Finding Endomorphisms of Drinfeld modules

Abstract

We give an effective algorithm to determine the endomorphism ring of a Drinfeld module, both over its field of definition and over a separable or algebraic closure thereof. Using previous results we deduce an effective description of the image of the adelic Galois representation associated to the Drinfeld module, up to commensurability. We also give an effective algorithm to decide whether two Drinfeld modules are isogenous, again both over their field of definition and over a separable or algebraic closure thereof.

Computing endomorphism rings and Frobenius matrices of Drinfeld modules

“…Remark 3.4. In [15], Kuhn and Pink gave a different algorithm for computing End k (φ). They work in the most general setting where A is a finitely generated normal integral domain of transcendence degree 1 over a finite field F q , k is an arbitrary finitely generated field, and no restrictions on D are imposed.…”

“…They work in the most general setting where A is a finitely generated normal integral domain of transcendence degree 1 over a finite field F q , k is an arbitrary finitely generated field, and no restrictions on D are imposed. On the other hand, the emphasis of [15] is on the existence of a deterministic algorithm that computes End k (φ) rather that its practicality, so some of the details of the algorithm are left out.…”

“…Next, one computes the action of φ T on on this basis, which gives an explicit matrix for the action of φ T as an R-linear transformation of End k (φ). It is then claimed in [15] (proof of Proposition 5.14) that this calculation yields a basis of End k (φ) as an A-module, although the details of this deduction are not explained. We have not pursued this line of calculations, so we are unable to say how complicated it is in practice, and whether it suffices for deducing the finer number-theoretic properties of End k (φ), such as its discriminant over A or the Frobenius index.…”

“…A completely different algorithm from ours for computing the endomorphism rings of Drinfeld modules was proposed by Kuhn and Pink in [15]. This algorithm works in all cases, without the restriction (1.1), and determines a basis of E as an F q [π]-module.…”