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.

让$\phi ：\mathrm{一个}\to ķ\{\tau \}$是秩的 Drinfeld 模块$2$具有一般特征，并假设自构环$\phi$引入 Drinfeld 模块$\psi ：乙\to ķ\{\tau \}$等级$1$. 让$\mathrm{一个}\in ķ$. 我们证明了一组地方$\wp$的$ķ$为此$\mathrm{一个}$生成$\phi ({\mathbb{𝔽}}_{\wp })$作为一个$\mathrm{一个}$-module 有一个密度。此外，我们表明，除了一些标准的例外情况外，这种密度是正的。

我们还重新审视了 Drinfeld 等级模块的 Artin 问题$1$，首先由 Hsu 和 Yu 考虑。一个关键点是我们的方法不需要$\mathrm{一个}$是一个主理想域。我们还能够为 Hsu 证明的函数域推广 Brun-Titchmarsh 定理。