A modular description of 𝒳0(n)

Česnavičius, Kęstutis

doi:10.2140/ant.2017.11.2001

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

Kęstutis Česnavičius

Vol. 11 (2017), No. 9, 2001–2089

DOI: 10.2140/ant.2017.11.2001

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 $Γ_{0} (n)$ -level modular stack $X_{0} (n)$ 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 $X_{0} (n)$ over $ℤ$ for all $n$ . The resulting modular description enables us to extend the regularity theorem of Katz and Mazur: $X_{0} (n)$ is also regular at the cusps. We also prove such regularity for $X_{1} (n)$ 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 ${\bar{ℰ ℓ ℓ}}_{m}$ of the stack $ℰ ℓ ℓ$ 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.

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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 Universität Bonn D-53115 Bonn Germany

Vol. 11, No. 9, 2017