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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Some results on finite Drinfeld modules

Author: Chih-Nung Hsu
Journal: Proc. Amer. Math. Soc. 126 (1998), 1955-1961
MSC (1991): Primary 11G09; Secondary 11A05
MathSciNet review: 1451807
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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.

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Chih-Nung Hsu
Affiliation: Department of Mathematics, National Taiwan Normal University, 88 Sec. 4 Ting-Chou Road, Taipei, Taiwan

Keywords: Drinfeld modules, Hilbert class field
Received by editor(s): July 23, 1996
Received by editor(s) in revised form: December 26, 1996
Communicated by: William W. Adams
Article copyright: © Copyright 1998 American Mathematical Society