# To prove that the Elliptic modular function is invariant under the modular transformation

I am not being able to understand that the Elliptic modular function $$J(\tau)=\frac{g_2(w_1,w_2)^3}{g_2(w_1,w_2)^3-27g_3(w_1,w_2)^2}$$ is invariant under the modular transformation $$\tau\mapsto \frac{a\tau +b}{c\tau +d}$$ where $$\begin{bmatrix} a&b\\ c&d\end{bmatrix}\in PSL(2,\mathbb{Z}), \tau=\frac{w_1}{w_2}\in\mathbb{H}\\ g_2(w_1,w_2)=60\sum\limits_{m,n\in\mathbb{Z}\\(m,n)\neq(0,0)}\frac{1}{(mw_1+nw_2)^4}\\g_3(w_1,w_2)=140\sum\limits_{m,n\in\mathbb{Z}\\(m,n)\neq(0,0)}\frac{1}{(mw_1+nw_2)^6}$$.

The author says that it should be obvious from the definition, am I missing something obvious? Some help would be appreciated.

## 1 Answer

$$g_2(w_1,w_2) =60\sum_{u \ne 0 \in\ \Bbb{Z}w_1+\Bbb{Z}w_2} u^{-4}$$ For $$a,b,c,d\in \Bbb{Z},ad-bc=\pm 1$$ then $$\Bbb{Z}(aw_1+bw_2)+\Bbb{Z}(cw_1+dw_2) = \Bbb{Z}w_1+\Bbb{Z}w_2$$ thus $$g_2(a w_1 +bw_2,cw_1+dw_2)=g_2(w_1 ,w_2)$$ Finally $$g_2(r w_1,rw_2) = r^{-4} g_2(w_1 ,w_2)$$.