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Computing submodules of points of general Drinfeld modules over finite fields

Antoine Leudière, Renate Scheidler

TL;DR

The paper tackles the problem of determining the submodule structure of the point module $\phi(K)$ for Drinfeld $A$-modules over finite fields, and, in the case $A=\mathbb{F}_q[T]$, provides a Frobenius decomposition of submodules. It develops a general framework based on Frobenius normal forms, module presentations, and Fitting ideals, augmented by fast Ore-polynomial arithmetic, to compute invariant factors and Frobenius decompositions efficiently; a SageMath implementation is provided. For $A=\mathbb{F}_q[T]$, the main algorithm achieves a complexity of $\widetilde{O}(dr + dn + d^\omega)$, with $n$ the $\tau$-degree and $r$ the module rank, and the results extend to the general case via a presentation-based approach. Additionally, the authors introduce an invariant $g$ via Anderson motives that encodes all $a\in \mathbb{F}_q[T]$ for which the $a$-torsion is rational, computable without prior invariant factors, and thereby offering a practical tool for torsion rationality questions. Collectively, the work provides a deterministic, implementable, and broadly applicable set of algorithms for Drinfeld modules that outperform elliptic-curve based methods in this context and opens avenues for efficient arithmetic in function-field settings.

Abstract

We present an algorithm for computing the structure of any submodule of the module of points of a Drinfeld $A$-module over a finite field, where $A$ is a function ring over $\mathbb F_q$. When the function ring is $A = \mathbb F_q[T]$, we additionally compute a Frobenius decomposition of said submodule. Our algorithms apply in particular to kernels of isogenies and torsion submodules. They are presented within the frameworks of Frobenius normal forms, presentations of modules, and Fitting ideals. They rely largely on efficient and classical linear algebra methods, combined with fast arithmetic of Ore polynomials. We analyze the complexity of our algorithms, explore optimizations, and provide an implementation in SageMath. Finally, we compute a simple invariant attached to a Drinfeld $\mathbb F_q[T]$-module that encodes all the polynomials in $\mathbb F_q[T]$ whose associated torsion is rational.

Computing submodules of points of general Drinfeld modules over finite fields

TL;DR

The paper tackles the problem of determining the submodule structure of the point module for Drinfeld -modules over finite fields, and, in the case , provides a Frobenius decomposition of submodules. It develops a general framework based on Frobenius normal forms, module presentations, and Fitting ideals, augmented by fast Ore-polynomial arithmetic, to compute invariant factors and Frobenius decompositions efficiently; a SageMath implementation is provided. For , the main algorithm achieves a complexity of , with the -degree and the module rank, and the results extend to the general case via a presentation-based approach. Additionally, the authors introduce an invariant via Anderson motives that encodes all for which the -torsion is rational, computable without prior invariant factors, and thereby offering a practical tool for torsion rationality questions. Collectively, the work provides a deterministic, implementable, and broadly applicable set of algorithms for Drinfeld modules that outperform elliptic-curve based methods in this context and opens avenues for efficient arithmetic in function-field settings.

Abstract

We present an algorithm for computing the structure of any submodule of the module of points of a Drinfeld -module over a finite field, where is a function ring over . When the function ring is , we additionally compute a Frobenius decomposition of said submodule. Our algorithms apply in particular to kernels of isogenies and torsion submodules. They are presented within the frameworks of Frobenius normal forms, presentations of modules, and Fitting ideals. They rely largely on efficient and classical linear algebra methods, combined with fast arithmetic of Ore polynomials. We analyze the complexity of our algorithms, explore optimizations, and provide an implementation in SageMath. Finally, we compute a simple invariant attached to a Drinfeld -module that encodes all the polynomials in whose associated torsion is rational.
Paper Structure (24 sections, 15 theorems, 23 equations, 6 algorithms)

This paper contains 24 sections, 15 theorems, 23 equations, 6 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Invariant factors and Frobenius decompositions
  4. Ore polynomials
  5. Drinfeld modules
  6. Computer algebra
  7. Cost functions for polynomials and matrices
  8. Ore polynomials
  9. Divide and conquer algorithms
  10. Modules from matrices
  11. The case $A = {\mathbb{F}_q}[T]$
  12. The general case
  13. Computer algebra
  14. Computer algebra of Ore polynomials
  15. Computer algebra for divisibility chains
  16. ...and 9 more sections

Key Result

Theorem 2.1

Let $A$ be a Dedekind domain and $W$ a finite $A$-module. There exist ideals $\mathfrak d_1, \dots, \mathfrak d_\ell \subset A$ and elements $x_1,\dots, x_\ell \in W$ such that the following hold. In particular, $W \simeq \prod_{i=1}^n A/\mathfrak d_i$, and the chain $(\mathfrak d_1, \dots, \mathfrak d_\ell)$ is unique. We call the ideals $\mathfrak d_1,\dots,\mathfrak d_\ell$ the invariant facto

Theorems & Definitions (37)

  • Theorem 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Remark 2.7
  • Definition 2.8
  • Definition 2.9
  • Proposition 2.10
  • ...and 27 more