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A modular description of $\mathscr{X}_0(n)$. (arXiv:1511.07475v2 [math.NT] UPDATED)
来源于:arXiv
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
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