ABC implies primitive prime divisors in arithmetic dynamics

Abstract

Let K be a number field, let ϕ(x) ∈ K(x) be a rational function of degree d > 1, and let α ∈ K be a wandering point such that ϕ n (α) = 0 for all n > 0. We prove that if the abc-conjecture holds for K, then for all but finitely many positive integers n, there is a prime p of K such that v p (ϕ n (α)) > 0 and vp(ϕ m (α)) 0 for all positive integers m < n. Under appropriate ramification hypotheses, we can replace the condition v p (ϕ n (α)) > 0 with the stronger condition vp(ϕ n (α)) = 1. We prove the same result unconditionally for function fields of characteristic 0 when ϕ is not isotrivial.

ABC implies a Zsigmondy principle for ramification

“…The fact that there are infinitely many such primes follows from [Sil93]. Various authors (see [Zsi92,Elk91,Ric07,Kri13,IS09,FG11,GNT13] for example) have sought to show that not only are there infinitely many primes that divide φ n (α) − β for some n, but the stronger statement that there exists an N such that for all n > N, there is a prime that divides φ n (α)−β that does not divide φ m (α)−β for any m < n. If this is true, one might say that there are infinitely many primes dividing φ n (α) − β for some n because after a certain point each "new iterate" φ n (α) − β gives a "new prime" dividing φ n (α) − β. This is sometimes referred as the "Zsigmondy principle", after Zsigmondy [Zsi92] who studied these questions in the context of primitive divisors of a n − b n .…”

“…As in [GNT13], normalizing by the degree of the number field will make it easier to state proofs in the same way for both number fields and function fields.…”

“…t − 1. Then Proposition 5.1 of [GNT13] asserts that if Z ′ (n) is the set of primes p such that min(v p (φ m+t (α)), v p (F (φ n (α)))) > 0, then for any δ > 0, there is a constant C δ such that…”

“…We will make use of the following estimate, sometimes called "Rothabc" as in [GNT13], which holds for number fields conditionally on the abc-conjecture and is true unconditionally for function fields of characteristic 0. The following combines Propositions 3.4 and 4.2 from [GNT13].…”

“…The strategy of our proof combines the approaches of [GNT13] and [BIJ + 15]. We begin with Lemma 3.1, which gives a necessary condition for K n to ramify over p; this is adapted from [BGH + 13].…”