2012/07/31 by Andrew Niles · 2 citations
Mathematics · #Abelian group #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Base (topology) #Elliptic curve #Homotopy and Cohomology in Algebraic Topology #Interpretation (philosophy) #Invertible matrix #Moduli #Stack (abstract data type) #Twists of curves #math.AG #math.NT #msc:11G18 #msc:14D23 #msc:14H10 #msc:14H52 #msc:14K10
paper · pdf · doi:10.2140/ant.2013.7.2141
published as Algebra Number Theory 7 (2013), no. 9, 2141-2202 · 46 pages; to appear in Algebra & Number Theory
arxiv created 2013/02/14 · openalex publication_date 2013/12/18 · arxiv updated 2014/11/11 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.