Vol. 14, No. 7, 2020

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Abelian extensions in dynamical Galois theory

Jesse Andrews and Clayton Petsche

Vol. 14 (2020), No. 7, 1981–1999

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 . 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.

arithmetic dynamics, dynamical Galois theory, arboreal representations, Weil height, small points, Arakelov–Zhang pairing
Mathematical Subject Classification 2010
Primary: 11R32
Secondary: 11G50, 11R18, 37P30
Received: 2 January 2020
Revised: 15 April 2020
Accepted: 23 May 2020
Published: 18 August 2020
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