Artin’s conjecture for Drinfeld modules

Abstract

Let $\varphi :A\to K\{\tau \}$ be a Drinfeld module of rank $2$ with generic characteristic, and suppose that the endomorphism ring of $\varphi$ induces a Drinfeld module $\psi :B\to K\{\tau \}$ of rank $1$. Let $a\in K$. We prove that the set of places $\wp$ of $K$ for which $a$ generates $\varphi ({\mathbb{𝔽}}_{\wp })$ 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.