#### Vol. 11, No. 9, 2017

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

### Kęstutis Česnavičius

Vol. 11 (2017), No. 9, 2001–2089
##### 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 ${\mathsc{X}}_{0}\left(n\right)$ 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 ${\mathsc{X}}_{0}\left(n\right)$ over $ℤ$ for all $n$. The resulting modular description enables us to extend the regularity theorem of Katz and Mazur: ${\mathsc{X}}_{0}\left(n\right)$ is also regular at the cusps. We also prove such regularity for ${\mathsc{X}}_{1}\left(n\right)$ 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 ${\overline{\mathsc{ℰ}\ell \ell }}_{m}$ of the stack $\mathsc{ℰ}\ell \ell$ 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.

##### Keywords
Elliptic curve, generalized elliptic curve, level structure, modular curve, moduli stack
##### Mathematical Subject Classification 2010
Primary: 11G18
Secondary: 14D22, 14D23, 14G35