On periods modulo <i>p</i> in arithmetic dynamics

Mei-Chu Chang

doi:10.1016/j.crma.2015.01.007

Number theory/Dynamical systems

On periods modulo p in arithmetic dynamics

Presented by: the Editorial Board

Mei-Chu Chang ¹

¹ Department of Mathematics, University of California, Riverside, CA 92521, USA

Comptes Rendus. Mathématique, Volume 353 (2015) no. 4, pp. 283-285

Abstract (Original language)
Résumé

We prove the following analogue of Silverman's results [9] for pairs of maps.

Let $d \geq 2$ be an integer, $K / Q$ a number field, and $N = N_{K / Q} (P)$ the norm of an ideal $P \subset O_{K}$ . Let $h (z) \in K [z]$ be non-constant and not of the form $h (z) = ξ z$ , $ξ^{d - 1} = 1$ . Denote $f_{t} (z) = z^{d} + t$ , $g_{t} (z) = z^{d} + h (t)$ , and $F^{(ℓ)}$ the ℓ-th iteration of F. There are constants $c_{1}$ , $c_{2}$ depending on d and h such that the following holds.

For almost all prime ideals $P \subset O_{K}$ , there is a finite subset $T \subset {\bar{F}}_{P}$ , $| T | \leq c_{1}$ such that if $t \in {\bar{F}}_{P} ∖ T$ at least one of the sets

{f_{t}^{(ℓ)} (0) : ℓ = 1, 2, \dots, [c_{2} \log N]}, {g_{t}^{(ℓ)} (0) : ℓ = 1, 2, \dots, [c_{2} \log N]}

(1)

consists of distinct elements.

Nous prouvons l'analogue suivant des résultats de Silverman [9] pour les paires d'applications.

Soit $d \geq 2$ un entier, $K / Q$ un corps de nombres, et $N = N_{K / Q} (P)$ la norme d'un idéal $P \subset O_{K}$ . Soit $h (z) \in K [z]$ un polynôme non constant qui n'est pas de la forme $h (z) = ξ z$ , $ξ^{d - 1} = 1$ . Posons $f_{t} (z) = z^{d} + t$ , $g_{t} (z) = z^{d} + h (t)$ et $F^{(ℓ)}$ les itérés de F. Il existe des constantes $c_{1}$ , $c_{2}$ , dépendant de d et h, possédant la propriété suivante : pour presque tout idéal premier $P \subset O_{K}$ , il y a un sous-ensemble $T \subset {\bar{F}}_{P}$ , $| T | \leq c_{1}$ tel que si $t \in {\bar{F}}_{P} ∖ T$ , au moins un des ensembles

{f_{t}^{(ℓ)} (0) : ℓ = 1, 2, \dots, [c_{2} \log N]}, {g_{t}^{(ℓ)} (0) : ℓ = 1, 2, \dots, [c_{2} \log N]}

se compose d'éléments distincts.

Received: 2014-08-18
Accepted: 2015-01-20
Published online: 2015-02-02

DOI: 10.1016/j.crma.2015.01.007

Author's affiliations:

Mei-Chu Chang ¹

¹ Department of Mathematics, University of California, Riverside, CA 92521, USA

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     author = {Mei-Chu Chang},
     title = {On periods modulo \protect\emph{p} in arithmetic dynamics},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {283--285},
     year = {2015},
     publisher = {Elsevier},
     volume = {353},
     number = {4},
     doi = {10.1016/j.crma.2015.01.007},
     language = {en},
}

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Mei-Chu Chang. On periods modulo p in arithmetic dynamics. Comptes Rendus. Mathématique, Volume 353 (2015) no. 4, pp. 283-285. doi: 10.1016/j.crma.2015.01.007

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