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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h1300601.png" /> be the vector space of (entire) modular forms of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h1300602.png" />, see [[Modular form|Modular form]] or [[#References|[a1]]]. Then the Hecke operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h1300603.png" /> is defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h1300604.png" /> by
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h1300605.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
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was used.
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If the TeX and formula formatting is correct and if all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h1300606.png" />, the upper half-plane. One (easily) proves that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h1300607.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h1300608.png" />.
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Out of 67 formulas, 65 were replaced by TEX code.-->
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h1300609.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006010.png" />, is the Fourier expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006011.png" />, then
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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 valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006012.png" /></td> </tr></table>
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\begin{equation}
 +
\tag{a1}
 +
(T_n f)(\tau) = n^{k-1} \sum_{d|n} d^{-k} \sum_{b=0}^{d-1} f \left( \frac{n\tau+bd}{d^2} \right),
 +
\end{equation}
 +
 
 +
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
 
with
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006013.png" /></td> </tr></table>
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\begin{equation*} \gamma _ { n } ( m ) = \sum _ { d | ( n , m ) } d ^ { k - 1 } c \left( \frac { m n } { d ^ { 2 } } \right). \end{equation*}
  
 
Note that
 
Note that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006014.png" /></td> </tr></table>
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\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006015.png" /> commute.
+
so that, in particular, the $T _ { n }$ commute.
  
 
The discriminant form
 
The discriminant form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006016.png" /></td> </tr></table>
+
\begin{equation*} \Delta ( z ) = ( 2 \pi ) ^ { 12 } \sum _ { m = 1 } ^ { \infty } \tau ( m ) q ^ { m } ( z ) \in M ( 12 ), \end{equation*}
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006017.png" /> is the Ramanujan function, is a simultaneous eigenfunction of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006018.png" />.
 
  
Formula (a1) can be regarded as coming from an operation on lattices in the complex plane, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006019.png" />, where the sum is over all sublattices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006020.png" /> of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006021.png" />. This geometric definition, [[#References|[a4]]], makes (a1) easier to understand.
+
where $\tau ( m )$ is the Ramanujan function, is a simultaneous eigenfunction of all $T _ { n }$.
  
There are Hecke operators in much more general settings, e.g. for suitable subgroups of the [[Modular group|modular group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006022.png" />. A quite abstract group setting follows, [[#References|[a6]]].
+
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.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006023.png" /> be a [[Group|group]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006024.png" /> a subgroup. Another subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006025.png" /> is commensurable with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006026.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006027.png" /> is of finite index in both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006029.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006030.png" />. This is a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006031.png" /> that contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006032.png" />.
+
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]]].
  
Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006033.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006034.png" />-module of all formal sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006035.png" />, i.e. the free Abelian group on the double cosets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006036.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006037.png" />. There is an associative multiplication on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006038.png" />, defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006040.png" />. Then the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006041.png" /> is clearly a (disjoint) union of double cosets. It gives a product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006042.png" />, provided multiplicities are taken into account. More precisely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006044.png" />. Then
+
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 $D$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006045.png" /></td> </tr></table>
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006046.png" /></td> </tr></table>
+
\begin{equation*} ( D \alpha D ) ( D \beta D ) = D \alpha D \beta D = D \alpha ( \bigcup _ { \beta ^ { \prime } } D \beta ^ { \prime } ) = \end{equation*}
  
Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006047.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006048.png" />. (The restriction of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006049.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006050.png" /> is needed to keep things, e.g. the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006052.png" />, finite.)
+
\begin{equation*} = \bigcup _ { \beta ^ { \prime } } D \alpha D \beta ^ { \prime } = \bigcup _ { \alpha ^ { \prime } , \beta ^ { \prime } } D \alpha ^ { \prime } \beta ^ { \prime }. \end{equation*}
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006053.png" /> be a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006054.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006055.png" /> and multiplicatively closed. Then one defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006056.png" /> as the submodule of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006057.png" /> spanned by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006058.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006059.png" />. This gives a subring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006060.png" />. Finally, one defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006061.png" />, the Hecke algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006062.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006063.png" />.
+
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.)
  
In many situations the double cosets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006064.png" /> 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
+
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 $R_0 ( X , D ) \otimes \mathbf{Q}$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006065.png" /></td> </tr></table>
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130060/h13006066.png" /></td> </tr></table>
+
\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.
 
which gives rise to the (usual) Hecke operators for modular forms.
Line 54: Line 64:
  
 
====References====
 
====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>
+
<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>

Latest revision as of 00:38, 15 February 2024

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

\begin{equation} \tag{a1} (T_n f)(\tau) = n^{k-1} \sum_{d|n} d^{-k} \sum_{b=0}^{d-1} f \left( \frac{n\tau+bd}{d^2} \right), \end{equation}

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, [a4], makes (a1) easier to understand.

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

Let $G$ be a 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 [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 } & { b } \\ { c } & { d } \end{array} \right) \in \operatorname{SL} ( 2 , \mathbf{Z} ) : \left( \begin{array} { c c } { a } & { b } \\ { c } & { d } \end{array} \right) \equiv \left( \begin{array} { l l } { 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right) ( \operatorname { mod } n ) \right\}, \end{equation*}

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

In [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

[a1] T.M. Apostol, "Modular functions and Dirichlet series in number theory" , Springer (1976) pp. 120ff
[a2] N. Hurt, "Quantum chaos and mesoscopic systems" , Kluwer Acad. Publ. (1997) pp. 101; 163ff
[a3] M.I. Knopp, "Modular functions in analytic number theory" , Markham Publ. (1970)
[a4] A. Ogg, "Modular forms and Dirichlet series" , Benjamin (1969) pp. Chap. II
[a5] R.A. Rankin, "Modular forms and functions" , Cambridge Univ. Press (1977) pp. Chap. 9
[a6] G. Shimura, "Euler products and Eisenstein series" , Amer. Math. Soc. (1997) pp. Sect. 11
[a7] A.B. Venkov, "Spectral theory of automorphic functions" , Kluwer Acad. Publ. (1990) pp. 34; 59
[a8] D. Bump, "Automorphic forms and representations" , Cambridge Univ. Press (1997)
[a9] N.E. Hurt, "Exponential sums and coding theory. A review" Acta Applic. Math. , 46 (1997) pp. 49–91
How to Cite This Entry:
Hecke operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hecke_operator&oldid=13264
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article