Artin’s conjecture for Drinfeld modules

Kuo, Wentang; Tweedle, David

doi:10.2140/ant.2022.16.1025

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Artin's conjecture for Drinfeld modules

Wentang Kuo and David Tweedle

Vol. 16 (2022), No. 5, 1025–1070

DOI: 10.2140/ant.2022.16.1025

Abstract

Let $ϕ : A \to K {τ}$ be a Drinfeld module of rank $2$ with generic characteristic, and suppose that the endomorphism ring of $ϕ$ induces a Drinfeld module $ψ : B \to K {τ}$ of rank $1$ . Let $a \in K$ . We prove that the set of places $℘$ of $K$ for which $a$ generates $ϕ (𝔽_{℘})$ 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

Milestones

Received: 22 February 2020

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

Vol. 16, No. 5, 2022