View source for Fourier coefficients of automorphic forms

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Many of the applications of automorphic forms (cf. also [[Automorphic form|Automorphic form]]) involve their Fourier coefficients. Here, the special case of holomorphic modular forms $f$ of weight $k$ for the full [[Modular group|modular group]] $\operatorname{SL} ( 2 , {\bf Z} )$ will be considered. If $f ( z )$ is a such a [[Modular form|modular form]], then

\begin{equation*} f \left( \frac { a z + b } { c z + d } \right) = ( c z + d ) ^ { k } f ( z ), \end{equation*}

for all integers $a$, $b$, $c$, $d$ such that $a d - b c = 1$ and $z$ in the upper half-plane. Therefore it has period $1$ in the real part of $z$ and must have a Fourier expansion

\begin{equation*} f ( z ) = \sum _ { n = 0 } ^ { \infty } c ( n ) e ^ { 2 \pi i n z }. \end{equation*}

The numbers $c ( n )$ are the Fourier coefficients of $f$. The modular form is a cusp form if $c ( 0 ) = 0$. Some references are [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]], [[#References|[a10]]], [[#References|[a11]]], [[#References|[a12]]], [[#References|[a15]]] (see also the extensive bibliography in this).

For Eisenstein series

\begin{equation*} G _ { k } ( z ) = \sum _ { c , d \in \mathbf{Z} ^ { 2 } \backslash 0 } ( c z + d ) ^ { - k } ,\, k = 4,6,8, \dots , \end{equation*}

the non-zero Fourier coefficients are essentially divisor functions

\begin{equation*} \sigma _ { k - 1 } ( n ) = \sum _ { 0 &lt; d | n } d ^ { k - 1 }. \end{equation*}

The theta-function associated to a positive-definite real symmetric $( n \times n )$-matrix $P$ and a point $z$ in the upper half-plane is defined 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/f/f120/f120100/f12010023.png"/></td> </tr></table>

where $\square ^ { t } a$ denotes the transpose of the column vector $a$. One says that $P$ is an even integral matrix when $\square ^ { t } a P a$ is an even integer, for all integral column vectors $a$. In this case, $\theta _ { n } ( P , z )$ is a modular form of weight $n / 2$. Then the $m$th Fourier coefficient $r ( P , m )$ of $\theta _ { n } ( P , z )$ is the number of representations of $2 m$ as $\square ^ { t } a P a$, for some integer vector $a$. It follows from the one-dimensionality of the space of modular forms of weight $4$ that when $n = 8$,

\begin{equation*} r ( I _ { 8 } , m ) = 240 \sigma _ { 3 } ( m ), \end{equation*}

where $I_8$ denotes the identity $( 8 \times 8 )$-matrix.

There are many more examples of number-theoretically interesting Fourier coefficients. The Fourier coefficients of the inverse of the [[Dedekind eta-function|Dedekind eta-function]] involve the [[partition function (number theory)|partition function]] $p ( n )$, which is the number of ways of writing $n$ as a sum of positive integers. The discriminant function $\Delta ( z ) = ( 60 G _ { 4 } ) ^ { 3 } - 27 ( 140 G _ { 6 } ) ^ { 2 }$ is a cusp form whose Fourier coefficients are $( 2 \pi ) ^ { 12 } \tau ( n )$, where $\tau ( n )$ is the Ramanujan tau-function (cf. also [[Ramanujan function]]). S. Ramanujan conjectured that $\tau ( m n ) = \tau ( m ) \tau ( n )$ when $m$ and $n$ are relatively prime. This was proved by L.J. Mordell. E. Hecke created the theory of Hecke operators, which gave a general framework for such results. Ramanujan also conjectured that for all prime numbers $p$ one has the inequality $| \tau ( p ) | \leq 2 p ^ { 11 / 2 }$. P. Deligne proved this conjecture as a special case of a much more general result holding for all cusp forms. This result has applications in extremal graph theory (cf. also [[Graph theory]]); [[#References|[a7]]], [[#References|[a10]]].

Two other conjectures are still (1998) open. Computations lead one to believe that $\tau ( n ) \neq 0$, for all $n &gt; 0$. Writing $\tau ( p ) = 2 p ^ { 11 / 2 } \operatorname { cos } ( \phi _ { p } )$, J.-P. Serre conjectured that the angles $\phi _ { p }$ are distributed in $[ 0 , \pi ]$ with respect to the measure $( 2 / \pi ) \operatorname { sin } ^ { 2 } \phi d \phi$ (the Sato–Tate or Wigner semi-circle distribution).

