From Lang's introduction to modular form

Question

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I have trouble dealing with the italicized sentence 'For ever element v...'. Lang says the relation between imaginary part of z and gamma_z plays a critical role. I searched other texts, but they are mostly discussing in a more general context, so they were of no help. Anyone could use Lang's approach(That is, to use the imaginary part relation) and prove this? Thanks in advance.

Answer 1

You can say it follows from $$\Im(\gamma z) = \Im(\frac{az+b}{cz+d}) = \Im(\frac{(az+b)(c\overline{z}+d)}{|cz+d|^2})=\frac{\Im(z)}{|cz+d|^2}$$ whenever $a,b,c,d \in \Bbb{R},ad-bc=1$.

That $\gamma z$ is very close to $z$ means $|cz+d|$ is very close to $\pm 1$.

Here we assume $z$ is in $\Im (z) \ge 1/2$. If it is not just replace $z$ by $\alpha z$ and $\gamma$ by $\gamma \alpha^{-1}$, the problem will stay the same.

Thus we can assume $\Im(z) \ge 1/2$, and since $a,b,c,d \in \Bbb{Z}$,

that $|cz+d|$ is very close to $\pm 1$ means $c=0,d=\pm 1$ or $c=\pm 1,d=0, \Im(z) \approx 1$, in the first case $\gamma z = z+b$ and $\gamma z = z$, in the former case $\gamma z = \frac{az-1}{z}= a-1/z$ and $\gamma z = z, z \approx e^{\pm 2i \pi / 3}$.