Cyclicity of Drinfeld modules modulo primes in arithmetic progressions

Abstract

Let $A=\mathbb{F}_q[T]$ and $K=\mathbb{F}_q(T)$, and let $\phi$ be a Drinfeld $A$-module over $K$ of rank $r\geq 2$. We study the reduction of $\phi$ at primes $\mathfrak{p} \in A$ lying in a prescribed arithmetic progression, for which $\phi(\mathbb{F}_{\mathfrak{p}})$ has at most $r-1$ cyclic components. We obtain asymptotic formulas for the number of such primes of a given degree, with error bounds uniform in the modulus of the arithmetic progression. Furthermore, when $\mathrm{End}_{\overline K}(\phi)=A$, we sharpen the error term and, under additional hypotheses, establish a necessary and sufficient condition for positivity of the resulting density.