Computing submodules of points of general Drinfeld modules over finite fields

TL;DR

This paper introduces algorithms for computing the structure of submodules of points of Drinfeld modules over finite fields, with applications to kernels of isogenies and torsion submodules, using linear algebra and Ore polynomial arithmetic.

Contribution

It presents new algorithms for analyzing submodules of Drinfeld modules, including Frobenius decomposition, with implementation and complexity analysis.

Findings

Algorithms effectively compute submodule structures.

Implementation in SageMath demonstrates practical utility.

Complexity analysis guides optimization of computations.

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.