Compactified Moduli Stack of Elliptic Curves

• The compactified moduli stack of elliptic curves is a Deligne–Mumford stack that parametrizes both smooth curves and their stable degenerations with full level structures.
• It employs twisted stable maps and generalized elliptic curves to construct canonical, proper, and flat compactifications aligning with classical modular models.
• Explicit boundary stratifications and modular isomorphisms in bad characteristics enable deeper insights into the geometry and arithmetic of elliptic curves.

The compactified moduli stack of elliptic curves is a Deligne–Mumford stack that provides a natural, algebraically robust modular interpretation of degenerating families of elliptic curves, including elliptic curves with prescribed level structures, over arbitrary bases. Through the frameworks of twisted stable maps and generalized elliptic curves, such stacks admit canonical compactifications encompassing not only smooth elliptic curves but also their stable degenerations, such as Néron polygons and twisted curves with stacky nodes. The modern theory recovers and extends the minimal regular models (in the sense of Katz–Mazur) over $\Spec\mathbb{Z}$, displays compatibility with boundary stratifications in bad characteristics, and unifies approaches to moduli of curves, their level structures, and associated degenerations.

1. Moduli Problem and Stack-Theoretic Formulation

Let $N\geq2$ and $G=(\mathbb{Z}/N)^2$ viewed as a finite diagonalizable group-scheme over $\mathbb{Z}$. The moduli problem is to parametrize elliptic curves (and their degenerations) equipped with a full level-$N$ structure—that is, a homomorphism $\phi: G^\vee\cong G \to E[N]$, where $E$ is an elliptic curve or a suitable degeneration. The essential moduli stack is the closure of the open locus of such smooth elliptic curves with $G$-structure within a larger stack which also includes stable singular curves.

The stack $K_{1,1}(\mathcal{B}G)$, built from the Kontsevich moduli stack of twisted stable maps from 1-marked, genus 1 twisted curves into the classifying stack $\mathcal{B}G$, is a proper Artin stack over $\mathbb{Z}$. The open substack $H(G)$, representing smooth curves with full $G$-level structure, embeds into $K_{1,1}(\mathcal{B}G)$. The scheme-theoretic closure $\overline{X}(N)$ of $H(G)$ inside $K_{1,1}(\mathcal{B}G)$ is the compactified moduli stack of elliptic curves with full level-$N$ structure, isomorphic to the regular Katz–Mazur model $X(N)$ over $\Spec\mathbb{Z}$ (Niles, 2012).

Equivalent formulations in terms of generalized elliptic curves and twisted curves yield essentially the same stack, as made precise by explicit modular isomorphisms (Niles, 2014).

2. Twisted Stable Maps and Picard Stack Construction

Given a 1-marked genus-1 twisted curve $\mathcal{C}\to S$, where $S=\Spec\mathbb{Z}$ (or its localizations), a representable map $f:\mathcal{C}\to\mathcal{B}G$ (a twisted stable map) is equivalent to the data of a $G$-torsor $P\to\mathcal{C}$. The moduli interpretation is via a homomorphism $\phi: G^\vee \longrightarrow \Pic^0_{\mathcal{C}/S}$. In the smooth case ($\mathcal{C}=E$ a smooth elliptic curve), $\Pic^0_{E/S} \cong E$, and $G$-torsors correspond bijectively to full level-$N$ structures.

To compactify, one allows $\mathcal{C}$ to degenerate to a twisted Néron $N$-gon, i.e., the stack quotient $[\text{Néron }N\text{-gon}/\mu_N]$ with a stacky node of index $N$. On such curves, the Picard scheme $\Pic^0_{\mathcal{C}/S}$ recovers the expected smooth part of a generalized elliptic curve, and level structures correspond accordingly. This construction is compatible with the valuative criterion for properness and produces a flat, proper Deligne–Mumford stack (Niles, 2012, Niles, 2014).

3. Comparison to Generalized Elliptic Curves and Modular Isomorphisms

The approach via twisted curves is naturally isomorphic to the classical moduli of generalized elliptic curves (Deligne–Rapoport semistable genus 1 curves with group law) equipped with level structures. The main result establishes an explicit canonical isomorphism between the stack compactified via twisted stable maps and the stack compactified using generalized elliptic curves—both encapsulating the same modular data on the boundary (Niles, 2014).

At the boundary, the stack of twisted genus-1 curves (with stacky cyclic stabilizers at nodes) is related to the Néron $d$-gons of generalized elliptic geometry by stack-quotient constructions and universal group-scheme correspondences. Drinfeld-type level structures are preserved under this correspondence, and both viewpoints yield modularly meaningful and proper compactifications.

4. Boundary Stratification and Degenerations in Bad Characteristic

In characteristics $p$ dividing $N$, the special fiber of the compactified moduli stack $\overline{X}(N)_k$ decomposes with crossings at supersingular points into boundary strata indexed by cyclicity data $(a,b)$, as described by Drinfeld and Katz–Mazur. Each stratum $X(N)_k^{(a,b)}$ parametrizes twisted curves (or generalized elliptic curves) whose level structure on $\Pic^0$ (or in the smooth locus) has component-label $(a,b)$, corresponding to specified extensions and cyclic subgroup data (Niles, 2012).

Such a stratification is made explicit via the moduli description in the twisted stable map framework and naturally organizes the boundary divisors intersecting at supersingular points, matching the classical description in integral models of modular curves.

5. Relation to Deligne–Rapoport and Katz–Mazur Models

When restricted to coarse families of stable genus-1 curves and non-stacky markings, the compactified moduli stack recovers precisely the classical Deligne–Rapoport compactification $M_{1,1}[N]$ and its extension to degenerate fibers, including its boundary of Néron polygons and component labels. The twisted perspective makes stacky structure at the cusps automatic and synthesizes level structure with degenerations (Niles, 2012).

For stacks associated to $\Gamma_0(n)$, a refined moduli problem is necessary when $n$ is not squarefree; naive moduli stacks fail to give the correct stack at the cusps. Deligne’s refinement provides a modular stack $X_0(n)$ with a correct moduli interpretation everywhere, matching the normalization construction and yielding regularity at the cusps (Cesnavicius, 2015).

6. Regularity, Functoriality, and Applications

The compactified moduli stacks are proper, flat, regular Deligne–Mumford stacks of relative dimension 1 over $\Spec\mathbb{Z}$, extending the Katz–Mazur regular models (Niles, 2012, Cesnavicius, 2015). The modular interpretations are compatible with Hecke correspondences and Galois-twist descriptions, and explicit equations for $X(N)$ in small level cases remain valid over regular excellent Noetherian bases (Studnia, 9 Apr 2025).

Further, compactifications admit uniformity across base changes, functorial extension of level structures, and facilitate reduction arguments for more intricate level structures via towers of compactifications and congruences, all of which are essential for modern research in the geometry and arithmetic of modular curves.

7. Extensions and Broader Moduli Theory

A full classification of modular compactifications of the space of pointed genus 1 curves with Gorenstein singularities, including Smyth’s $m$-stable spaces and their combinatorial generalizations indexed by $Q$ (downward-closed subsets of the partition poset), elucidates the range of possible compactified moduli stacks beyond the classical Deligne–Mumford and twisted stable map settings (Bozlee et al., 2021). Their organization into a cube complex reflects the wall-and-chamber structure anticipated by log minimal model program theory, indicating a rich combinatorial and geometric landscape for modular compactifications of elliptic curves and related objects.