# Definition of modular forms

First Definition. A modular form of level n and dimension -k is an analytic function $$F$$ of $$\omega_1$$ and $$\omega_2$$ satisfying the following properties :

1. $$F(\omega_1,\omega_2)$$ is holomorphic and unique for all $$\omega_1,\omega_2$$ , where $$Im(\omega_1/\omega_2)>0$$ .

2.$$F(\lambda\omega_1,\lambda\omega_2)=\lambda^{-k}F(\omega_1,\omega_2)$$ for all $$\lambda\neq0$$ .

3.$$F(a\omega_1+b\omega_2,c\omega_1+d\omega_2)=F(\omega_1,\omega_2)$$ , if $$a,b,c,d$$ are rational integers with $$ad-bc=1$$ $$a\equiv d \equiv1 ,b\equiv c \equiv0 (mod \ n)$$

1. The function $$F(\tau)=\omega_2^kF(\omega_1,\omega_2)$$ has a power series in $$\tau=\infty$$ $$F(\tau)=\sum_{m=0}^{\infty} c_m e^{2\pi im\tau/n} \ ,Im(\tau)>0$$

Second Definition .

Let $$k \in \mathbb{Z}$$ . A function $$f:\mathbb{H} \rightarrow\mathbb{C}$$ is is a modular form of weight k for $$SL_2(\mathbb{Z})$$ if it satisfies the following properties :

1) $$f$$ is holomorphic on $$\mathbb{H} .$$

2)$$f|_k M=f$$ for all $$M \in SL_2(\mathbb{Z})$$ .

3) $$f$$ has a Fourier expansion $$f(\tau)=\sum_{n=0}^{\infty} a_n e^{2\pi i n\tau}$$

I do not understand why these two definitions are equivalent .

Thanks for the help .

The recipe to convert:

Given $$f$$, set $$F(\omega_1, \omega_2) = f(\omega_1/\omega_2)$$.

Given $$F$$, set $$f(\tau) = F(\tau, 1)$$.

(You'll have to check that if $$f$$ has all the properties it's supposed to then the corresponding $$F$$ does and vice-versa.)

The "big picture" is that it is a "function" on the space of isomorphism classes of lattice (up to homothety-- this isn't literally true unless $$k=0$$). You can either specify the lattice by giving two generators, or by scaling such that one of the generators is $$1$$ and giving the other generator.

For $$n=1$$, $$F$$ is a function of lattices.

For $$a,b,c,d\in\Bbb{Z},ad-bc=1$$ $$f(z)=F(\Bbb{Z}+z \Bbb{Z}) = F((az+b)\Bbb{Z}+(cz+d) \Bbb{Z})$$ $$=(cz+d)^{-k} F(\frac{az+b}{cz+d}\Bbb{Z}+ \Bbb{Z})=(cz+d)^{-k}f(\frac{az+b}{cz+d})$$