Moduli Spaces of Elliptic Curves

• Moduli spaces of elliptic curves are geometric frameworks that encode families of curves along with their degenerations, markings, and level structures.
• They utilize compactifications via stability conditions and wall-and-chamber decompositions to systematically classify singularities and degenerations.
• These spaces underpin key theories in modular forms, enumerative geometry, and arithmetic studies through explicit arithmetic stratifications and stack-theoretic methods.

A moduli space of elliptic curves, or more generally a moduli stack, encodes families of elliptic curves and their degenerations, organized up to isomorphism with additional structure such as markings, level structures, or affine connections. The rigorous study of such moduli spaces forms a central part of algebraic geometry, arithmetic geometry, and related areas, serving as the foundation for the theory of modular forms, enumerative geometry, and arithmetic of elliptic curves. Recent advances provide a comprehensive classification of modular compactifications of moduli of pointed elliptic curves by Gorenstein curves, wall-and-chamber structures on the moduli, stack-theoretic refinements relevant for arithmetic counts, and moduli interpretations for both congruence and noncongruence modular curves.

1. Fundamental Definitions and Moduli Stacks

The Deligne–Mumford stack $M_{1,n}$ over $\mathbb{Z}[1/6]$ parametrizes smooth, connected, projective genus-1 curves equipped with $n$ ordered, pairwise disjoint marked points. S-points are families $(C\to S; \sigma_1,\ldots,\sigma_n)$, with each fibre $C$ a smooth genus-1 curve and $\sigma_i$ disjoint sections (Bozlee et al., 2021). The moduli stack $M_{1,1} = [\mathcal{H}/SL_2(\mathbb{Z})]$ is a smooth Deligne–Mumford stack of complex dimension 1, with coarse moduli space a weighted projective line $\mathbb{P}^1_{2,3}\setminus\{\infty\}$ featuring two orbifold points of orders 2 and 3 (at $j=1728, 0$) and a cusp at infinity (Gu et al., 2016).

Over $\mathbb{Z}[1/6]$, the compactified coarse moduli space $\overline{M}_{1,1}$ can be presented as the weighted projective stack $$P(4,6) = [(\mathbb{A}^2 \setminus \{0\})/\mathbb{G}_m]$$, with $\mathbb{G}_m$ acting as $(a_4,a_6)\mapsto (\zeta^4 a_4, \zeta^6 a_6)$ for $\zeta\in\mathbb{G}_m$ and $j=1728\,\frac{4a_4^3}{4a_4^3 + 27a_6^2}$ (Bejleri et al., 2022). Stack-theoretic points with nontrivial inertia correspond to elliptic curves with extra automorphisms (e.g., at $j=0, 1728$).

2. Modular Compactifications and Gorenstein Curves

A one-dimensional, reduced, connected, projective curve $C$ is Gorenstein if its dualizing sheaf $\omega_C$ is invertible. The compactification of $M_{1,n}$ is governed by stability conditions $Q \subset \text{Part}(n)$, where $\text{Part}(n)$ denotes the set of partitions of $\{1,\ldots,n\}$ ordered by refinement (Bozlee et al., 2021).

Key Definitions:

• Level partitions: For a genus-1 Gorenstein curve with distinct markings, each connected genus-one subcurve $Z\subset C$ (or each elliptic Gorenstein singularity $q$) is assigned a partition $\mathrm{lev}(Z)\in \mathrm{Part}(n)$, encoding the distribution of markings and attached rational tails outside $Z$.
• Elliptic Gorenstein singularities are classified by the genus of the singularity $g(p) = \delta(p) - m(p) + 1 = 1$ ($m(p)$ is the number of branches, $\delta(p)$ the delta invariant). Smyth's classification shows the possible singularities are determined by $m(p)$: $m=1$ corresponds to cusps, $m=2$ to tacnodes, $m\ge 3$ to $m$-fold Gorenstein points.

