Gamma 1 level structure on a complex lattice (${\Gamma }_1(N)$-level structure)

A Gamma 1 level structure on a complex lattice $\Lambda$ is a point of exact order $N$ in $\mathbb{C}/\Lambda$. Similarity classes of such pairs are parametrized by ${\Gamma }_1(N)\backslash \mathfrak{h}$.

A lattice has $p+1$ index-$p$ overlattices. All preserve the exact order of a Gamma 1 level structure of order $N$ when $p\nmid N$; exactly $p$ do so when $p\mid N$.