Asymptotic cyclicity conjecture for rank-2 Drinfeld modules

Let $L=F_{q^n}$ and let $P$ be the $A$-characteristic of $L$. Set $m=[L,A/P]$ and let $d=\operatorname{deg} P$.

Asymptotic cyclicity conjecture.

$$\lim_{q\rightarrow\infty} C(d,m,q)=\lim_{q\rightarrow\infty} C_0(d,m,q)=1.$$

The quantities $C(d,m,q)$ and $C_0(d,m,q)$ measure cyclicity statistics for rank-2 Drinfeld $A$-modules over $L$. The paper verifies the analogous limits in the cases $md\leq 2$ and proposes the displayed asymptotic statement in general; no resolution is given here.