Classification Theorem: Every proper Deligne–Mumford modular compactification with Gorenstein geometric points and distinct markings is isomorphic to $\overline{M}_{1,n}(Q)$ for a unique $Q$ in the set $\mathcal{Q}_n$ of downward-closed subsets of $\text{Part}(n)$ not containing the discrete partition (Bozlee et al., 2021).

• Q-stability: A flat, proper family $\pi: C\to S$ with $n$ disjoint smooth sections is $Q$-stable if (i) every genus-one subcurve $Z$ has $\mathrm{lev}(Z)\not\in Q$; (ii) every elliptic Gorenstein singularity $q$ has $\mathrm{lev}(q)\in Q$; (iii) $H^0(C,\omega_C^\vee(-\sum \sigma_i)) = 0$ (no infinitesimal automorphisms).

3. Wall-and-Chamber Structures and Artin Stack Interpolation

$\text{Part}(n)$ naturally equips the set of compactifications with a cube complex structure $X_{1,n}\subset [0,1]^{\text{Part}(n)}$. Each $Q\in\mathcal{Q}_n$ corresponds to a vertex (with $c_P\in\{0,1\}$ for all $P$), and non-integer coordinates $0<c_P<1$ parametrize Artin stacks interpolating between DM stacks, allowing for half-contracted degenerations (Bozlee et al., 2021).

• Face inclusions and specialization: For $c,d\in X_{1,n}$ with $d$ a specialization of $c$ (i.e., $d_P = c_P$ if $c_P\in\{0,1\}$), there exists a fully faithful inclusion of stacks $$\overline{M}_{1,n}((d))\hookrightarrow \overline{M}_{1,n}((c))$$, mirroring the wall-and-chamber decomposition of the log-MMP for $\overline{M}_g$.
• Radially aligned curves and contractions: The stack $M_{1,n}^{\mathrm{rad}}$ of log radially aligned $n$-marked genus-1 curves is birationally contracted to $\overline{M}_{1,n}(Q)$ by simultaneously collapsing rational trees at radii below the universal radius defined by $Q$ (Bozlee et al., 2021).
• Examples: For $n=3$, there are $2^3+1=9$ distinct $Q$'s, which interpolate between the classical DM-Knudsen space $\overline{M}_{1,3}$ and the Smyth $m$-stable spaces for $m=0,1,2$.

4. Level Structures, Noncongruence Moduli, and Automorphic Aspects

Level structures: The classical concept fixes a finite Galois cover via the $\ell$-torsion subgroup $E[\ell]$. In the present context, $Q$-stability encompasses a combinatorial (geometric) analogue, with certain choices of $Q$ yielding compactifications dominating the full modular curve $X(\ell)\to M_{1,1}$ (Bozlee et al., 2021).

Noncongruence modular curves: The moduli stack $\mathcal{M}(G)$, for a finite 2-generated $G$, parametrizes elliptic curves with $G$-structures: surjective homomorphisms from the (profinite) fundamental group of the punctured curve to $G$ modulo conjugation (Chen, 2015). If $G$ is nonabelian, the stabilizer subgroup $\Gamma$ of $SL_2(\mathbb{Z})$ is typically noncongruence, yielding moduli spaces isomorphic (over $\mathbb{C}$) to $H/\Gamma$. These play a role in the inverse Galois problem and the arithmetic of modular forms with unbounded denominators.

Structure	Moduli Interpretation	Stack Type
Level–$\ell$	Principal $\ell$-torsion structure	Classical modular curve, DM stack
$Q$-stable	Combinatorial degeneration control	DM stack, Artin stack for interpolations
$G$-structure	(Non)abelian Galois covers	DM stack, possibly noncongruence quotient

5. Stacky Arithmetic and Point Counting

The cyclotomic stack $P(4,6)$ governs the compactified moduli of elliptic curves over $\mathbb{Z}[1/6]$, with stacky points at $j=0,1728$ reflecting enhanced automorphisms (Bejleri et al., 2022). For elliptic curves over function fields, rational points of fixed height correspond to twisted stable maps to $P(4,6)$. The stacky height function is $H(P) = \deg(L)$ for Weierstrass data $(a_4,a_6)$ with $a_4\in H^0(C,L^4)$ and $a_6\in H^0(C,L^6)$.

