Endomorphism rings of reductions of Drinfeld modules

Abstract

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 $r\geq 2$ 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.