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 Fq. 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 fα(M)={f∈A|ψf(α)≡0(modM)} be the ideal in A. We denote by ω(fα(M)) the number of distinct prime ideal divisors of fα(M). If q≠2, we prove that there exists a normal distribution for the quantityω(fα(M))−12(log⁡deg⁡M)213(log⁡deg⁡M)3/2.