A special modular function: $ j $-invariant. It is known that j invariant $$j(\tau)= 1728 \frac{g_2^3(\tau)}{\Delta(\tau)} $$
$\tau \in \mathbb{H}$ 
attains every complex value  , Can someone guide me its proof.??
where 
 $L(\tau ) = \{\tau m + n | \quad  \tau \in \mathbb{C} ; m,n \in \mathbb{Z}\}  $
 $G_k=\sum_{\omega \in L , \omega \neq 0}\frac{1}{\omega^k} $ 
$g_2(\tau)=60G_4(\tau);g_3(\tau)=140G_6(\tau);\Delta = g_2^3 -27g_3^2 $
There is a Theorem given in Advance topics in arithmetic in elliptic curves it states that :$$j:X(1)\rightarrow \mathbb{P}^1 (\mathbb{C}) $$ So do is this question linked to mine??
Also what is cusp $\infty$?
There is an argument written  " The j has a simple pole at the cusp $\infty \in X(1)$, so the map is analytic with degree 1 between compact riemman surface. It is therefore isomorphism  " 
Can some one explain this ?

This is not true. For instance $j(\tau)=j(\tau+1)=j(-1/\tau)$ for every $\tau\in\mathbb{H}$. More generally,
$$j(\tau) = j\left(\frac{a\tau+b}{c\tau+d}\right)$$
for any matrix $\left(\begin{array}[cc] aa & b\\ c& d\end{array}\right)$ in $\Gamma(1)=\operatorname{SL}(2,\mathbb{Z})$. It is true, however, that $j:\mathbb{H}/\Gamma(1)\to \mathbb{C}$ is a bijection. See Chapter 1, Section 3, of Silverman's "Advanced Topics in the Theory of Elliptic Curves".