The modular invariant is $J = 60 G _ { 4 } ^ { 3 } / \Delta$. The function $j = 1728 J$ has integer Fourier coefficients (writing $q = e ^ { 2 \pi i z }$)

\begin{equation*} j ( z ) = q ^ { - 1 } + 744 + 196884 q + 21493760 q ^ { 2 } +\dots . \end{equation*}

There is a surprising connection of these Fourier coefficients with the representations of the largest of the $26$ sporadic finite simple groups (the Monster; cf. also [[Simple finite group|Simple finite group]]). All of the early Fourier coefficients of $j ( z )$ are simple linear combinations of degrees of characters of the Monster group. This led J.H. Conway [[#References|[a1]]] to refer to the "moonshine" properties of the Monster. In 1998, R.E. Borcherds won the Fields Medal in part because he proved the Conway–Norton "[[moonshine conjectures]]" ; see [[#References|[a17]]].

The Hecke correspondence maps a modular form $f$ with Fourier coefficients $c ( n )$ to an $L$-function (cf. also [[Dirichlet L-function|Dirichlet $L$-function]]) given by

\begin{equation*} L ( s ) = \sum _ { n = 1 } ^ { \infty } c ( n ) n ^ { - s } , \operatorname { Re } s &gt; k. \end{equation*}

This $L$-function has an analytic continuation and a functional equation. The [[Riemann zeta-function|Riemann zeta-function]] corresponds to the simplest theta-function $\theta _ { 1 } ( 1 , z )$. But here the weight of the modular form is $1/2$ and $\theta _ { 1 }$ is not invariant under the full modular group.

If $f$ is a form of integer weight for the modular group which is an eigenfunction for all the Hecke operators, then $L ( s )$ has an [[Euler product|Euler product]]. There are also converse theorems. The result is that many of the important zeta- and $L$-functions of number theory correspond to modular forms.

The Taniyama–Shimura–Weil conjecture says that the zeta-function of an [[elliptic curve]] over the rational numbers should come via the Hecke correspondence from a cusp form of weight $2$ (for a congruence subgroup of the modular group). This conjecture is intrinsically linked with the proof by A. Wiles [[#References|[a16]]] of Fermat's last theorem (cf. also [[Fermat last theorem|Fermat last theorem]]). In 1999, B. Conrad, F. Diamond and R. Taylor proved the Taniyama–Shimura–Weil conjecture when the conductor is not divisible by $27$ (see [[#References|[a18]]]) and announced with C. Breuil a proof of the conjecture in general.

Maass wave forms (see [[#References|[a8]]]) are eigenfunctions of the non-Euclidean Laplacian on the upper half-plane which are invariant under the modular group. Their Fourier expansions involve $K$-Bessel functions and the Fourier coefficients are more mysterious, except for those of the Eisenstein series, which are still divisor functions. Computer studies have been carried out in, e.g., [[#References|[a4]]], [[#References|[a14]]]. The analogue of the Ramanujan conjecture is still (1998) open.

Siegel modular forms are holomorphic functions $f ( Z )$, where $Z$ is a complex symmetric $( n \times n )$-matrix with positive-definite imaginary part, such that

\begin{equation*} f ( ( A Z + B ) ( C Z + D ) ^ { - 1 } ) = \operatorname { det } ( C Z + D ) ^ { k } f ( Z ), \end{equation*}

for all matrices $A$, $B$, $C$, $D$ with integer entries and such that with

\begin{equation*} g = \left( \begin{array} { c c } { A } &amp; { B } \\ { C } &amp; { D } \end{array} \right) \end{equation*}

one has $\square ^ { t } g  J g  = J$, where

\begin{equation*} J = \left( \begin{array} { c c } { 0 } &amp; { I _ { n } } \\ { - I _ { n } } &amp; { 0 } \end{array} \right), \end{equation*}

with $ { I } _ { n }$ the identity $( n \times n )$-matrix. Then $f ( Z )$ has a Fourier expansion

\begin{equation*} f ( Z ) = \sum _ { \begin{array} { c c } { 0 \leq T = \square ^ { t } T}\\{\text{semi}\square\text{integral} } \end{array}} c ( T ) e ^ { 2 \pi i \operatorname { Tr } ( T Z ) }. \end{equation*}

Here, "semi-integral" means that the diagonal entries of $T$ are integers while the off-diagonal entries are either integers or halves of integers. Examples of Siegel modular forms include Eisenstein series and theta-functions. The Siegel main theorem on quadratic forms is a consequence of a formula relating the two. The analogue of the Hecke correspondence can be discussed. See [[#References|[a13]]], for applications to the theory of quadratic forms, or [[#References|[a9]]].

When looking at modular forms for groups with more than one cusp, such as congruence subgroups of the modular group or Hilbert modular forms for $\operatorname{SL} ( 2 , O _ { K } )$, with $O _ { K }$ the ring of integers of a totally real algebraic number field $K$, there will be Fourier expansions at each cusp. Once more, Eisenstein series give examples. It is also possible to consider modular forms over imaginary quadratic number fields or even arbitrary number fields. See [[#References|[a2]]]. The adelic version of the theory can be found in [[#References|[a3]]] and [[#References|[a7]]], for example.

