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

Abstract

As we explain, when a positive integer $n$ is not squarefree, even over $ℂ$ 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 ${\Gamma }_0\left(n\right)$-level modular stack ${\mathcal{X}}_0\left(n\right)$ 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 ${\mathcal{X}}_0\left(n\right)$ over $\mathbb{Z}$ for all $n$. The resulting modular description enables us to extend the regularity theorem of Katz and Mazur: ${\mathcal{X}}_0\left(n\right)$ is also regular at the cusps. We also prove such regularity for ${\mathcal{X}}_1\left(n\right)$ 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 ${\overline{\mathcal{ℰ}\ell \ell }}_m$ of the stack $\mathcal{ℰ}\ell \ell$ 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.