Unified Moduli Solution for X₀(N)

• The paper introduces a unified moduli solution for X₀(N) by refining stack data to restore representability and coherence for every level.
• Explicit modular invariants, such as twisted Weierstrass functions, are constructed to generate the function field and compute isogenies.
• The approach establishes regularity at cusps and unifies both classical and sporadic cases with a rigorous stack-theoretic framework.

The modular curve $X_0(N)$ encodes the isomorphism classes of pairs $(E, C)$, with $E$ an elliptic curve and $C \subset E$ a cyclic subgroup of order $N$. The moduli problem for $X_0(N)$ centers on providing a comprehensive, uniform, and explicit description of the objects and morphisms parametrized by $X_0(N)$, valid for all $N$ and at all fibers, including cuspidal and non-squarefree cases. Previous treatments resolved this problem only in special cases (e.g., squarefree $N$, genus-$0$), but recent developments provide a unified approach that addresses all levels, with explicit modular generators, stack-theoretic precision, and regularity at the cusps.

1. Classical and Naïve Moduli Interpretations

The coarse moduli problem for $X_0(N)$ seeks to parametrize isomorphism classes of pairs $(E, C)$ where $E/\mathbb{C}$ is an elliptic curve and $C \subset E$ is a cyclic subgroup of order $N$. In geometric terms, isogenies $\varphi: E \to E'$ of degree $N$ with cyclic kernel $C \simeq \mathbb{Z}/N\mathbb{Z}$ correspond to points of $X_0(N)$. The analogous moduli space $X_1(N)$, classifying a chosen point of order $N$, admits explicit uniformizations, such as the Tate normal form. However, for $X_0(N)$, the direct moduli stack of generalized elliptic curves with ample cyclic subgroup of order $N$—often called $X_0(N)^{\mathrm{naive}}$—fails to coincide with the Deligne–Rapoport normalization $X_{0}(N)$ when $N$ is not squarefree. Specifically, over non-squarefree levels (e.g., $N = p^2$), automorphism pathologies arise at the "cusps", where degeneracy in the fibers creates representability issues for the forgetful functor to $X(1)$, distinguishing $X_0(N)^{\mathrm{naive}}$ from the correct moduli stack $X_0(N)$ (Cesnavicius, 2015).

2. Refined Moduli Stack via Decontraction and Coherence

Resolution of the moduli problem for all $N$ requires refining the moduli data near each cusp by incorporating the structure of "decontractions" corresponding to finer $m$-gon degenerations. For each divisor $m \mid N$, define $d(m) = m / \gcd(m, N/m)$ and consider $S_{(m)}$ as the locus where degenerate fibers are $d(m)$-gons. The universal decontraction stack $\mathcal{E}_{(m)}$ classifies $m$-gon curves together with a contraction isomorphism to a fixed fiber. Over each $S_{(m)}$, one specifies a cyclic subgroup $G_{(m)} \subset \mathcal{E}_{(m)}^{\mathrm{sm}}$ of order $N$; coherence demands that this data matches under pullback across overlapping decontractions. The moduli functor is then

$$(E \to S, \; G \subset E^{\mathrm{sm}}[N]\ \mathrm{ample~cyclic},\ \{ S_{(m)} \}_{m \mid N},\ \{ G_{(m)} \}_{m \mid N}\, \mathrm{coherent}).$$

This modular description ensures isomorphism with the Deligne–Rapoport stack $X_{0}(N)$ for all $N$, restoring representability and functorial glueing across degeneracies. For each $m \mid N$, the local moduli ($X_0(N)_{(m)}$) fits into a Cartesian diagram with $X_m$ and $X_{d(m)}$ as base, preserving the required stack-theoretic properties (representability, flatness, separatedness) (Cesnavicius, 2015).

3. Explicit Modular Function Generators and Uniform Construction

For all $N$, a unified, explicit description of $X_0(N)$ and its isogenies is achieved by constructing concrete generators—"twisted Weierstrass invariants"—for the function field $\mathbb{C}(X_0(N))$. Let

$$\begin{aligned} a_4(\tau) &= -\frac{E_4(\tau)}{48 E_2^{(N)}(\tau)^2},\ a_6(\tau) &= \frac{E_6(\tau)}{864 E_2^{(N)}(\tau)^3},\ a_4'(\tau) &= -\frac{E_4(N\tau)}{48 E_2^{(N)}(\tau)^2},\ a_6'(\tau) &= \frac{E_6(N\tau)}{864 E_2^{(N)}(\tau)^3}, \end{aligned}$$

where $E_4$, $E_6$ are Eisenstein series and $E_2^{(N)}$ is the standard weight-2 Eisenstein series on $\Gamma_0(N)$. These modular functions satisfy the transformation laws making them genuine $\Gamma_0(N)$-invariants, and the discriminant ensures that smooth fibers are parametrized away from zeros of $E_2^{(N)}$. Jeon–Kwon prove injectivity on the open $Y_0'(N)$ (excluding zeros of $E_2^{(N)}$), so $(a_4(\tau), a_6(\tau))$ separate points and

