Abstract
As we explain, when a positive integer is not squarefree, even over the moduli stack that parametrizes generalized elliptic curves equipped with an ample cyclic subgroup of order does not agree at the cusps with the -level modular stack defined by Deligne and Rapoport via normalization. Following a suggestion of Deligne, we present a refined moduli stack of ample cyclic subgroups of order that does recover over for all . The resulting modular description enables us to extend the regularity theorem of Katz and Mazur: is also regular at the cusps. We also prove such regularity for 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 of the stack that parametrizes elliptic curves—the ability to vary in the tower permits robust reductions of the analysis of Drinfeld level structures on generalized elliptic curves to elliptic curve cases via congruences.
Citation
Kęstutis Česnavičius. "A modular description of $\mathscr{X}_0(n)$." Algebra Number Theory 11 (9) 2001 - 2089, 2017. https://doi.org/10.2140/ant.2017.11.2001
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