A Carlitz Module Analogue of a Conjecture of Erdős and Pomerance

Let A = Fq [T ] be the ring of polynomials over the finite field Fq and 0 = a ∈ A. Let C be the A-Carlitz module. For a monic polynomial m ∈ A, let C(A/mA) and ā be the reductions of C and a modulo mA respectively. Let fa(m) be the monic generator of the ideal {f ∈ A,Cf (ā) = 0̄} on C(A/mA). We denote by ω(fa(m)) the number of distinct monic irreducible factors of fa(m). If q = 2 or q = 2 and a = 1, T , or (1 + T ), we prove that there exists a normal distribution for the quantity ω(fa(m))− 12 (log degm) 2 1 √ 3 (log degm)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.