A Carlitz module analogue of a conjecture of Erdos and Pomerance

Abstract

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