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.

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