Concrete Moduli of Cyclic N-Isogenies

• Concrete moduli of cyclic N-isogenies are explicit models that parameterize elliptic curves with a cyclic subgroup of order N via the modular curve X₀(N).
• They employ universal families, explicit Weierstrass models, and modular functions such as Hauptmoduln to compute and construct isogenies uniformly.
• Applications span arithmetic geometry and isogeny-based cryptography, providing detailed classification over number fields and practical computational methods.

A cyclic $N$-isogeny is a morphism between elliptic curves preserving group structure and with cyclic kernel of order $N$. The modular curve $X_0(N)$ provides a coarse moduli space parameterizing isomorphism classes of pairs $(E, C)$ consisting of an elliptic curve $E$ and a cyclic subgroup $C \subset E$ of order $N$. Concrete moduli interpretations of cyclic $N$-isogenies are established by explicit models and formulas leveraging modular, arithmetic, and geometric structures on $X_0(N)$, with recent advances yielding uniformly explicit constructions for all $N$ and detailed arithmetic descriptions over number fields.

1. Functor-of-Points for $X_0(N)$

The modular curve $X_0(N)$, over a base scheme $S/\operatorname{Spec}\mathbb{Z}$, represents the functor

$$X_0(N)(S) = \{ \text{isomorphism classes of pairs } (E \to S, C) \}$$

where $E \to S$ is an elliptic curve and $C \subset E$ is a finite locally free $S$-subgroup scheme of rank $N$ that is étale-locally isomorphic to $\mathbb{Z}/N\mathbb{Z}$; i.e., $C$ is a cyclic $S$-subgroup of order $N$ (Bruin et al., 2014, Jeon et al., 24 Dec 2025). In this formulation, $X_0(N)$ is the coarse moduli scheme for such pairs, with the moduli problem encoded via the functor-of-points formalism.

Analogous formalism applies in other settings, such as the Sekiguchi–Suwa theory for cyclic $p^n$-isogenies of smooth group schemes, where the representing object is an explicit affine scheme classifying filtered group schemes and cyclic isogenies unifying Kummer and Artin–Schreier–Witt theory (Mézard et al., 2011).

2. Universal Family and Explicit Weierstrass Models

Over $X_0(N)$, there exists:

• A universal elliptic curve $\pi: E_{\text{univ}} \to X_0(N)$,
• A universal cyclic subgroup scheme $C_{\text{univ}} \subset E_{\text{univ}}$ of order $N$,
• A universal differential $\omega_{\text{univ}}$ (given a Néron differential).

Concrete Weierstrass models are constructed as

$y^2 = x^3 - g_4(t) \, x - g_6(t)$

where $g_4, g_6$ are weight-4 and weight-6 modular forms on $X_0(N)$. The cyclic order $N$ subgroup $C_{\text{univ}}$ is characterized by the vanishing locus of the $N$-th division polynomial $\psi_N(x; t)$, so

$$C_{\text{univ}} = \operatorname{Spec} (\mathcal{O}_{X_0(N)}[x] / \psi_N(x))$$

on the appropriate affine chart (Bruin et al., 2014). These universal formulas extend to both characteristic zero and positive characteristic, and their explicit construction solves the moduli problem in concrete terms for all points of $X_0(N)$ (Jeon et al., 24 Dec 2025).

On the analytic side, the Weierstrass coefficients for the isogeny pairs can be written as modular functions in terms of Eisenstein series and the weight-2 form $$E_2^{(N)}(\tau) = \frac{1}{2\pi i} \frac{d}{d\tau} \log(\eta(N\tau)/\eta(\tau))$$ (Jeon et al., 24 Dec 2025, Dowd, 2021). Explicit principal moduli (Hauptmoduln) are used for genus-0 $X_0(N)$ levels, yielding rational presentations of the moduli space and Weierstrass coefficients (Dowd, 2021).

3. Explicit Construction and Parameterization of Cyclic $N$-Isogenies

Every noncuspidal point $P \in X_0(N)(K)$ (over a number field $K$) corresponds to a pair of elliptic curves $(E, E')$ related by a unique cyclic $N$-isogeny, explicitly constructed as follows (Jeon et al., 24 Dec 2025, Dowd, 2021):

1. Compute the explicit modular functions $a_4, a_6, a_4', a_6'$ as rational functions (in a Hauptmodul or other generators) on the chosen model of $X_0(N)$:

$$a_4 = -\frac{E_4(\tau)}{48 \, E_2^{(N)}(\tau)^2}, \quad a_6 = \frac{E_6(\tau)}{864 \, E_2^{(N)}(\tau)^3}$$

with analogous expressions for $a_4', a_6'$ after substitution $\tau \to N\tau$.

