The \(abcd\) conjecture, uniform boundedness, and dynamical systems

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.