Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 137 / 3 / b

The point $1/N+{\Lambda }_{\tau }$ has exact order $N$, and changing $\tau$ by ${\Gamma }_1(N)$ changes the pair only by a complex scaling, so the stated map is well-defined.

Conversely, scale a lattice to write it as $\mathbb{Z}\tau \oplus \mathbb{Z}$. A point of exact order $N$ is represented by $(r\tau +s)/N$, where $(r,s)$ is primitive modulo $N$. The group $S{L}_2(\mathbb{Z})$ acts transitively on primitive vectors modulo $N$, so a basis change carries this point to $1/N$. This proves surjectivity. Two resulting normalized pairs are similar precisely when their basis-change matrix fixes $(0,1)$ modulo $N$, namely when it lies in ${\Gamma }_1(N)$. This proves injectivity and the Gamma 1 level structure on a complex lattice bijection