Integral points in orbits in characteristic $p$

TL;DR

This paper extends Silverman's theorem to characteristic p, proving results about integral points in polynomial orbits over function fields, and explores applications like arboreal representations.

Contribution

It introduces a characteristic p analogue of Silverman's theorem and establishes a primitive prime divisor theorem in this setting, with applications to arboreal representations.

Findings

Proved a characteristic p version of Silverman's theorem.

Established a primitive prime divisor theorem for polynomials in characteristic p.

Applied results to finite index theorems for arboreal representations.

Abstract

We prove a characteristic $p$ version of a theorem of Silverman on integral points in orbits over number fields and establish a primitive prime divisor theorem for polynomials in this setting. We provide some applications of these results, including a finite index theorem for arboreal representations coming from quadratic polynomials over function fields of odd characteristic.