The $abcd$ conjecture, uniform boundedness, and dynamical systems

TL;DR

This paper surveys Vojta's generalizations of the $abc$ conjecture and explores their implications in arithmetic dynamics, including uniform boundedness of torsion points and lower bounds for canonical heights.

Contribution

It connects the $abcd$ conjecture to key problems in arithmetic dynamics, highlighting new implications and conjectural analogues.

Findings

The $abcd$ conjecture implies a dynamical uniform boundedness conjecture.

It suggests a dynamical analogue of Lang's conjecture on canonical heights.

Recent developments apply these conjectures to various problems in arithmetic dynamics.

Abstract

We survey Vojta's higher-dimensional generalizations of the $abc$ conjecture and Szpiro's conjecture as well as recent developments that apply them to various problems in arithmetic dynamics. In particular, the "$abcd$ conjecture" implies a dynamical analogue of a conjecture on the uniform boundedness of torsion points and a dynamical analogue of Lang's conjecture on lower bounds for canonical heights.