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Isotriviality, integral points, and primitive primes in orbits in characteristic $p$

Alexander Carney, Wade Hindes and Thomas J. Tucker

Vol. 17 (2023), No. 9, 1573–1594
DOI: 10.2140/ant.2023.17.1573
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. In characteristic p, the Thue–Siegel–Dyson–Roth theorem is false, so the proof requires new techniques from those used by Silverman. The problem is largely that isotriviality can arise in subtle ways, and we define and compare three different definitions of isotriviality for maps, sets, and curves. Using results of Favre and Rivera-Letelier on the structure of Julia sets, we prove that if φ is a nonisotrivial rational function and β is not exceptional for φ, then φn(β) is a nonisotrivial set for all sufficiently large n; we then apply diophantine results of Voloch and Wang that apply for all nonisotrivial sets. When φ is a polynomial, we use the nonisotriviality of φn(β) for large n along with a partial converse to a result of Grothendieck in descent theory to deduce the nonisotriviality of the curve y = φn(x) β for large n and small primes p whenever β is not postcritical; this enables us to prove stronger results on Zsigmondy sets. 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.

Keywords
arithmetic dynamics, integral points, arboreal representations, Zsigmondy sets
Mathematical Subject Classification
Primary: 37P15
Secondary: 11G50, 11R32, 14G25, 37P05, 37P30
Milestones
Received: 31 December 2021
Revised: 2 September 2022
Accepted: 4 October 2022
Published: 9 September 2023
Authors
Alexander Carney
Department of Mathematics
University of Rochester
Rochester, NY
United States
Wade Hindes
Department of Mathematics
Texas State University
San Marcos, TX
United States
Thomas J. Tucker
Department of Mathematics
University of Rochester
Rochester, NY
United States

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