Let ${\mathbb{F}}_q[T]$ be the polynomial ring over a finite field ${\mathbb{F}}_q$. We study the endomorphism rings of Drinfeld ${\mathbb{F}}_q[T]$-modules of arbitrary rank over finite fields. We compare the endomorphism rings to their subrings generated by the Frobenius endomorphism and deduce from this a refinement of a reciprocity law for division fields of Drinfeld modules proved in our earlier paper. We then use these results to give an efficient algorithm for computing the endomorphism rings and discuss some interesting examples produced by our algorithm.

中文翻译：

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计算 Drinfeld 模块的自同态环和 Frobenius 矩阵

让${\mathbb{F}}_q[吨]$是有限域上的多项式环${\mathbb{F}}_q$. 我们研究 Drinfeld 的自同态环${\mathbb{F}}_q[吨]$- 有限域上任意等级的模块。我们将自同态环与其由 Frobenius 自同态产生的子环进行比较，并由此推导出在我们之前的论文中证明的 Drinfeld 模块的划分域的互易律的改进。然后我们使用这些结果给出一个计算自同态环的有效算法，并讨论我们的算法产生的一些有趣的例子。