A Carlitz module analogue of a conjecture of Erdos and Pomerance

Abstract

Abstract. 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 ¯a be the reductions of C and a modulo mA respectively. Let fa(m) be the monic generator of the ideal {f ∈ A, Cf (¯a) = ¯0} 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)) − 1 2 (log deg m)2 √1 3 (log deg m)3/2 . This result is analogous to an open conjecture of Erd˝os 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.