Isomorphism classes of Drinfeld modules over finite fields

TL;DR

This paper classifies isomorphism classes of Drinfeld modules over finite fields within a fixed isogeny class, revealing conditions for endomorphism rings and describing the action of fractional ideals.

Contribution

It provides a complete description of isomorphism classes of Drinfeld modules over finite fields, especially for ordinary or prime field cases, using ideal class actions.

Findings

Characterization of when the endomorphism ring is maximal at 

Description of the ideal class monoid action on isomorphism classes

Explicit classification for ordinary and prime field cases

Abstract

We study isogeny classes of Drinfeld $A$-modules over finite fields $k$ with commutative endomorphism algebra $D$, in order to describe the isomorphism classes in a fixed isogeny class. We study when the minimal order $A[\pi]$ of $D$ occurs as an endomorphism ring by proving when it is locally maximal at $\pi$, and show that this happens if and only if the isogeny class is ordinary or $k$ is the prime field. We then describe how the monoid of fractional ideals of the endomorphism ring $\mathcal{E}$ of a Drinfeld module $\phi$ up to $D$-linear equivalence acts on the isomorphism classes in the isogeny class of $\phi$, in the spirit of Hayes. We show that the action is free when restricted to kernel ideals, of which we give three equivalent definitions, and determine when the action is transitive. In particular, the action is free and transitive on the isomorphism classes in an isogeny class which is either ordinary or defined over the prime field, yielding a complete and explicit description in these cases.