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

The Gamma 1 congruence subgroup is

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

Index-$p$ overlattices ${\Lambda }^{\prime }$ correspond to order-$p$ subgroups of $(p^{-1}\Lambda )/\Lambda \cong {\mathbb{F}}_p^2$, hence to the $p+1$ lines in that vector space.

If $p\nmid N$, the order of $v+{\Lambda }^{\prime }$ cannot decrease: its decrease would have a factor dividing both $p=[{\Lambda }^{\prime }:\Lambda ]$ and $N$. Thus all $p+1$ overlattices are counted.

If $p\mid N$, the element $(N/p)v+\Lambda$ is a nonzero point of order $p$ in $\mathbb{C}/\Lambda$. Exactly one of the $p+1$ overlattices contains it; in that overlattice the image of $v$ has order $N/p$, while in every other one it retains order $N$. ThereforeThis is the Prime-index overlattices preserving a Gamma 1 level structure count.