Vol. 11, No. 9, 2017

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

Kęstutis Česnavičius

Vol. 11 (2017), No. 9, 2001ā€“2089
Abstract

As we explain, when a positive integer n is not squarefree, even over 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 Γ0(n)-level modular stack X0(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 X0(n) over for all n. The resulting modular description enables us to extend the regularity theorem of Katz and Mazur: X0(n) is also regular at the cusps. We also prove such regularity for X1(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 ¯m of the stack 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
Milestones
Received: 6 January 2016
Revised: 13 July 2017
Accepted: 5 October 2017
Published: 2 December 2017
Authors
Kęstutis Česnavičius
Mathematisches Institut
UniversitƤt Bonn
D-53115 Bonn
Germany