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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 .

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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.

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

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

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