Isogeny Classes of Drinfeld Modules

• Isogeny classes of Drinfeld modules are equivalence classes of modules connected by surjective A-homomorphisms with finite kernels, analogously extending the concept of elliptic curve isogenies.
• They are classified using Weil polynomials, J-invariants, and modular polynomials under precise local and global conditions, forming a key framework in arithmetic geometry over function fields.
• The computational approach employs algorithms for enumeration and isogeny graph traversal, enabling advances in explicit class field theory and emerging cryptographic constructions.

A Drinfeld module over a finite field or function field generalizes the notion of elliptic curves in the arithmetic of function fields. The isogeny class of a Drinfeld module captures the modules connected by isogenies—surjective A-homomorphisms with finite kernel—mirroring the abelian variety context but governed by the structure of noncommutative T- and τ-polynomials. The study of isogeny classes of Drinfeld modules provides both a classification framework (Honda–Tate–Yu theory, modular polynomials, endomorphism algebras, and invariants) and a computational toolkit (algorithms for enumeration, explicit relations, and structure of isogeny graphs), and is essential for arithmetic geometry, explicit class field theory, and emerging cryptographic constructions.

1. Formal Definition and Structure of Drinfeld Modules and Isogenies

Let $A = \mathbb{F}_q[T]$. A Drinfeld $A$-module of rank $r\ge2$ over an $A$-field $(L,\gamma:A\to L)$ is an $\mathbb{F}_q$-algebra homomorphism

$$\varphi: A \longrightarrow \operatorname{End}_{\mathbb{F}_q}(L) \cong L\{\tau\}$$

where $L\{\tau\}$ is the Ore polynomial ring with $\tau(x) = x^q$, and $\tau c = c^q \tau$ for $c\in L$. Explicitly, for $T\in A$,

$$\varphi_T = \gamma(T) + a_1\tau + \dotsb + a_{r-1}\tau^{r-1} + a_r\tau^r, \qquad a_r \ne 0.$$

An isogeny $f: \varphi \to \psi$ is a nonzero $f \in L\{\tau\}$ such that $f \circ \varphi_a = \psi_a \circ f$ for all $a \in A$. The isogeny class of $\varphi$ is its equivalence class under isogenies. The kernel of an isogeny is a finite $A$-submodule of $L^{sep}$; the degree or type of an isogeny is governed by the τ-degree of $f$ and the structure of the kernel (e.g., for monic $N$-isogeny, $\deg_\tau f = q^{r \deg N}$, $f$ is monic in $\tau$ and $\ker f \subset \varphi[N]$) (Breuer et al., 2013, Breuer et al., 2024, Karemaker et al., 2022).

2. Invariants and Classification of Isogeny Classes

The isogeny class of a rank-$r$ Drinfeld module over a finite field $L$ ($\deg L = m$ over its $A$-characteristic residue field) is classified by a Weil polynomial, the minimal polynomial $M(x)$ of the Frobenius $\pi_\varphi = \tau^s$ ($s = [L:\mathbb{F}_q]$), subject to Yu's conditions:

• $M(x) \in A[x]$ is monic, degree divides $r$,
• $M(x)$ is separable unless $p\,|\,r$,
• Satisfies certain local conditions at characteristic place and at infinity regarding ramification, place splitting, and eigenvalue size.

There is a bijection between isogeny classes and Weil polynomials meeting these constraints (Assong, 2020). For rank 2, these reduce to explicit analogues of the classical Hasse–Weil bounds; for higher rank, new factorizations and inseparable phenomena arise (e.g., $M(x) = f(x^{p^n})$ for inseparable cases).

Within each isogeny class, isomorphism classes are uniquely determined by:

• Basic $J$-invariants (Potemine): rational functions in the coefficients of the $\tau$-expansion subject to specific monomial and weight conditions.
• Fine isomorphy invariants: constructed from Bézout relations among the exponents $\{q^i-1\}$, representing multiplicative relations among the coefficients modulo suitable $d$th powers.

Two modules $\varphi$, $\psi$ in the same isogeny class are $L$-isomorphic if and only if all their $J$- and fine invariants coincide (Assong, 2020).

3. Modular Polynomials and Kronecker Congruences

Modular polynomials encode relations between isomorphism invariants of Drinfeld modules linked by isogenies of prescribed type. For rank $r\ge2$, fix algebraically independent $g_1,\dotsc,g_{r-1}$ over $k=\mathbb{F}_q(T)$ and define the generic Drinfeld module $$y_T = T + g_1\tau + \cdots + g_{r-1}\tau^{r-1} + \tau^r$$. The full modular polynomial of level $P$ and invariant $J\in C = B^{\mathbb{F}_q^*}$ is

$$\Psi_{J,P}(X) = \prod_{f \in I_P} (X - J(y^{(f)})),$$

where $I_P$ is the finite set of monic $P$-isogenies of $y$. By partitioning the kernel types, "partial" modular polynomials $\Phi^H_{J,P}$ are defined, whose roots enumerate $P$-isogenous modules with kernel in a specified $GL_r$-orbit.

