Moduli of elliptic curves via twisted stable maps

TL;DR

This paper extends the theory of moduli stacks of elliptic curves with level structures to arbitrary base schemes using twisted stable maps, connecting to Katz-Mazur models and analyzing characteristic-dependent interactions.

Contribution

It generalizes existing results on modular compactifications of elliptic curves with level structures to arbitrary base schemes via twisted stable maps.

Findings

Recovered Katz-Mazur regular models with a new moduli interpretation.

Extended twisted stable maps to non-DM tame stacks over arbitrary schemes.

Analyzed interactions of moduli stacks in characteristics dividing the level.

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.