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Abramovich, Corti and Vistoli have studied modular compactifications ofstacks of curves equipped with abelian level structures arising as substacks ofthe 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. RecentlyAbramovich, Olsson and Vistoli extended the notion of twisted stable maps toallow arbitrary base schemes, where the target is a tame stack, not necessarilyDeligne-Mumford. We use this to extend the results of Abramovich, Corti andVistoli to the case of elliptic curves with level structures over arbitrarybase schemes; we prove that we recover the compactified Katz-Mazur regularmodels, with a natural moduli interpretation in terms of level structures onPicard schemes of twisted curves. Additionally, we study the interactions ofthe different such moduli stacks contained in a stack of twisted stable maps incharacteristics dividing the level.

Moduli of elliptic curves via twisted stable maps