Moduli of elliptic curves via twisted stable maps

Niles, Andrew

doi:10.2140/ant.2013.7.2141

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Moduli of elliptic curves via twisted stable maps

Andrew Niles

Vol. 7 (2013), No. 9, 2141–2202

DOI: 10.2140/ant.2013.7.2141

Abstract

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.

Keywords

generalized elliptic curve, twisted curve, Drinfeld structure, moduli stack

Mathematical Subject Classification 2010

Primary: 11G18

Secondary: 14K10, 14H10, 14D23, 14H52

Milestones

Received: 1 August 2012

Revised: 4 January 2013

Accepted: 9 February 2013

Published: 18 December 2013

Authors

Andrew Niles
Department of Mathematics University of California, Berkeley Berkeley, CA 94720 United States
http://math.berkeley.edu/~andrew/

Vol. 7, No. 9, 2013