On periods modulo \(p\) in arithmetic dynamics
type: Item
instance of: scholarly article
title: On periods modulo \(p\) in arithmetic dynamics
zbMATH Open document ID: 1323.37055
author: Mei-Chu Chang
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2015-04-13T00:00:00Z
review text: This work deals with the problem of simultaneous pre-periodicity in parameterised families of algebraic dynamical systems. The author considers two parametrised families of polynomial maps over an algebraic number field, and investigates the size modulo \(p\) of the orbit of their critical point. The polynomials have a specific form: \(z^d+t\) and \(z^d+h(t)\), where \(d\geqslant 2\), \(t\) is the parameter, and \(h\) is an arbitrary polynomial -- as long as it is not linear and of finite order. One then considers the local dynamics in the (algebraic closure of the) residue field of a prime ideal of good reduction, allowing the parameter to range in that field. The result is a lower bound for the size of the critical orbit of at least one of the two maps. The author shows that there are two constants \(c_1\) and \(c_2\) depending only on \(d\) and \(h\), such that for almost all prime ideals (in the sense of density) and all but \(c_1\) parameter values, the critical orbit of at least one of the two maps consists of at least \(c_2\log N\) points, where \(N\) is the absolute norm of the ideal. This result is related to earlier work of \textit{J. H. Silverman} [New York J. Math. 14, 601--616 (2008; Zbl 1153.11028)], of \textit{A. Akbary} and \textit{D. Ghioca} [J. Number Theory 129, No. 11, 2831--2842 (2009; Zbl 1254.37065)], and of \textit{D. Ghioca}, \textit{H. Krieger} and \textit{K. Nguyen} [``A case of the dynamical Andre-Oort conjecture'', Preprint, \url{arXiv:1404.6011}].
Mathematics Subject Classification ID: 37P05
Mathematics Subject Classification ID: 11G35
zbMATH DE Number: 6424577
zbMATH Keywords: arithmetic dynamics
zbMATH Keywords: pre-periodic points
zbMATH Keywords: good reduction
reviewed by: Franco Vivaldi
MaRDI profile type: Publication
full work available at URL: j.crma.2015.01.007
OpenAlex ID: W1966541947
cites work: Periods of orbits modulo primes
cites work: Preperiodic points and unlikely intersections
cites work: Special curves and postcritically finite polynomials
cites work: Effective Bezout identities in \({\mathbb{Q}}[z_ 1,\dots ,z_ n]\)
cites work: Elements of large order in prime finite fields
cites work: Preperiodic points for families of polynomials
cites work: Preperiodic points for families of rational maps
cites work: Variation of periods modulo \(p\) in arithmetic dynamics
DOI: 10.1016/J.CRMA.2015.01.007
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description: scientific article; zbMATH DE number 6424577