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 $\varphi$ be a Drinfeld $A$-module of rank $r\ge 2$ over $F$. For all but finitely many primes $\mathfrak{p}⊲A$, one can reduce $\varphi$ modulo $\mathfrak{p}$ to obtain a Drinfeld $A$-module $\varphi \otimes {\mathbb{F}}_{\mathfrak{p}}$ of rank $r$ over ${\mathbb{F}}_{\mathfrak{p}}=A/\mathfrak{p}$. The endomorphism ring ${\mathcal{E}}_{\mathfrak{p}}={\mathrm{End}}_{{\mathbb{F}}_{\mathfrak{p}}}(\varphi \otimes {\mathbb{F}}_{\mathfrak{p}})$ 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 $\varphi \otimes {\mathbb{F}}_{\mathfrak{p}}$. 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 ${\mathrm{End}}_{F^{\mathrm{alg}}}(\varphi )=A$, 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 $\varphi$. 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.

Journal of Number Theory (David Goss Memorial Issue), to appear