l-Isogenous Elliptic Curves

• l-Isogenous elliptic curves are defined via rational cyclic isogenies of prime degree, highlighting key Galois and arithmetic structures.
• They exhibit uniformity properties and explicit parameterizations that provide computable bounds on torsion growth and isogeny descent.
• Their study informs isogeny-based cryptography and local-global principles by quantifying invariants such as discriminants and Tamagawa numbers.

An $\ell$-isogeny of elliptic curves is a central object in the arithmetic of elliptic curves and the study of their Galois representations, with deep implications for the fields of arithmetic geometry, number theory, and cryptography. The theory focuses on cyclic isogenies of prime degree $\ell$, the fields over which such isogenies are defined, and the associated algebraic and arithmetic structures. Progress in the understanding of $\ell$-isogenous pairs has been driven by developments in uniformity results, explicit parameterizations, Galois-theoretic obstructions, and the analysis of local and global invariants.

1. Definition and Fundamental Properties

Let $E$ be an elliptic curve defined over a number field $F_0$ of characteristic zero, and let $\ell$ be a prime. A cyclic $\ell$-isogeny is an $F_0$-rational isogeny $\varphi: E \to E'$ with kernel $C = \ker\varphi$ a cyclic subgroup of order $\ell$. Equivalently, $C \subset E(\overline{\mathbb{Q}})[\ell]$ is cyclic of order $\ell$ and Galois-stable under $$G_{F_0}=\operatorname{Gal}(\overline{\mathbb{Q}}/F_0)$$. The quotient $E' = E/C$ inherits a natural structure as an elliptic curve defined over $F_0$.

Given any cyclic subgroup $C \subset E[\ell]$, there is a minimal extension $F_0(C)/F_0$ where $C$ is stable—this field is the fixed field of the stabilizer of $C$ in $G_{F_0}$. The isogeny (and $C$) is said to be $F_0$-rational if $F_0(C)=F_0$.

The dual of a cyclic $\ell$-isogeny $\varphi$ is the unique isogeny $\widehat{\varphi}: E' \to E$ of degree $\ell$ such that $\widehat{\varphi}\circ\varphi = [\ell]_E$, where $[\ell]_E$ denotes multiplication by $\ell$, and similarly for the other direction. This duality controls the structure of the isogeny graph for a fixed isogeny class.

2. Uniformity and Field of Definition Results

The study of which fields admit new $\ell$-isogenies, or rational points on $X_0(\ell)$, has seen breakthrough results. Under the so-called "LV-hypotheses"—that the Generalized Riemann Hypothesis (GRH) holds for all Dedekind zeta functions of subfields of $F_0$, and that $F_0$ contains no elliptic curve with complex multiplication (CM) defined over $F_0$—the following uniformity theorem holds (Genao, 2024):

There exists an integer $B = B(F_0)>0$ such that for any finite extension $L/F_0$ with $[L:F_0]$ coprime to $B$, and any elliptic curve $E/F_0$ with $j(E)\neq 0,1728$, every $L$-rational cyclic isogeny of $E$ is already $F_0$-rational. In particular, the set of primes $\ell$ for which a new $\ell$-isogeny first appears over some extension of $F_0$ is uniformly bounded in terms of $F_0$.

The constant $B(F_0)$ is effectively computable as the product of all primes $p\leq c(F_0)$, where $c(F_0)$ depends on the ramification in $F_0$ and the finite Larson–Vaintrob set of exceptional primes.

Key consequences:

• For $F_0=\mathbb{Q}$, this recovers the Mazur-Kenku bound: rational elliptic curves have rational $\ell$-isogenies only for $\ell\leq 163$, and over odd-degree extensions only for $\ell\leq 37$.
• For general $F_0$, except for a finite set of small primes, no new prime degree isogenies appear in extensions of degree coprime to $B(F_0)$.

When the LV-hypotheses are relaxed, similar strong statements hold for sufficiently large $\ell$ compared to the degree $[F_0:\mathbb{Q}]$ (Genao, 2024). If $\ell$ is large and unramified in $F_0$, then for any $E/F_0$:

• Either the image of the Galois representation $\rho_{E,\ell}$ is the full $\mathrm{GL}_2(\mathbb{Z}/\ell\mathbb{Z})$;
• Or it is contained in a Borel subgroup (i.e., reducible, corresponding to the existence of an $\ell$-isogeny over $F_0$), with index dividing a small explicitly bounded integer.

