Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 137 / 2 / a / Solution

For $\tau =x+iy\in \mathfrak{h}$ and $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \Gamma (1)$,The set $\{c\tau +d:(c,d)\in {\mathbb{Z}}^2\setminus \{0\}\}$ is a lattice in $\mathbb{C}$, so it has a shortest nonzero vector. Dividing $(c,d)$ by their greatest common divisor can only shorten it; hence a minimizing pair may be chosen coprime and completed to the bottom row of some $\gamma \in S{L}_2(\mathbb{Z})$. Consequently the orbit contains a point ${\tau }_0$ of maximal imaginary part.

Apply a power of $T$ so that $-1/2\le \mathrm{Re}{\tau }_0\le 1/2$. If $|{\tau }_0|<1$, thencontradicting maximality. Thus $|{\tau }_0|\ge 1$, and ${\tau }_0$ lies in the standard fundamental domain of the modular group.