} ''(See chapter 8.)''
*{{springer|title=Hecke operator|id=p/h130060}}
*{{Citation | last1=Hecke | first1=E. | title=Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. I. | language=German | doi=10.1007/BF01594160 | zbl=0015.40202 | year=1937a | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=114 | pages=1–28}} 
*{{Citation | last1=Hecke | first1=E. | title=Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. II. | language=German | doi=10.1007/BF01594180 | zbl=0016.35503 | year=1937b | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=114 | pages=316–351}}
*{{Citation | last1=Mordell | first1=Louis J. | author1-link=Louis Mordell | title=On Mr.  Ramanujan's empirical expansions of modular functions. | url=https://archive.org/stream/proceedingsofcam1920191721camb#page/n133 | jfm=46.0605.01 | year=1917 | journal=[[Proceedings of the Cambridge Philosophical Society]] | volume=19 | pages=117–124}}
* [[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 {{ISBN|978-3-540-74117-6}}

[[Category:Modular forms]]