On an Erdős–Pomerance conjecture for rank one Drinfeld modules ☆

Abstract Let k be a global function field of characteristic p which contains a prime divisor of degree one and the field of constants F q . Let ∞ be a fixed place of degree one and A be the ring of elements of k which have only ∞ as a pole. Let ψ be an sgn-normalized rank one Drinfeld A -module defined over O , the integral closure of A in the Hilbert class field of A . We prove an analogue of a conjecture of Erdős and Pomerance for ψ . Given any α ∈ O ∖ { 0 } and an ideal M in O , let f α ( M ) = { f ∈ A | ψ f ( α ) ≡ 0 ( mod M ) } be the ideal in A . We denote by ω ( f α ( M ) ) the number of distinct prime ideal divisors of f α ( M ) . If q ≠ 2 , we prove that there exists a normal distribution for the quantity ω ( f α ( M ) ) − 1 2 ( log ⁡ deg ⁡ M ) 2 1 3 ( log ⁡ deg ⁡ M ) 3 / 2 .