For Borel image, for all order-$\ell$ subgroups $C' \subset E[\ell]$, $[F_0(C'):F_0] \in \{1,\ell\}$; thus, isogenies are either $F_0$-rational or defined over a degree-$\ell$ extension.

3. Galois Representations and Failure of Local-Global Principles

The existence of an $\ell$-isogeny over a number field $K$ is intimately tied to the structure of the mod-$\ell$ Galois representation. A $K$-rational isogeny of degree $\ell$ exists if and only if $\rho_{E, \ell}(G_K)$ is contained, up to conjugacy, in a Borel subgroup of $\mathrm{GL}_2(\mathbb{F}_\ell)$.

Failure of the local-global principle for $\ell$-isogenies—that is, the existence of curves such that for almost all finite places $v$ of $K$, $E_{/v}$ admits an $\mathbb{F}_v$-rational $\ell$-isogeny, but $E/K$ does not—has been classified:

• Over $\mathbb{Q}$, the only such failure occurs for $(\ell, j) = (7,2268945/128)$; all other cases satisfy the local-global principle (Banwait et al., 2013).
• Over general number fields, further failures arise when the image $H_{E,\ell} \subset \mathrm{PGL}_2(\mathbb{F}_\ell)$ is dihedral ($H_{E,\ell} \cong D_{2n}$ with $n$ odd dividing $(\ell-1)/2$ and $\ell\equiv 3\!\mod 4$) or isomorphic to an exceptional group ($A_4$, $S_4$, $A_5$) for $\ell$ congruent to $1$ modulo $12$, $24$, $60$ respectively, provided the relevant quadratic subfields are contained in $K$.

Examples include infinite families for $\ell=5$ over $\mathbb{Q}(\sqrt{5})$ and for $\ell=13$ over $\mathbb{Q}(\sqrt{13})$, corresponding to special points on $X_{\text{split}}(5)$ and $X_{S_4}(13)$.

4. Local and Global Invariants under Isogeny

The passage from $E$ to an $\ell$-isogenous curve $E'$ alters arithmetic and geometric invariants in explicit ways (Dokchitser et al., 2012):

• Discriminant: If $\varphi: E \to E'$ is of prime degree $p=\ell>3$, $\Delta(E')/\Delta(E)^p \in K^{\times 12}$; for $p=2,3$ the power is $3,4$ respectively. The valuation satisfies $\delta' \equiv p \delta \pmod{12}$.
• Kodaira types and Tamagawa numbers: For semistable reduction, if reduction is good, $c'/c=1$; for multiplicative reduction, $c'/c=p^{\pm1}$. For additive, $c'/c=1$ if $\ell>3$, with precisely described exceptions for $\ell=2,3$.
• Periods: The ratio of periods satisfies $\Omega(E)/\Omega(E') \in \{1,p\}$ depending on signatures of real embeddings and embedding of the kernel into $\hat{E}$.
• Wild potentially supersingular reduction ($\ell=p$): Only partial results exist for $\alpha=|\phi^* \omega'/\omega|_K^{-1}$, $c'/c$, and period quotients; structure is known for tame and ordinary cases.

Tabular summary of discriminant and Tamagawa behavior for $p=\ell>3$: | Reduction Type | $\delta'$ formula | $c'/c$ | |----------------------|--------------------------|---------------| | good ordinary | $\delta'=\delta$ | 1 | | good supersingular | $\delta'=\delta$ | ? | | split multiplicative | $p\delta', \delta=p\delta'$ | $p,1/p$ | | non-split multiplicative | $p\delta', \delta=p\delta'$ | $1$ | | additive | $\delta'=\delta\pm\frac{p-1}{p}v(j)$ | $1$ |

5. Explicit Parametrization, Isogeny Graphs, and Descent

For primes $\ell$ with $X_0(\ell)$ genus zero ($\ell = 2,3,5,7,13$), all $\ell$-isogenous pairs over $\mathbb{Q}$ (up to quadratic twist) arise from two-parameter families $\mathcal{C}_{\ell,1}(t,d)$ and $\mathcal{C}_{\ell,2}(t,d)$ in short Weierstrass form, built from classical Fricke parameterizations. For each $(t,d)$, the curves $\mathcal{C}_{\ell,1}(t,d)$ and $\mathcal{C}_{\ell,2}(t,d)$ are linked by a unique explicit rational cyclic $\ell$-isogeny specified via kernel polynomials and formal expressions (Barrios, 2022).

