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Let $M ( k )$ be the vector space of (entire) modular forms of weight $k$, see [[Modular form|Modular form]] or [[#References|[a1]]]. Then the Hecke operator $T _ { n }$ is defined for $f \in M ( k )$ by

<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h1300605.png"/></td> <td style="width:5%;text-align:right;" valign="top">(a1)</td></tr></table>

where $\tau \in H$, the upper half-plane. One (easily) proves that $T _ { n } f \in M ( k )$ if $f \in M ( k )$.

If $f ( z ) = \sum _ { m = 0 } ^ { \infty } c ( m ) q ^ { m } ( z )$, $q ( z ) = e ^ { 2 \pi i z }$, is the Fourier expansion of $f$, then

\begin{equation*} T _ { n } f ( z ) = \sum _ { m = 0 } ^ { \infty } \gamma _ { n } ( m ) q ^ { m } ( z ), \end{equation*}

with

\begin{equation*} \gamma _ { n } ( m ) = \sum _ { d | ( n , m ) } d ^ { k - 1 } c \left( \frac { m n } { d ^ { 2 } } \right). \end{equation*}

Note that

\begin{equation*} T _ { n } T _ { m } = \sum _ { d  | ( n , m ) } d ^ { k - 1 } T _ { m n / d^2 } , \end{equation*}

so that, in particular, the $T _ { n }$ commute.

The discriminant form

\begin{equation*} \Delta ( z ) = ( 2 \pi ) ^ { 12 } \sum _ { m = 1 } ^ { \infty } \tau ( m ) q ^ { m } ( z ) \in M ( 12 ), \end{equation*}

where $\tau ( m )$ is the Ramanujan function, is a simultaneous eigenfunction of all $T _ { n }$.

Formula (a1) can be regarded as coming from an operation on lattices in the complex plane, $\widetilde{T} _ { n } ( L ) = \sum L ^ { \prime }$, where the sum is over all sublattices of $L$ of index $n$. This geometric definition, [[#References|[a4]]], makes (a1) easier to understand.

There are Hecke operators in much more general settings, e.g. for suitable subgroups of the [[Modular group|modular group]] $\Gamma$. A quite abstract group setting follows, [[#References|[a6]]].

Let $G$ be a [[Group|group]] and $D$ a subgroup. Another subgroup $D ^ { \prime }$ is commensurable with $D$ if $D \cap D ^ { \prime }$ is of finite index in both $D$ and $D ^ { \prime }$. Let $\widetilde { D } = \{ \alpha \in G : \alpha D \alpha ^ { - 1 } \text { is commensurable with} \ D\}$. This is a subgroup of $G$ that contains $D$.

Now, let $R$ be the $\bf Z$-module of all formal sums $\sum c _ { \alpha } D \alpha D$, i.e. the free Abelian group on the double cosets of $D$ in $\widetilde { D }$. There is an associative multiplication on $R$, defined as follows. Let $u = D \alpha D$, $v = D \beta D$. Then the product $u v = D \alpha D \beta D$ is clearly a (disjoint) union of double cosets. It gives a product $u . v$, provided multiplicities are taken into account. More precisely, let $D \alpha D = \coprod _ { \alpha ^ { \prime } \in A } D \alpha ^ { \prime }$, $D \beta D = \coprod _ { \beta ^ { \prime } \in A } D \beta ^ { \prime }$. Then

\begin{equation*} ( D \alpha D ) ( D \beta D ) = D \alpha D \beta D = D \alpha ( \bigcup _ { \beta ^ { \prime } } D \beta ^ { \prime } ) = \end{equation*}

\begin{equation*} = \bigcup _ { \beta ^ { \prime } } D \alpha D \beta ^ { \prime } = \bigcup _ { \alpha ^ { \prime } , \beta ^ { \prime } } D \alpha ^ { \prime } \beta ^ { \prime }. \end{equation*}

Now, let $\mu ( u \cdot v , w ) = \# \{ ( \alpha ^ { \prime } , \beta ^ { \prime } ) \in A \times B : D \alpha ^ { \prime } \beta ^ { \prime } = D \xi \,\text { with } w = D \xi D \}$. Then $u\cdot v = \sum _ { w } \mu ( u  \cdot v  , w ) w$. (The restriction of the $D \alpha D$ to $\alpha \in \widetilde{ D }$ is needed to keep things, e.g. the sets $A$, $B$, finite.)

Let $X$ be a subset of $\widetilde { D }$ containing $D$ and multiplicatively closed. Then one defines $R _ { 0 } ( X , D )$ as the submodule of $R$ spanned by the $D \xi D$ for $\xi \in X$. This gives a subring of $R$. Finally, one defines $R ( X , D )$, the Hecke algebra of $( X , D )$ as $R_0 ( X , D ) \otimes \mathbf{Q}$.

In many situations the double cosets $D \xi D$ act on forms, functions, etc., which gives Hecke operators. See [[#References|[a5]]] for an example in the case of double cosets with respect to the principal congruence subgroup

\begin{equation*} \Gamma ( n ) = \left\{ \left( \begin{array} { l l } { a } &amp; { b } \\ { c } &amp; { d } \end{array} \right) \in \operatorname{SL} ( 2 , \mathbf{Z} ) : \left( \begin{array} { c c } { a } &amp; { b } \\ { c } &amp; { d } \end{array} \right) \equiv \left( \begin{array} { l l } { 1 } &amp; { 0 } \\ { 0 } &amp; { 1 } \end{array} \right) ( \operatorname { mod } n ) \right\}, \end{equation*}

which gives rise to the (usual) Hecke operators for modular forms.

In [[#References|[a6]]] this setting is used to define Hecke operators for the case of adelic groups.

Modular forms turn up all over mathematics and physics and, hence, so do the Hecke operators. See the references for a variety of uses of them.

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  T.M. Apostol,   "Modular functions and Dirichlet series in number theory" , Springer  (1976)  pp. 120ff</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  N. Hurt,   "Quantum chaos and mesoscopic systems" , Kluwer Acad. Publ.  (1997)  pp. 101; 163ff</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  M.I. Knopp,   "Modular functions in analytic number theory" , Markham Publ.  (1970)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  A. Ogg,   "Modular forms and Dirichlet series" , Benjamin  (1969)  pp. Chap. II</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  R.A. Rankin,   "Modular forms and functions" , Cambridge Univ. Press  (1977)  pp. Chap. 9</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  G. Shimura,   "Euler products and Eisenstein series" , Amer. Math. Soc.  (1997)  pp. Sect. 11</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  A.B. Venkov,   "Spectral theory of automorphic functions" , Kluwer Acad. Publ.  (1990)  pp. 34; 59</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  D. Bump,   "Automorphic forms and representations" , Cambridge Univ. Press  (1997)</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  N.E. Hurt,   "Exponential sums and coding theory. A review"  ''Acta Applic. Math.'' , '''46'''  (1997)  pp. 49–91</td></tr></table>