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 ϕ : A 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 K{τ} of rank 1. Let a 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