View source for Fourier coefficients of automorphic forms

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f1201001.png" /> of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f1201002.png" /> for the full [[Modular group|modular group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f1201003.png" /> will be considered. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f1201004.png" /> is a such a [[Modular form|modular form]], 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/f/f120/f120100/f1201005.png" /></td> </tr></table>

for all integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f1201006.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f1201007.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f1201008.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f1201009.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010011.png" /> in the upper half-plane. Therefore it has period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010012.png" /> in the real part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010013.png" /> and must have a Fourier expansion

<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/f/f120/f120100/f12010014.png" /></td> </tr></table>

The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010015.png" /> are the Fourier coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010016.png" />. The modular form is a cusp form if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010017.png" />. 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

<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/f/f120/f120100/f12010018.png" /></td> </tr></table>

the non-zero Fourier coefficients are essentially divisor functions

<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/f/f120/f120100/f12010019.png" /></td> </tr></table>

The theta-function associated to a positive-definite real symmetric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010020.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010021.png" /> and a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010022.png" /> in the upper half-plane is defined 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/f/f120/f120100/f12010023.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010024.png" /> denotes the transpose of the column vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010025.png" />. One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010026.png" /> is an even integral matrix when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010027.png" /> is an even integer, for all integral column vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010028.png" />. In this case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010029.png" /> is a modular form of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010030.png" />. Then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010031.png" />th Fourier coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010032.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010033.png" /> is the number of representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010034.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010035.png" />, for some integer vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010036.png" />. It follows from the one-dimensionality of the space of modular forms of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010037.png" /> that when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010038.png" />,

<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/f/f120/f120100/f12010039.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010040.png" /> denotes the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010041.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010042.png" />, which is the number of ways of writing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010043.png" /> as a sum of positive integers. The discriminant function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010044.png" /> is a cusp form whose Fourier coefficients are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010045.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010046.png" /> is the Ramanujan tau-function (cf. also [[Ramanujan function|Ramanujan function]]). S. Ramanujan conjectured that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010047.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010049.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010050.png" /> one has the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010051.png" />. 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|Graph theory]]); [[#References|[a7]]], [[#References|[a10]]].

Two other conjectures are still (1998) open. Computations lead one to believe that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010052.png" />, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010053.png" />. Writing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010054.png" />, J.-P. Serre conjectured that the angles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010055.png" /> are distributed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010056.png" /> with respect to the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010057.png" /> (the Sato–Tate or Wigner semi-circle distribution).

The modular invariant is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010058.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010059.png" /> has integer Fourier coefficients (writing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010060.png" />)

<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/f/f120/f120100/f12010061.png" /></td> </tr></table>

There is a surprising connection of these Fourier coefficients with the representations of the largest of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010062.png" /> sporadic finite simple groups (the Monster; cf. also [[Simple finite group|Simple finite group]]). All of the early Fourier coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010063.png" /> 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 conjecture" ; see [[#References|[a17]]].

The Hecke correspondence maps a modular form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010064.png" /> with Fourier coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010065.png" /> to an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010066.png" />-function (cf. also [[Dirichlet-L-function|Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010067.png" />-function]]) given 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/f/f120/f120100/f12010068.png" /></td> </tr></table>

This <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010069.png" />-function has an analytic continuation and a functional equation. The [[Riemann zeta-function|Riemann zeta-function]] corresponds to the simplest theta-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010070.png" />. But here the weight of the modular form is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010071.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010072.png" /> is not invariant under the full modular group.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010073.png" /> is a form of integer weight for the modular group which is an eigenfunction for all the Hecke operators, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010074.png" /> has an [[Euler product|Euler product]]. There are also converse theorems. The result is that many of the important zeta- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010075.png" />-functions of number theory correspond to modular forms.

The Taniyama–Shimura–Weil conjecture says that the zeta-function of an [[Elliptic curve|elliptic curve]] over the rational numbers should come via the Hecke correspondence from a cusp form of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010076.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010077.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010079.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010080.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010081.png" /> is a complex symmetric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010082.png" />-matrix with positive-definite imaginary part, such 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/f/f120/f120100/f12010083.png" /></td> </tr></table>

for all matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010084.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010086.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010087.png" /> with integer entries and such that 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/f/f120/f120100/f12010088.png" /></td> </tr></table>

one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010089.png" />, where

<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/f/f120/f120100/f12010090.png" /></td> </tr></table>

with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010091.png" /> the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010092.png" />-matrix. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010093.png" /> has a Fourier expansion

<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/f/f120/f120100/f12010094.png" /></td> </tr></table>

Here,  "semi-integral"  means that the diagonal entries of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010095.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010096.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010097.png" /> the ring of integers of a totally real algebraic number field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010098.png" />, 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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Elstrodt,   F. Grunewald,   J. Mennicke,   "Groups acting on hyperbolic space" , Springer  (1998)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Gelbart,   F. Shahidi,   "Analytic properties of automorphic L-functions" , Acad. Press  (1988)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D. Hejhal,   B. Rackner,   "On the topography of Maass waveforms for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120100/f12010099.png" />"  ''Experim. Math.'' , '''4'''  (1992)  pp. 275–305</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Koecher,   A. Krieg,   "Elliptische Funktionen und Modulformen" , Springer  (1998)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  S. Lang,   "Introduction to modular forms" , Springer  (1976)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  W.C.W. Li,   "Number theory with applications" , World Sci.  (1996)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  H. Maass,   "Lectures on modular functions of one complex variable" , Springer  (1983)</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)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  P. Sarnak,   "Some applications of modular forms" , Cambridge Univ. Press  (1990)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  J.-P. Serre,   "A course in arithmetic" , Springer  (1973)</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)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  C.L. Siegel,   "Gesammelte Abhandlungen" , '''I–II''' , Springer  (1966)</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)</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)</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</TD></TR></table>