Vol. 11, No. 9, 2017

Download this article
Download this article For screen
For printing
Recent Issues

Volume 11
Issue 10, 2213–2445
Issue 9, 1967–2212
Issue 8, 1739–1965
Issue 7, 1489–1738
Issue 6, 1243–1488
Issue 5, 1009–1241
Issue 4, 767–1007
Issue 3, 505–765
Issue 2, 253–503
Issue 1, 1–252

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Editors' Addresses
Editors' Interests
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Author Index
To Appear
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
A modular description of $\mathscr{X}_0(n)$

Kęstutis Česnavičius

Vol. 11 (2017), No. 9, 2001ā€“2089

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.

Elliptic curve, generalized elliptic curve, level structure, modular curve, moduli stack
Mathematical Subject Classification 2010
Primary: 11G18
Secondary: 14D22, 14D23, 14G35
Received: 6 January 2016
Revised: 13 July 2017
Accepted: 5 October 2017
Published: 2 December 2017
Kęstutis Česnavičius
Mathematisches Institut
UniversitƤt Bonn
D-53115 Bonn