2. The domain and codomain curves are then:

$$E : y^2 = x^3 + a_4 x + a_6, \quad E' : y^2 = x^3 + a_4' x + a_6'$$

3. The unique normalized isogeny $\phi: E \to E'$ with cyclic kernel of order $N$ is constructed algebraically by Vélu's formulas, using explicit points of $C$ in terms of $\psi_N(x)$.

This construction is uniform for all $N$, and in the genus-0 case, the explicit rational parametrizations in terms of Hauptmoduln are tabulated for all such $N$ (Dowd, 2021). For positive-genus $X_0(N)$, the explicit expressions are computed using models due to Yang, Galbraith, or canonical embeddings in projective space (Jeon et al., 24 Dec 2025).

4. Classification of Quadratic Points and $N$-Isogenies Over Number Fields

For $N$ such that $X_0(N)$ is hyperelliptic of genus $\ge 2$ and the Jacobian $J_0(N)(\mathbb{Q})$ has rank $0$, all points of $X_0(N)$ defined over quadratic fields can be classified explicitly (Bruin et al., 2014). Apart from a finite set of exceptional points, every quadratic point arises from the inverse image under the hyperelliptic map $X_0(N) \to \mathbb{P}^1$ of a rational point, giving

$$P = (x, \pm \sqrt{f_N(x)}) \in X_0(N)(K), \quad K = \mathbb{Q}(\sqrt{f_N(x)})$$

and the corresponding $N$-isogeny is concretely described by evaluating $g_4, g_6$ and $\psi_N(x)$ at $P$ and applying Vélu's formula (Bruin et al., 2014).

There is a finite, explicitly tabulated set of exceptional quadratic points which do not arise in this way, corresponding to elliptic curves over quadratic fields with an $N$-isogeny that are not $\mathbb{Q}$-curves. The complete data—including field, coordinates, and complex multiplication status—is given in the referenced tables (Bruin et al., 2014).

5. Relation to $\mathbb{Q}$-Curves and Isogeny Twists

A key arithmetic outcome is that, up to finitely many exceptions, every elliptic curve over a quadratic field admitting a cyclic $N$-isogeny is a $\mathbb{Q}$-curve (Bruin et al., 2014). For non-CM curves arising from non-exceptional points, there exists $d \mid N$ (determined by the involution structure) and a quadratic extension $L/K$ such that $E$ is $d$-isogenous over $K$ to the quadratic twist of its Galois conjugate by a parameter $\mu$. After base extension to $L$, $E$ becomes isogenous to all its conjugates, thereby satisfying the $\mathbb{Q}$-curve property. Explicit formulas for $d$ and $\mu$ are given in terms of the modular invariants $g_4, g_6, c_4, c_6$ at the relevant moduli point (Bruin et al., 2014).

6. Applications and Further Generalizations

These explicit moduli interpretations are central for:

• Arithmetic geometry of elliptic curves over number fields and the study of rational points on modular curves (Bruin et al., 2014);
• Construction of explicit models for isogeny-based cryptography, especially for small $N$, since rational parameterizations in terms of Hauptmoduln provide practical equations for isogeny computations (Dowd, 2021);
• The explicit Sekiguchi–Suwa theory, which constructs a fine moduli space for cyclic $p^n$-isogenies of affine smooth group schemes, realizing both the Kummer and Artin–Schreier–Witt cases in a unified framework, with universal families and universal isogeny built over an explicit parameter scheme $M = \operatorname{Spec} R$ (Mézard et al., 2011).

The uniformity and explicitness of the constructions, particularly for positive-genus and sporadic cases, provide both a complete theoretical solution and a computational pathway for all cyclic $N$-isogeny moduli problems (Jeon et al., 24 Dec 2025).

7. Summary Table: Explicit Moduli for Cyclic $N$-Isogenies

Aspect	Explicit Construction	Reference
Functor-of-points moduli	$(E, C)$ over $S$	(Bruin et al., 2014, Jeon et al., 24 Dec 2025)
Universal family	$y^2 = x^3 - g_4(t) x - g_6(t)$, $C$ via $\psi_N$	(Bruin et al., 2014)
Genus-0 parametrization	Hauptmodul eta products, rational Weierstrass coefficients	(Dowd, 2021)
Construction for all $N$	Modular $a_4, a_6$ via $E_2^{(N)}$, explicit in function field	(Jeon et al., 24 Dec 2025)
Quadratic points/arithmetic	Classification, exceptional sets, $\mathbb{Q}$-curve analysis	(Bruin et al., 2014)
Unification of group schemes	Fine moduli via Sekiguchi–Suwa, affine smooth case	(Mézard et al., 2011)

This synthesis demonstrates that the moduli problem for cyclic $N$-isogenies is resolved uniformly and explicitly across all modular levels, with both algebraic and analytic models, and comprehensive arithmetic classification over number fields.