Kronecker congruence relations (Breuer–Rück): Under reduction modulo a prime $P$, the modular polynomial factors into ordinary and special loci. Explicit congruences

$$\Phi^{ord}_{J,P,H_s}(X) \equiv \Phi^{spec}_{J,P,H_{s+1}}(X^{|P|}) \mod P$$

describe the transition of isogeny types under Frobenius, reflecting how ordinary isogenies at level $s$ "fuse" into special isogenies at level $s+1$ (Breuer et al., 2013).

4. Endomorphism Rings and Orders within an Isogeny Class

Given a Drinfeld module $\varphi$ in an isogeny class defined by $M(x)$, the endomorphism ring $\mathcal{E} = \operatorname{End}_k(\varphi)$ is an $A$-order in the endomorphism algebra $D = \mathcal{E}\otimes_A k$ (a field in the CM case). The possible endomorphism rings occurring within an isogeny class are those $A$-orders $\mathcal{O} \subset D$ that:

• Contain $\pi$ (the Frobenius),
• Are locally maximal at the unique place $v_0$ above the characteristic place $v$ of $A$ (i.e., $\mathcal{O}\otimes_A A_v$ is a maximal $A_v$-order in $D\otimes_k k_v$).

For rank 3, explicit forms for all such $A$-orders can be computed. In the ordinary case, every order containing $\pi$ is locally maximal at $v_0$ (Assong, 2020, Karemaker et al., 2022). In the supersingular or non-ordinary case, local conditions determine the realizable orders.

5. Structure, Enumeration, and Graphs of Isogeny Classes

The set of Drinfeld modules in an isogeny class and a given endomorphism ring is parameterized by classes of fractional ideals in the order up to $D$-linear equivalence. In the ordinary or prime field case, there is a free and transitive action of the class group of $A[\pi]$ on the set of isomorphism classes in the isogeny class, and the number of isomorphism classes equals $|\operatorname{Pic}(A[\pi])|$ (Karemaker et al., 2022).

Computationally, enumeration can be effected by explicit algorithms: solving systems in $\alpha_i$ against $M(\tau^s)=0$, then grouping by $J$- and fine invariants (Assong, 2020). Modular polynomials, interpreted via elimination processes and explicit symmetric functions in the roots of parameterizing polynomials, encode the isogeny structure efficiently (Breuer et al., 2024). In CM cases, the isogeny graph exhibits volcano structures analogous to those in the theory of ordinary and supersingular elliptic curves (Chen, 26 Nov 2025).

6. Heights, Finiteness, and Arithmetic Properties

Taguchi’s isogeny estimate gives for $f:\varphi\to\psi$,

$$-\frac{1}{r} \log \deg f \leq h_{Tag}(\psi) - h_{Tag}(\varphi) \leq \frac{1}{r} \log \deg f,$$

furnishing a height gap within isogeny classes (Breuer et al., 2019). Finiteness follows: for given $K/F$ finite and fixed $r$, every $K$-isogeny class contains only finitely many $K$-isomorphism classes.

Canonical bounds can be placed on the coefficients of modular polynomials (e.g., in rank 2, explicit exponential bounds in terms of the degree of the involved ideal) (Breuer et al., 2019). For higher rank, similar but more intricate degree formulas are proven (Breuer et al., 2024).

7. Applications and Computational Aspects

Explicit understanding of isogeny classes of Drinfeld modules underpins a variety of computational schemes:

• Modular polynomials for explicit class field theory and algorithmic number theory.
• Isogeny graphs (including volcanoes in the CM case) inform potential isogeny-based cryptographic protocols, with structure reminiscent of CSIDH/SIDH schemes, though current attacks on Drinfeld module DLPs and isogeny enumeration suggest necessary caution (Assong, 2020).
• Algorithms, often implemented in SageMath, build on Ore-polynomial arithmetic, elimination techniques, and class group computations (Armana et al., 5 Jan 2026).
• Structural results enable effective partitioning and traversal of isogeny graphs, point counting, and the explicit realization of class field theoretical correspondences.

Isogeny classes of Drinfeld modules thus serve as a central organizing principle for arithmetic geometry over function fields, unifying theoretical, computational, and applied aspects (Breuer et al., 2013, Assong, 2020, Karemaker et al., 2022, Breuer et al., 2024, Chen, 26 Nov 2025).