#### Vol. 16, No. 5, 2022

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Artin's conjecture for Drinfeld modules

### Wentang Kuo and David Tweedle

Vol. 16 (2022), No. 5, 1025–1070
##### Abstract

Let $\varphi :A\to K\left\{\tau \right\}$ be a Drinfeld module of rank $2$ with generic characteristic, and suppose that the endomorphism ring of $\varphi$ induces a Drinfeld module $\psi :B\to K\left\{\tau \right\}$ of rank $1$. Let $a\in K$. We prove that the set of places $\wp$ of $K$ for which $a$ generates $\varphi \left({\mathbb{𝔽}}_{\wp }\right)$ as an $A$-module has a density. Furthermore, we show that this density is positive other than in some standard exceptional cases.

We also revisit Artin’s problem for Drinfeld modules of rank $1$, first considered by Hsu and Yu. A key point is that our methods do not require that $A$ be a principal ideal domain. We are also able to generalize a Brun–Titchmarsh theorem for function fields proved by Hsu.

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##### Keywords
Artin's primitive root conjecture, function fields, Drinfeld modules
##### Mathematical Subject Classification 2010
Primary: 11G09
Secondary: 11G15, 11R45
##### Milestones
Revised: 13 April 2021
Accepted: 22 August 2021
Published: 16 August 2022
##### Authors
 Wentang Kuo Department of Pure Mathematics University of Waterloo Waterloo ON Canada David Tweedle Department of Mathematics and Statistics The University of the West Indies, St. Augustine Campus St. Augustine Trinidad and Tobago