PlanetMath: modular form

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (204)
Orphanage
Unclass'd
Unproven (490)
Corrections (7)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

modular form

(Definition)

Let $\textrm{SL}_{2}(\mathbb{R})$ be the group of real $2\times 2$ matrices with determinant (see entry on special linear groups). The group $\textrm{SL}_{2}(\mathbb{R})$ acts on , the upper half plane, through fractional linear transformations. That is, if

$\displaystyle \gamma = \begin{pmatrix}a & b \\ c & d\end{pmatrix},$

and $\tau\in H$ , then we let

$\displaystyle \gamma \tau=\frac{a\tau+b}{c\tau+d}.$

(1)

For any natural number $N \geq 1$ , define the congruence subgroup $\Gamma_0(N)$ of level to be the following subgroup of the group $\textrm{SL}_{2}(\mathbb{Z})$ of integer coefficient matrices of determinant :

$\displaystyle \Gamma_0(N) := \left\{ \left. \begin{pmatrix} a & b \ c & d \en... ...ix}\in \textrm{SL}_{2}(\mathbb{Z})\ \right\vert\ c \equiv 0 \pmod{N} \right\}.$

Fix an integer . For $\gamma\in\textrm{SL}_{2}(\mathbb{Z})$ and a function defined on , we define

$\displaystyle f_{\mid\gamma}(\tau)=\frac{f(\gamma \tau)}{(c\tau+d)^k}.$

For a finite index subgroup $\Gamma$ of $\textrm{SL}_{2}(\mathbb{Z})$ containing a congruence subgroup, a function

defined on

is said to be a weight

modular form if:

$f=f_{\mid \gamma}$ for $\gamma \in \Gamma$ .
is holomorphic on .
is holomorphic at the cusps.

This last condition requires some explanation. First observe that the element

$\displaystyle \mu = \begin{pmatrix} 1 & m \ 0 & 1 \end{pmatrix}\in \Gamma_0(N),$

and $\mu z = z + m$ , while if

satisfies all the other conditions above, $f_{\mid \mu} = f$ . In other words,

is periodic with period

. Thus, convergence permitting,

admits a Fourier transform. Therefore, we say that

is holomorphic at the cusps if, for all $\gamma \in \Gamma$ , $f_{\mid \gamma}$ admits a a Fourier expansion

$\displaystyle f_{\mid \gamma}(\tau)=\sum_{n=0}^{\infty}a_{n}q^{n},$

(2)

where $q=e^{2i\pi \tau}$ .

If all the are zero for $n\le 0$ , then a modular form is said to be a cusp form. The set of modular forms for $\Gamma$ (respectively cusp forms for $\Gamma$ ) is often denoted by $M_{k}(\Gamma)$ (respectively $S_{k}(\Gamma)$ ). Both $M_{k}(\Gamma)$ and $S_{k}(\Gamma)$ are finite dimensional vector spaces.

The space of modular forms for $\textrm{SL}_{2}(\mathbb{Z})$ (respectively cusp forms) is non-trivial for any even and greater than 4 (respectively greater than and not ). Examples of modular forms for $\textrm{SL}_{2}(\mathbb{Z})$ are:

The Eisenstein series $E_{m}$ , where is even and greater than , is a modular form of weight . Here $B_{m}$ denotes the -th Bernoulli number and, as usual, $q=e^{2i\pi \tau}$ :

$\displaystyle E_{m}(\tau)=1-\frac{2m}{B_{m}}\underset{n=1}{\overset{\infty}{\sum}}\sigma_{m- 1}(n)q^n.$ (3)

For instance,

$\displaystyle E_{4}(\tau)=1+240\underset{n=1}{\overset{\infty}{\sum}}\sigma_{3}(n)q^n$ (4)

and

$\displaystyle E_{6}(\tau)=1-504\underset{n=1}{\overset{\infty}{\sum}}\sigma_{5}(n)q^n.$ (5)
The Weierstrass $\Delta$ function, also called the modular discriminant, is a modular form of weight :

$\displaystyle \Delta(\tau)=q\underset{n=1}{\overset{\infty}{\prod}}(1-q^n)^{24}.$ (6)