Show $-\arg j_{AB}(z)+ \arg j_A(Bz)+\arg j_B(z)$ does not depends on $z$

by J.Shim   Last Updated January 16, 2018 14:20 PM

In Iwaniec's Topics in Classical Automorphic forms, page 40, he says

Since $j_{AB}(z)=j_A(Bz)j_B(z)$,

$2\pi \omega(A,B)=-\arg j_{AB}(z)+ \arg j_A(Bz)+\arg j_B(z)$ does not depend on $z\in\mathbb{H}$

where $A,B\in SL_2(\mathbb{R})$ and $j_A(z)=cz+d$ for $A= \begin{bmatrix} a & b \\ c & d \end{bmatrix}$.

I can easily see that $\omega(A,B)$ can only take three values but it's not clear to me why it does not depends on $z$ at all. Can anyone provide me a proof?

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