A Carlitz module analogue of a conjecture of Erdos and Pomerance

Abstract:

Let $A=\mathbb {F}_q[T]$ be the ring of polynomials over the finite field $\mathbb {F}_q$ and $0 \neq a \in A$. Let $C$ be the $A$-Carlitz module. For a monic polynomial $m\in A$, let $C(A/mA)$ and $\bar {a}$ be the reductions of $C$ and $a$ modulo $mA$ respectively. Let $f_a(m)$ be the monic generator of the ideal $\{f\in A, C_f(\bar {a}) =\bar {0}\}$ on $C(A/mA)$. We denote by $\omega (f_a(m))$ the number of distinct monic irreducible factors of $f_a(m)$. If $q\neq 2$ or $q=2$ and $a\neq 1, T$, or $(1+T)$, we prove that there exists a normal distribution for the quantity \[ \frac {\omega (f_a(m))-\frac {1}{2}(\log \deg m)^2}{\frac {1}{\sqrt {3}}{(\log \deg m)^{3/2}}}.\] This result is analogous to an open conjecture of Erdős and Pomerance concerning the distribution of the number of distinct prime divisors of the multiplicative order of $b$ modulo $n$, where $b$ is an integer with $|b|>1$, and $n$ a positive integer.