  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.
