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

Robin Zhang

doi:10.5802/pmb.58

The

a b c d

conjecture, uniform boundedness, and dynamical systems

Robin Zhang ^1,²

¹ Department of Mathematics, Columbia University, USA
² Department of Mathematics, Massachusetts Institute of Technology, USA

Publications mathématiques de Besançon. Algèbre et théorie des nombres (2024), pp. 119-134.

Abstract
Résumé

We survey Vojta’s higher-dimensional generalizations of the $a b c$ conjecture and Szpiro’s conjecture as well as recent developments that apply them to various problems in arithmetic dynamics. In particular, the “ $a b c d$ 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.

Nous décrivons des généralisations en dimension supérieure dues à Vojta de la conjecture $a b c$ et de la conjecture de Szpiro, ainsi que des avancées récentes qui les utilisent dans des problèmes variés de dynamique arithmétique. En particulier, la « conjecture $a b c d$ » implique un analogue dynamique de la conjecture de torsion et un analogue dynamique de la conjecture de Lang sur les minorations de hauteurs canoniques.

Received: 2021-12-31
Revised: 2022-06-10
Published online: 2024-04-22

DOI: 10.5802/pmb.58

Author's affiliations:

Robin Zhang ^{1, 2}

¹ Department of Mathematics, Columbia University, USA
² Department of Mathematics, Massachusetts Institute of Technology, USA

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Robin Zhang. The $abcd$ conjecture, uniform boundedness, and dynamical systems. Publications mathématiques de Besançon. Algèbre et théorie des nombres (2024), pp. 119-134. doi : 10.5802/pmb.58. https://pmb.centre-mersenne.org/articles/10.5802/pmb.58/

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