Let $N \ge 1$ and let $n \ge 1$ satisfy $\gcd(n,N)=1$. Over the base $\operatorname{Spec}\mathbb{Z}[1/Nn]$, there is a compact coarse modular curve $X_0(N;n)$ whose open elliptic-curve locus parameterises triples $(E,C,D)$, where $(E,C)$ is a point of $X_0(N)$, the subgroup $C \subset E$ is cyclic of order $N$, and $D \subset E$ is a cyclic subgroup of order $n$ with $D \cap C = 0$. The two maps

\begin{align*} \pi_1(E,C,D) &= (E,C), & \pi_2(E,C,D) &= (E/D,(C+D)/D) \end{align*}

define a finite correspondence on $X_0(N)$, denoted $T_n$.