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.

Keywords
Artin's primitive root conjecture, function fields, Drinfeld modules
Mathematical Subject Classification 2010
Primary: 11G09
Secondary: 11G15, 11R45