PlanetMath: modular form

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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)