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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Average Zsigmondy sets, dynamical Galois groups, and the Kodaira-Spencer map


Author: Wade Hindes
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 37P15, 11R32; Secondary 11B37, 14G05, 11G99
DOI: https://doi.org/10.1090/tran/7125
Published electronically: March 20, 2018
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Abstract: Let $ K$ be a global function field and let $ \phi (x)\in K[x]$. For all wandering basepoints $ b\in K$, we show that there is a bound on the size of the elements of the dynamical Zsigmondy set $ \mathcal {Z}(\phi ,b)$ that depends only on $ \phi $, the poles of the $ b$, and $ K$. Moreover, when we order $ b\in \mathcal {O}_{K,S}$ by height, we show that $ \mathcal {Z}(\phi ,b)$ is empty on average. As an application, we prove that the inverse limit of the Galois groups of iterates of $ \phi (x)=x^d+f$ is a finite index subgroup of an iterated wreath product of cyclic groups. In particular, since our methods translate to rational function fields in characteristic zero, we establish the inverse Galois problem for these groups via specialization.


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Additional Information

Wade Hindes
Affiliation: Department of Mathematics, CUNY Graduate Center, 365 Fifth Avenue, New York, New York 10016-4309
Email: whindes@gc.cuny.edu

DOI: https://doi.org/10.1090/tran/7125
Keywords: Arithmetic dynamics, rational points on curves, Galois theory
Received by editor(s): May 11, 2016
Received by editor(s) in revised form: June 18, 2016, and November 15, 2016
Published electronically: March 20, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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