Vol. 7, No. 9, 2013

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Moduli of elliptic curves via twisted stable maps

Andrew Niles

Vol. 7 (2013), No. 9, 2141–2202

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.

generalized elliptic curve, twisted curve, Drinfeld structure, moduli stack
Mathematical Subject Classification 2010
Primary: 11G18
Secondary: 14K10, 14H10, 14D23, 14H52
Received: 1 August 2012
Revised: 4 January 2013
Accepted: 9 February 2013
Published: 18 December 2013
Andrew Niles
Department of Mathematics
University of California, Berkeley
Berkeley, CA 94720
United States