A finiteness theorem for specializations of dynatomic polynomials

Abstract

Let t and x be indeterminates, let φ(x) = x 2 + t ∈ Q(t) [x], and for every positive integer n let Φn(t, x) denote the n th dynatomic polynomial of φ. Let Gn be the Galois group of Φn over the function field Q(t), and for c ∈ Q let Gn,c be the Galois group of the specialized polynomial Φn(c, x). It follows from Hilbert's irreducibility theorem that for fixed n we have Gn ∼ = Gn,c for every c outside a thin set En ⊂ Q. By earlier work of Morton (for n = 3) and the present author (for n = 4), it is known that En is infinite if n ≤ 4. In contrast, we show here that En is finite if n ∈ {5, 6, 7, 9}. As an application of this result we show that, for these values of n, the following holds with at most finitely many exceptions: for every c ∈ Q, more than 81% of prime numbers p have the property that the polynomial x 2 + c does not have a point of period n in the p-adic field Qp.