# modular form

Let $\textrm{SL}_{2}(\mathbb{R})$ be the group of real $2\times 2$ matrices with determinant $1$ (see entry on special linear groups). The group $\textrm{SL}_{2}(\mathbb{R})$ acts on $H$, the upper half plane, through fractional linear transformations. That is, if

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

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

 $\gamma\tau=\frac{a\tau+b}{c\tau+d}.$ (1)

For any natural number $N\geq 1$, define the congruence subgroup $\Gamma_{0}(N)$ of level $N$ to be the following subgroup of the group $\textrm{SL}_{2}(\mathbb{Z})$ of integer coefficient matrices of determinant $1$:

 $\Gamma_{0}(N):=\left\{\left.\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\textrm{SL}_{2}(\mathbb{Z})\ \right|\ c\equiv 0\pmod{N}% \right\}.$

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

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

For a finite index subgroup $\Gamma$ of $\textrm{SL}_{2}(\mathbb{Z})$ containing a congruence subgroup, a function $f$ defined on $H$ is said to be a weight $k$ modular form if:

1. 1.

$f=f_{\mid\gamma}$ for $\gamma\in\Gamma$.

2. 2.

$f$ is holomorphic on $H$.

3. 3.

$f$ is holomorphic at the cusps.

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

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

and $\mu z=z+m$, while if $f$ satisfies all the other conditions above, $f_{\mid\mu}=f$. In other words, $f$ is periodic with period $1$. Thus, convergence permitting, $f$ admits a Fourier expansion. Therefore, we say that $f$ is holomorphic at the cusps if, for all $\gamma\in\Gamma$, $f_{\mid\gamma}$ admits a a Fourier expansion

 $f_{\mid\gamma}(\tau)=\sum_{n=0}^{\infty}a_{n}q^{n},$ (2)

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

If all the $a_{n}$ are zero for $n\leq 0$, then a modular form $f$ 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 $k$ even and greater than 4 (respectively greater than $12$ and not $14$). Examples of modular forms for $\textrm{SL}_{2}(\mathbb{Z})$ are:

1. 1.

The Eisenstein series $E_{m}$, where $m$ is even and greater than $4$, is a modular form of weight $m$. Here $B_{m}$ denotes the $m$-th Bernoulli number and, as usual, $q=e^{2i\pi\tau}$:

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

For instance,

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

and

 $E_{6}(\tau)=1-504\underset{n=1}{\overset{\infty}{\sum}}\sigma_{5}(n)q^{n}.$ (5)
2. 2.

The Weierstrass $\Delta$ function, also called the modular discriminant, is a modular form of weight $12$:

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

Every modular form is expressible as

 $f(\tau)=\underset{n=0}{\overset{\lfloor{k/12}\rfloor}{\sum}}{a_{n}}{E_{k-12n}(% \tau)}{(\Delta(\tau))^{n}},$ (7)

where the $a_{n}$ are arbitrary constants, $E_{0}(\tau)=1$ and $E_{2}(\tau)=0$. Cusp forms are the forms with $a_{0}=0$.

Title modular form ModularForm 2013-03-22 14:07:37 2013-03-22 14:07:37 olivierfouquetx (2421) olivierfouquetx (2421) 31 olivierfouquetx (2421) Definition msc 11F11 TaniyamaShimuraConecture HeckeAlgebra AlgebraicNumberTheory RamanujanTauFunction cusp form