Some results on finite Drinfeld modules

Let $\operatorname{K}$ be a global function field, $\infty$ a degree one prime divisor of $\operatorname{K}$ and let $\operatorname{A}$ be the Dedekind domain of functions in $\operatorname{K}$ regular outside $\infty$. Let $\operatorname{H}$ be the Hilbert class field of $\operatorname{A}$, $\operatorname{B}$ the integral closure of $\operatorname{A}$ in $\operatorname{H}$. Let $\psi$ be a rank one normalized Drinfeld $\operatorname{A}$ -module and let $\mathfrak P$ be a prime ideal in $\operatorname{B}$. We explicitly determine the finite $\operatorname{A}$-module structure of $\psi(\operatorname{B} /\mathfrak P^N)$. In particular, if $\operatorname{K} =\mathbb F_q(t)$, $q$ is an odd prime number and $\psi$ is the Carlitz $\mathbb F_q[t]$-module, then the finite $\mathbb F_q[t]$-module $\psi(\mathbb F_q[t]/\mathfrak P^N)$ is always cyclic.