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Some results on finite Drinfeld modules

Authors:	Chih-Nung Hsu

Institution:	Department of Mathematics, National Taiwan Normal University, 88 Sec. 4 Ting-Chou Road, Taipei, {Taiwan}

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 $\operatorname{K} =\mathbb F_q(t)$ , is an odd prime number and $\psi$ is the Carlitz $\mathbb F_qt]$ -module, then the finite $\mathbb F_qt]$ -module $\psi(\mathbb F_qt]/\mathfrak P^N)$ is always cyclic.

Keywords:	Drinfeld modules Hilbert class field

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