====References====
<table>
<tr><td valign="top">[a1]</td> <td valign="top"> J.H. Conway, "Monsters and moonshine" ''Math. Intelligencer'' , '''2''' (1980) pp. 165–171 {{MR|0600222}} {{ZBL|0452.20021}} </td></tr>
<tr><td valign="top">[a2]</td> <td valign="top"> J. Elstrodt, F. Grunewald, J. Mennicke, "Groups acting on hyperbolic space" , Springer (1998) {{MR|1483315}} {{ZBL|0888.11001}} </td></tr>
<tr><td valign="top">[a3]</td> <td valign="top"> S. Gelbart, F. Shahidi, "Analytic properties of automorphic L-functions" , Acad. Press (1988) {{MR|0951897}} {{ZBL|0654.10028}} </td></tr>
<tr><td valign="top">[a4]</td> <td valign="top"> D. Hejhal, B. Rackner, "On the topography of Maass waveforms for $\operatorname { PSL } ( 2 , \bf Z )$" ''Experim. Math.'' , '''4''' (1992) pp. 275–305 {{MR|1257286}} {{ZBL|0813.11035}} </td></tr>
<tr><td valign="top">[a5]</td> <td valign="top"> M. Koecher, A. Krieg, "Elliptische Funktionen und Modulformen" , Springer (1998) {{MR|1711085}} {{ZBL|0895.11001}} </td></tr>
<tr><td valign="top">[a6]</td> <td valign="top"> S. Lang, "Introduction to modular forms" , Springer (1976) {{MR|0429740}} {{ZBL|0344.10011}} </td></tr>
<tr><td valign="top">[a7]</td> <td valign="top"> W.C.W. Li, "Number theory with applications" , World Sci. (1996) {{MR|1390759}} {{ZBL|0849.11006}} </td></tr>
<tr><td valign="top">[a8]</td> <td valign="top"> H. Maass, "Lectures on modular functions of one complex variable" , Springer (1983) {{MR|0734485}} {{ZBL|0539.10021}} </td></tr>
<tr><td valign="top">[a9]</td> <td valign="top"> H. Maass, "Siegel's modular forms and Dirichlet series" , ''Lecture Notes Math.'' , '''216''' , Springer (1971) {{MR|344198}} {{ZBL|}} </td></tr>
<tr><td valign="top">[a10]</td> <td valign="top"> P. Sarnak, "Some applications of modular forms" , Cambridge Univ. Press (1990) {{MR|1102679}} {{ZBL|0721.11015}} </td></tr>
<tr><td valign="top">[a11]</td> <td valign="top"> J.-P. Serre, "A course in arithmetic" , Springer (1973) {{MR|0344216}} {{ZBL|0256.12001}} </td></tr>
<tr><td valign="top">[a12]</td> <td valign="top"> G. Shimura, "Introduction to the arithmetic theory of automorphic functions" , Princeton Univ. Press (1971) {{MR|0314766}} {{ZBL|0221.10029}} </td></tr>
<tr><td valign="top">[a13]</td> <td valign="top"> C.L. Siegel, "Gesammelte Abhandlungen" , '''I–II''' , Springer (1966) {{MR|0197270}} {{ZBL|0143.00101}} </td></tr>
<tr><td valign="top">[a14]</td> <td valign="top"> H. Stark, "Fourier coefficients of Maass wave forms" R.A. Rankin (ed.) , ''Modular Forms'' , Horwood (1984) pp. 263–269</td></tr>
<tr><td valign="top">[a15]</td> <td valign="top"> A. Terras, "Harmonic analysis on symmetric spaces and applications" , '''I–II''' , Springer (1985/1988) {{MR|0955271}} {{MR|0791406}} {{ZBL|0668.10033}} {{ZBL|0574.10029}} </td></tr>
<tr><td valign="top">[a16]</td> <td valign="top"> A. Wiles, "Modular elliptic curves and Fermat's last theorem" ''Ann. of Math. (2)'' , '''141''' (1995) pp. 443–551</td></tr>
<tr><td valign="top">[a17]</td> <td valign="top"> R.E. Borcherds, "What is Moonshine" , ''Proc. Internat. Congress Mathem. (Berlin, 1998)'' (1998) pp. 607–615 (Doc. Math. Extra Vol. 1; see also pp. 99–108) {{MR|1660657}} {{ZBL|1003.11017}} </td></tr>
<tr><td valign="top">[a18]</td> <td valign="top"> B. Conrad, F. Diamond, R. Taylor, "Modularity of certain potentially Barsotti–Tate Galois representations" ''J. Amer. Math. Soc.'' , '''12''' (1999) pp. 521–567 {{MR|1639612}} {{ZBL|0923.11085}} </td></tr>
</table>