ABC implies a Zsigmondy principle for ramification

TL;DR

This paper establishes a Zsigmondy-type theorem for ramified primes in preimage fields of non-postcritically finite rational functions over number or function fields, assuming the abc-conjecture in the number field case.

Contribution

It proves a new ramification result for preimage fields of rational functions, extending Zsigmondy's theorem to a dynamical setting under the abc-conjecture.

Findings

Existence of new ramified primes in large preimage fields

Ramified primes do not appear in earlier preimage fields

Results depend on the abc-conjecture for number fields

Abstract

Let $K$ be a number field or a function field of characteristic 0. If $K$ is a number field, assume the $abc$-conjecture for $K$. We prove a variant of Zsigmondy's theorem for ramified primes in preimage fields of rational functions in $K(x)$ that are not postcritically finite. For example, suppose $K$ is a number field and $f\in K[x]$ is not postcritically finite, and let $K_n$ be the field generated by the $n$th iterated preimages under $f$ of $\beta\in K$. We show that for all large $n$, there is a prime of $K$ that ramifies in $K_n$ and does not ramify in $K_m$ for any $m<n$.