Cyclicity of Drinfeld modules modulo primes in arithmetic progressions

Let A=Fq[T] and K=Fq(T), and let $ϕ$ be a Drinfeld A-module over K of rank r≥ 2. We study the reduction of $ϕ$ at primes p ∈ A lying in a prescribed arithmetic progression, for which ϕ(Fₚ) 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 EndK(ϕ)=A, we sharpen the error term and, under additional hypotheses, establish a necessary and sufficient condition for positivity of the resulting density.