Integral points for Drinfeld modules

TL;DR

This paper establishes finiteness results for S-integral points in the backward orbits of non-preperiodic points under Drinfeld modules, confirming analogues of classical conjectures in positive characteristic.

Contribution

It proves finiteness of S-integral points in backward orbits and torsion points for Drinfeld modules, addressing questions by Sookdeo and Tucker.

Findings

Finiteness of S-integral points in backward orbits

Finiteness of torsion points S-integral to non-preperiodic points

Confirmation of analogues of Ih's conjecture for Drinfeld modules

Abstract

We prove that in the backward orbit of a non-preperiodic point under the action of a Drinfeld module of generic characteristic there exist at most finitely many points S-integral with respect to another nonpreperiodic point. This provides the answer (in positive characteristic) to a question raised by Sookdeo. We also prove that for each nontorsion point z, there exist at most finitely many torsion points which are S-integral with respect to z. This proves a question raised by Tucker and the author, and it gives the analogue of Ih's conjecture for Drinfeld modules.