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 $a_0=0$.