Isogeny graphs in this setting consist of two vertices joined by a single edge labeled $\ell$. The explicit isogeny formulas are determined using Vélu's formulas: for each $\ell$-isogeny $$\psi_\ell: \mathcal{C}_{\ell,1}(t,d) \to \mathcal{C}_{\ell,2}(t,d)$$, the map has the form

$\psi_\ell(x,y) = (N(x)/D(x), y\, d/dx(N(x)/D(x))),$

with $D(x)$ the kernel polynomial and $N(x)$ determined by the formal identity.

Explicit isogeny descent, as developed by Miller and Stoll, translates $\ell$-isogeny Selmer groups to kernels of maps between finite-dimensional $\mathbb{F}_\ell$-vector spaces determined by the splitting fields of the kernels. The $\ell$-Selmer group sits inside $$H^1(\mathbb{Q}, E[\varphi]) \cong (K^\times/(K^\times)^\ell)(1)$$, where $K=\mathbb{Q}(E[\varphi])$. This facilitates explicit computation of the $\ell$-part of the Tate–Shafarevich group, yielding bounds and confirming the Birch–Swinnerton–Dyer conjecture in many cases (Miller et al., 2010).

6. Distribution of Invariants and Isomorphism Classes Modulo Primes

Given two $\ell$-isogenous elliptic curves $E, E'/\mathbb{Q}$, the proportion of primes $p$ such that $E(\mathbb{F}_p) \cong E'(\mathbb{F}_p)$ can be explicitly computed (Cullinan et al., 30 Dec 2025). This density is

$C(\ell) = \frac{\ell^4 - 2\ell^2 - 1}{\ell^4 - 1},$

for non-CM, "generic" cases (e.g., $C(2) = 7/15$, $C(3) = 31/40$, $C(5) = 287/312$).

The computation uses the image of the $\ell$-adic Galois representation, the action of Frobenius on the Tate modules, and the field extensions generated by torsion points. By Chebotarev and conditional probability arguments, the density of primes where $E(\mathbb{F}_p)$ and $E'(\mathbb{F}_p)$ are not isomorphic is described by a rapidly convergent series, its sum giving the stated formula.

This exact density quantifies, in the context of isogeny-based cryptography and random walks on isogeny graphs, the similarity of reductions of isogenous pairs over finite fields.

7. Broader Context and Applications

The structure of $\ell$-isogenies and their fields of definition has direct implications:

• For the uniform boundedness of torsion growth in families of elliptic curves, especially non-CM $\mathbb{Q}$-curves and their behavior over odd-degree or special extensions.
• For the explicit determination of rational points on modular curves $X_0(\ell)$ and the related modular curves arising from exceptional subgroups.
• For the surjectivity of Galois representations modulo large $\ell$ (the Serre uniformity question), as surjectivity failure is shown not to introduce new isogenies over odd extensions (Genao, 2024).
• For the arithmetic of the Tate–Shafarevich group and explicit verification of the Birch–Swinnerton–Dyer conjecture, notably via isogeny descent and kernel calculations (Miller et al., 2010).

The classification of the possible failures of local-global principle for $\ell$-isogenies exhausts all Galois-theoretic sources: generic, dihedral, and exceptional (icosahedral, tetrahedral, and octahedral) images (Banwait et al., 2013). The explicit parameterization and uniformity theorems inform both the theoretical landscape and algorithmic applications, including isogeny-based cryptography and explicit point counting.

References:

• (Genao, 2024): New isogenies of elliptic curves over number fields
• (Cullinan et al., 30 Dec 2025): The probability of isomorphic group structures of isogenous elliptic curves over finite fields
• (Miller et al., 2010): Explicit isogeny descent on elliptic curves
• (Banwait et al., 2013): Tetrahedral Elliptic Curves and the local-global principle for Isogenies
• (Dokchitser et al., 2012): Local invariants of isogenous elliptic curves
• (Barrios, 2022): Explicit classification of isogeny graphs of rational elliptic curves