$\mathbb{C}(X_0(N)) = \mathbb{C}(a_4, a_6)$

for all $N$. Consequently, explicit rational functions in the affine coordinates $(X,Y)$ of any chosen algebraic model of $X_0(N)$ recover the Weierstrass invariants and thus the moduli objects $(E, C)$ for any given moduli point (Jeon et al., 24 Dec 2025).

4. Tower of Compactifications and Reductions

The theory is situated within a tower of compactification stacks $\overline{\mathcal{E}ll}_m$ for $m \ge 1$, with each stack parameterizing generalized elliptic curves whose degenerate fibers are $m$-gons. For divisors $d \mid m$, contraction maps $$c_{m \to d}\colon \overline{\mathcal{E}ll}_m \to \overline{\mathcal{E}ll}_d$$ normalize and collapse components in a controlled manner. This tower

$$\ldots \to \overline{\mathcal{E}ll}_{m'} \to \overline{\mathcal{E}ll}_m \to \ldots \to \overline{\mathcal{E}ll}_1 = X(1)$$

facilitates reduction of modular problems to the classical case via congruences and supports the analysis of Drinfeld level structures, which are shown to be finite flat of the expected rank regardless of degeneracy. This structure underpins the algebraic and topological properties necessary for the stack-theoretic uniformization of $X_0(N)$ (Cesnavicius, 2015).

5. Regularity at Cusps and Coarse Moduli Spaces

With the refined modular description, regularity at the cusps for $X_0(N)$ is established. On the open $j \ne 0, 1728$, the stack is tame Deligne–Mumford with stabilizer $\{\pm 1\}$, and its coarse moduli space is an étale quotient of a regular stack, ensuring regularity at those points. Over $\mathbb{Z}[1/N]$ the stack is everywhere smooth; singularities arise only at supersingular $j$-invariants in characteristic dividing $N$. This extends and generalizes classical results of Katz–Mazur, Edixhoven, and Gross–Zagier. The same argument extends to $X_1(N)$ and mixed-level curves $X_1(n; n')$, $X_0(n; n')$, via similar contraction-decontraction and coherence devices, ensuring regularity even where prior flatness-by-deformation arguments fail (Cesnavicius, 2015).

6. Explicit Examples and Rational Points

Jeon–Kwon’s construction permits fully explicit treatment of all rational points, including the eleven sporadic positive-genus levels with non-cuspidal $\mathbb{Q}$-points. For instance, at $N = 11$, Yang’s model

$X_0(11):\ Y^2+Y = X^3 - X^2 - 10X - 20$

along with specific $(X, Y)$-coordinates for rational points, gives Weierstrass coefficients via the explicit modular generators, recovering classical isogenies such as $121a1 \leftrightarrow 121c1$ up to quadratic twist. This procedure, detailed in Algorithm 4.1 of (Jeon et al., 24 Dec 2025), is effective: starting from a model for $X_0(N)$, one computes rational maps to the modular invariants and substitutes arbitrary moduli points to recover associated isogeny data.

7. Synthesis and Unification

The approaches of Česnavičius and Jeon–Kwon yield a genuinely unified and explicit solution to the moduli problem for $X_0(N)$. The modular stack structure with decontraction and coherence recovers representable, regular stacks for all $N$. Simultaneously, explicit modular generators $(a_4, a_6; a_4', a_6')$ generate the function field and provide an effective recipe for realizing any isogeny parametrized by $X_0(N)$. Together these results establish that, for every $N$, $X_0(N)$ is available as a proper, regular stack over $\mathbb{Z}$, with explicit invariants canonically describing the universal cyclic $N$-isogeny structure, achieving a unified, algebraically constructive moduli interpretation.

• Refined modular stacks resolve cusp pathologies for non-squarefree $N$ (Cesnavicius, 2015).
• Explicit modular invariants generate $\mathbb{C}(X_0(N))$ for all $N$ and enable computation of isogenies (Jeon et al., 24 Dec 2025).
• Regularity and representability at all fibers, including wild degeneration, are proved.
• The unified construction applies uniformly in genus 0, all finite-genus levels, and sporadic rational-point cases.