Algebra & Number Theory

Moduli of elliptic curves via twisted stable maps

Andrew Niles

Full-text: Open access


Abramovich, Corti and Vistoli have studied modular compactifications of stacks of curves equipped with abelian level structures arising as substacks of the stack of twisted stable maps into the classifying stack of a finite group, provided the order of the group is invertible on the base scheme. Recently Abramovich, Olsson and Vistoli extended the notion of twisted stable maps to allow arbitrary base schemes, where the target is a tame stack, not necessarily Deligne–Mumford. We use this to extend the results of Abramovich, Corti and Vistoli to the case of elliptic curves with level structures over arbitrary base schemes; we prove that we recover the compactified Katz–Mazur regular models, with a natural moduli interpretation in terms of level structures on Picard schemes of twisted curves. Additionally, we study the interactions of the different such moduli stacks contained in a stack of twisted stable maps in characteristics dividing the level.

Article information

Algebra Number Theory, Volume 7, Number 9 (2013), 2141-2202.

Received: 1 August 2012
Revised: 4 January 2013
Accepted: 9 February 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35]
Secondary: 14K10: Algebraic moduli, classification [See also 11G15] 14H10: Families, moduli (algebraic) 14D23: Stacks and moduli problems 14H52: Elliptic curves [See also 11G05, 11G07, 14Kxx]

generalized elliptic curve twisted curve Drinfeld structure moduli stack


Niles, Andrew. Moduli of elliptic curves via twisted stable maps. Algebra Number Theory 7 (2013), no. 9, 2141--2202. doi:10.2140/ant.2013.7.2141.

Export citation


  • D. Abramovich, “Raynaud's group-scheme and reduction of coverings”, pp. 1–18 in Number theory, analysis and geometry, edited by D. Goldfeld et al., Springer, New York, 2012.
  • D. Abramovich and B. Hassett, “Stable varieties with a twist”, pp. 1–38 in Classification of algebraic varieties (Schiermonnikoog, 2009), edited by C. Faber et al., Eur. Math. Soc., Zürich, 2011.
  • D. Abramovich and M. Romagny, “Moduli of Galois $p$-covers in mixed characteristics”, Algebra Number Theory 6:4 (2012), 757–780.
  • D. Abramovich and A. Vistoli, “Compactifying the space of stable maps”, J. Amer. Math. Soc. 15:1 (2002), 27–75.
  • D. Abramovich, A. Corti, and A. Vistoli, “Twisted bundles and admissible covers”, Comm. Algebra 31:8 (2003), 3547–3618.
  • D. Abramovich, T. Graber, and A. Vistoli, “Gromov–Witten theory of Deligne–Mumford stacks”, Amer. J. Math. 130:5 (2008), 1337–1398. 1193.14070
  • D. Abramovich, M. Olsson, and A. Vistoli, “Tame stacks in positive characteristic”, Ann. Inst. Fourier $($Grenoble$)$ 58:4 (2008), 1057–1091.
  • D. Abramovich, M. Olsson, and A. Vistoli, “Twisted stable maps to tame Artin stacks”, J. Algebraic Geom. 20:3 (2011), 399–477.
  • E. Arbarello, M. Cornalba, and P. A. Griffiths, Geometry of algebraic curves, vol. 2, Grundlehren Math. Wiss. 268, Springer, Heidelberg, 2011.
  • N. Bourbaki, Éléments de mathématique, algèbre commutative, chapitre 10: Profondeur, régularité, dualité, Masson, Paris, 1998.
  • C.-L. Chai and P. Norman, “Bad reduction of the Siegel moduli scheme of genus two with $\Gamma\sb 0(p)$-level structure”, Amer. J. Math. 112:6 (1990), 1003–1071.
  • B. Conrad, “Arithmetic moduli of generalized elliptic curves”, J. Inst. Math. Jussieu 6:2 (2007), 209–278.
  • P. Deligne and D. Mumford, “The irreducibility of the space of curves of given genus”, Inst. Hautes Études Sci. Publ. Math. 36:1 (1969), 75–109.
  • P. Deligne and M. Rapoport, “Les schémas de modules de courbes elliptiques”, pp. 143–316 in Modular functions of one variable (Antwerp, 1972), vol. 2, edited by P. Deligne and W. Kuyk, Lecture Notes in Mathematics 349, Springer, Berlin, 1973.
  • W. Fulton, Intersection theory, 2nd ed., Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin, 1998.
  • N. M. Katz and B. Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies 108, Princeton University Press, 1985.
  • S. Keel and S. Mori, “Quotients by groupoids”, Ann. of Math. $(2)$ 145:1 (1997), 193–213.
  • D. Knutson, Algebraic spaces, Lecture Notes in Mathematics 203, Springer, Berlin, 1971.
  • D. Petersen, “Cusp form motives and admissible $G$-covers”, Algebra Number Theory 6:6 (2012), 1199–1221.
  • M. Pikaart and A. J. de Jong, “Moduli of curves with non-abelian level structure”, pp. 483–509 in The moduli space of curves (Texel Island, 1994), edited by R. Dijkgraaf et al., Progress in Mathematics 129, Birkhäuser, Boston, MA, 1995.
  • S. S. Shatz, “Group schemes, formal groups, and $p$-divisible groups”, pp. 29–78 in Arithmetic geometry (Storrs, CT, 1984), edited by G. Cornell and J. H. Silverman, Springer, New York, 1986.