Stacky Tate algorithm: The local vanishing data $(\nu(a_4),\nu(a_6))$ determines the monodromy stabilizer and the Kodaira type of the special fibre. The moduli stack stratifies according to twisting data and admits a finite-type, separated Deligne–Mumford structure with Northcott property for heights.

Counting results: Asymptotic point counts for bounded stacky height over $\mathbb{F}_q(t)$ are governed by the main term $B^{5/6}$ and lower-order terms precisely corresponding to the stacky loci with extra automorphisms at $j=0,1728,\infty$: $$N(B) = 2\,\frac{q^9-1}{q^8-q^7}B^{5/6} +4\,\frac{q^5-1}{q^5-q^4}B^{1/2} + 2\,\frac{q^3-1}{q^3-q^2}B^{1/3} - 2B^{1/6} + O(1),$$ explaining the geometric origin of the secondary terms (Bejleri et al., 2022).

6. Line Bundles, Gerbes, and Universal Structures

The Hodge bundle $\mathcal{L}_H$ over $M_{1,1}$ is the determinant of the relative cohomology of the universal elliptic curve. The Bagger–Witten line bundle, a formal square root of $\mathcal{L}_H$, is a fractional (i.e., twisted, projective) line bundle classified by a non-trivial two-cocycle torsion class in $H^2(M_{1,1},\mathbb{C}^*)$ (Gu et al., 2016). Its twelfth tensor power trivializes, but on excising the orbifold points, the fractionality persists.

Passing to the metaplectic cover,

$$1 \to \mathbb{Z}_2 \to Mp(2,\mathbb{Z}) \to SL_2(\mathbb{Z}) \to 1,$$

one can realize the Bagger–Witten bundle as an honest line bundle on the stack $[\mathcal{H}/Mp(2,\mathbb{Z})]$. Physically, the moduli of superconformal field theories correspond to flat connections on this bundle, reflecting the torsion and flatness constraints inherent in worldsheet consistency.

Universal elliptic curves and Poincaré bundles exist only on stacks, not on the coarse spaces, due to enhanced stabilizers at special $j$-values. The existence of global universal SCFTs is similarly obstructed, reappearing in the structure of gerbes and stacks (Gu et al., 2016).

7. Topological and Analytic Moduli: Lamé Functions

The moduli spaces $M_m$ of Lamé pairs, i.e., pairs $(E, \omega)$ with $E$ a complex elliptic curve and $\omega$ an even Abelian differential of the second kind with a unique zero of order $2m$ at the origin and $m$ double poles with vanishing residues, are Riemann surfaces of finite type (Eremenko et al., 2020). These spaces are biholomorphic to the moduli of Lamé functions of order $m$, and their components, $L_m^I$ and $L_m^{II}$ for $m\ge 2$, have explicitly computable genera and Euler characteristics.

Degeneration loci corresponding to spherical metrics with a single cone point (Lin-Wang curves) are unions of $m(m+1)/2$ real-analytic arcs, each related to the real periods of underlying Abelian integrals.

• Classification and combinatorial foundations: (Bozlee et al., 2021)
• Stacky moduli and arithmetic properties: (Bejleri et al., 2022, Gu et al., 2016)
• Noncongruence structures and modular interpretations: (Chen, 2015)
• Topological and analytic moduli: (Eremenko et al., 2020)
• Explicit models for congruence relations: (Fisher, 2018)

These advances systematically describe the fine geometry, arithmetic, and stack theory underlying moduli spaces of elliptic curves, providing explicit structures, compactifications, wall-crossing phenomena, and arithmetic stratifications indispensable for research across algebraic geometry, modular forms, and arithmetic geometry.