Some results on finite Drinfeld modules

Abstract:

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.