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Abelian extensions in
dynamical Galois theory
Jesse Andrews and Clayton Petsche |
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Vol. 14 (2020), No. 7, 1981–1999
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Abstract |
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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 -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. |
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Keywords
arithmetic dynamics, dynamical Galois theory, arboreal
representations, Weil height, small points, Arakelov–Zhang
pairing
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Mathematical Subject Classification 2010
Primary: 11R32
Secondary: 11G50, 11R18, 37P30
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Milestones
Received: 2 January 2020
Revised: 15 April 2020
Accepted: 23 May 2020
Published: 18 August 2020
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Authors |
| Jesse Andrews | |
| Department of Mathematics and
Computer Science Washington College Department of Mathematics and Computer Science Chestertown, MD United States |
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| Clayton Petsche | |
| Department of Mathematics Oregon State University Department of Mathematics Corvallis, OR United States |
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