Computing submodules of points of general Drinfeld modules over finite fields

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.