Abelian extensions in dynamical Galois theory

Andrews, Jesse; Petsche, Clayton

doi:10.2140/ant.2020.14.1981

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Abelian extensions in dynamical Galois theory

Jesse Andrews and Clayton Petsche

Vol. 14 (2020), No. 7, 1981–1999

DOI: 10.2140/ant.2020.14.1981

Abstract

We propose a conjectural characterization of when the dynamical Galois group associated to a polynomial is abelian, and we prove our conjecture in several cases, including the stable quadratic case over $ℚ$ . In the postcritically infinite case, the proof uses algebraic techniques, including a result concerning ramification in towers of cyclic $p$ -extensions. In the postcritically finite case, the proof uses the theory of heights together with results of Amoroso and Zannier and Amoroso and Dvornicich, as well as properties of the Arakelov–Zhang pairing.

Keywords

arithmetic dynamics, dynamical Galois theory, arboreal representations, Weil height, small points, Arakelov–Zhang pairing

Mathematical Subject Classification 2010

Primary: 11R32

Secondary: 11G50, 11R18, 37P30

Milestones

Received: 2 January 2020

Revised: 15 April 2020

Accepted: 23 May 2020

Published: 18 August 2020

Authors

Jesse Andrews
Department of Mathematics and Computer Science Washington College Department of Mathematics and Computer Science Chestertown, MD United States

Clayton Petsche
Department of Mathematics Oregon State University Department of Mathematics Corvallis, OR United States

Vol. 14, No. 7, 2020