On an Erdos-Pomerance conjecture for rank one Drinfeld modules

摘要

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 no be a fixed place of degree one and A be the ring of elements of k which have only infinity as a pole. Let psi 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 Erd8s and Pomerance for psi. Given any alpha is an element of O{0} and an ideal m in O, let f(alpha)(m) = {f is an element of A vertical bar psi(f) (alpha) equivalent to 0 (mod m)} be the ideal in A. We denote by w(f(alpha) (m) ) the number of distinct prime ideal divisors of f(alpha) (m). If q not equal 2, we prove that there exists a normal distribution for the quantity