Let $A=\mathbb{F}_q[T]$ be the polynomial ring over $\mathbb{F}_q$, and $F$ be the field of fractions of $A$. Let $\phi$ be a Drinfeld $A$-module of rank $r$ over $F$. For all but finitely many primes $\mathfrak{p}\lhd A$, one can reduce $\phi$ modulo $\mathfrak{p}$ to obtain a Drinfeld $A$-module $\phi\otimes\mathbb{F}_\mathfrak{p}$ of rank $r$ over $\mathbb{F}_\mathfrak{p}=A/\mathfrak{p}$. The endomorphism ring $\mathcal{E}_\mathfrak{p}=\mathrm{End}_{\mathbb{F}_\mathfrak{p}}(\phi\otimes\mathbb{F}_\mathfrak{p})$ is an order in an imaginary field extension $K$ of $F$ of degree $r$. Let $\mathcal{O}_\mathfrak{p}$ be the integral closure of $A$ in $K$, and let $\pi_\mathfrak{p}\in \mathcal{E}_\mathfrak{p}$ be the Frobenius endomorphism of $\phi\otimes\mathbb{F}_\mathfrak{p}$. Then we have the inclusion of orders $A[\pi_\mathfrak{p}]\subset \mathcal{E}_\mathfrak{p}\subset \mathcal{O}_\mathfrak{p}$ in $K$. We prove that if $\mathrm{End}_{F^\mathrm{alg}}(\phi)=A$, then for arbitrary non-zero ideals $\mathfrak{n}, \mathfrak{m}$ of $A$ there are infinitely many $\mathfrak{p}$ such that $\mathfrak{n}$ divides the index $\chi(\mathcal{E}_\mathfrak{p}/A[\pi_\mathfrak{p}])$ and $\mathfrak{m}$ divides the index $\chi(\mathcal{O}_\mathfrak{p}/\mathcal{E}_\mathfrak{p})$. We show that the index $\chi(\mathcal{E}_\mathfrak{p}/A[\pi_\mathfrak{p}])$ is related to a reciprocity law for the extensions of $F$ arising from the division points of $\phi$. In the rank $r=2$ case we describe an algorithm for computing the orders $A[\pi_\mathfrak{p}]\subset \mathcal{E}_\mathfrak{p} \subset \mathcal{O}_\mathfrak{p}$, and give some computational data.

中文翻译：

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Drinfeld 模归约的自同态环

设 $A=\mathbb{F}_q[T]$ 是 $\mathbb{F}_q$ 上的多项式环，$F$ 是 $A$ 的分数域。令 $\phi$ 是一个 Drinfeld $A$ 模块，其等级为 $r$ 超过 $F$。对于除有限多个素数 $\mathfrak{p}\lhd A$ 之外的所有素数，可以将 $\phi$ 取模 $\mathfrak{p}$ 得到一个 Drinfeld $A$-module $\phi\otimes\mathbb{F }_\mathfrak{p}$ 的排名 $r$ 超过 $\mathbb{F}_\mathfrak{p}=A/\mathfrak{p}$。自同态环 $\mathcal{E}_\mathfrak{p}=\mathrm{End}_{\mathbb{F}_\mathfrak{p}}(\phi\otimes\mathbb{F}_\mathfrak{p })$ 是度数为 $r$ 的 $F$ 的假想字段扩展 $K$ 中的一个命令。令 $\mathcal{O}_\mathfrak{p}$ 是 $A$ 在 $K$ 中的积分闭包，令 $\pi_\mathfrak{p}\in \mathcal{E}_\mathfrak{p} $ 是 $\phi\otimes\mathbb{F}_\mathfrak{p}$ 的 Frobenius 自同态。然后我们在 $K$ 中包含订单 $A[\pi_\mathfrak{p}]\subset \mathcal{E}_\mathfrak{p}\subset \mathcal{O}_\mathfrak{p}$。我们证明如果 $\mathrm{End}_{F^\mathrm{alg}}(\phi)=A$，那么对于任意的非零理想 $\mathfrak{n}，\mathfrak{m}$ 的 $ A$ 有无穷多个 $\mathfrak{p}$ 使得 $\mathfrak{n}$ 除以索引 $\chi(\mathcal{E}_\mathfrak{p}/A[\pi_\mathfrak{p} ])$ 和 $\mathfrak{m}$ 将索引 $\chi(\mathcal{O}_\mathfrak{p}/\mathcal{E}_\mathfrak{p})$ 相除。我们表明索引 $\chi(\mathcal{E}_\mathfrak{p}/A[\pi_\mathfrak{p}])$ 与由除法引起的 $F$ 扩展的互易律有关$\phi$ 的点。在排名 $r=2$ 的情况下，我们描述了一种计算订单 $A[\pi_\mathfrak{p}]\subset \mathcal{E}_\mathfrak{p} \subset \mathcal{O}_\ mathfrak{p}$,