A modular description of $\mathscr{X}_0(n)$

As we explain, when a positive integer $n$ is not squarefree, even over $\mathbb{C}$ the moduli stack that parametrizes generalized elliptic curves equipped with an ample cyclic subgroup of order $n$ does not agree at the cusps with the $\Gamma_0(n)$-level modular stack $\mathscr{X}_0(n)$ defined by Deligne and Rapoport via normalization. Following a suggestion of Deligne, we present a refined moduli stack of ample cyclic subgroups of order $n$ that does recover $\mathscr{X}_0(n)$ over $\mathbb{Z}$ for all $n$. The resulting modular description enables us to extend the regularity theorem of Katz and Mazur: $\mathscr{X}_0(n)$ is also regular at the cusps. We also prove such regularity for $\mathscr{X}_1(n)$ and several other modular stacks, some of which have been treated by Conrad by a different method. For the proofs we introduce a tower of compactifications $\overline{Ell}_m$ of the stack $Ell$ that parametrizes elliptic curves---the ability to vary $m$ in the tower permits robust reductions of the analysis of Drinfeld level structures on generalized elliptic curves to elliptic curve cases via congruences.