modular form

Let ${\text{SL}}_2(ℝ)$ be the group of real $2\times 2$ matrices with determinant $1$ (see entry on special linear groups). The group ${\text{SL}}_2(ℝ)$ acts on $H$, the upper half plane, through fractional linear transformations. That is, if

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

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

For any natural number $N\ge 1$, define the congruence subgroup ${\mathrm{\Gamma }}_0(N)$ of level $N$ to be the following subgroup of the group ${\text{SL}}_2(\mathbb{Z})$ of integer coefficient matrices of determinant $1$:

Fix an integer $k$. For $\gamma \in {\text{SL}}_2(\mathbb{Z})$ 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 $\mathrm{\Gamma }$ of ${\text{SL}}_2(\mathbb{Z})$ containing a congruence subgroup, a function $f$ defined on $H$ is said to be a weight $k$ modular form if:

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

2. $f$ is holomorphic on $H$.

3. $f$ is holomorphic at the cusps.

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

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 $\gamma \in \mathrm{\Gamma }$, $f_{\mid \gamma }$ admits a a Fourier expansion

$$f_{\mid \gamma }(\tau )=\sum _{n=0}^{\mathrm{\infty }}{a}_n{q}^n,$$ (2)

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

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

The space of modular forms for ${\text{SL}}_2(\mathbb{Z})$ (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 ${\text{SL}}_2(\mathbb{Z})$ are:

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, $q=e^{2i\pi \tau }$:

   $$E_m(\tau )=1-\frac{2m}{B_m}\sum _{n=1}^{\infty }{\sigma }_{m-1}(n)q^n.$$ (3)

   For instance,

   $$E_4(\tau )=1+240\sum _{n=1}^{\infty }{\sigma }_3(n)q^n$$ (4)

   and

   $$E_6(\tau )=1-504\sum _{n=1}^{\infty }{\sigma }_5(n)q^n.$$ (5)

2. The Weierstrass $\mathrm{\Delta }$ function, also called the modular discriminant, is a modular form of weight $12$:

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

Every modular form is expressible as

$$f(\tau )=\sum _{n=0}^{\lfloor k/12\rfloor }{a}_n{E}_{k-12n}(\tau ){(\mathrm{\Delta }(\tau ))}^n,$$ (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
Canonical name	ModularForm
Date of creation	2013-03-22 14:07:37
Last modified on	2013-03-22 14:07:37
Owner	olivierfouquetx (2421)
Last modified by	olivierfouquetx (2421)
Numerical id	31
Author	olivierfouquetx (2421)
Entry type	Definition
Classification	msc 11F11
Related topic	TaniyamaShimuraConecture
Related topic	HeckeAlgebra
Related topic	AlgebraicNumberTheory
Related topic	RamanujanTauFunction
Defines	cusp form