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

Chang, Mei-Chu

doi:10.1016/j.crma.2015.01.007

Number theory/Dynamical systems

On periods modulo p in arithmetic dynamics
[Sur les périodes modulo p des systèmes dynamiques arithmétiques]

Présenté par : the Editorial Board

Chang, Mei-Chu ¹

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

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

Résumé
Abstract

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.

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.

Reçu le : 2014-08-18
Accepté le : 2015-01-20
Publié le : 2015-02-02

DOI : 10.1016/j.crma.2015.01.007

Affiliations des auteurs :

Chang, Mei-Chu ¹

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

@article{CRMATH_2015__353_4_283_0,
     author = {Chang, Mei-Chu},
     title = {On periods modulo \protect\emph{p} in arithmetic dynamics},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {283--285},
     publisher = {Elsevier},
     volume = {353},
     number = {4},
     year = {2015},
     doi = {10.1016/j.crma.2015.01.007},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.crma.2015.01.007/}
}

TY  - JOUR
AU  - Chang, Mei-Chu
TI  - On periods modulo p in arithmetic dynamics
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 283
EP  - 285
VL  - 353
IS  - 4
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.crma.2015.01.007/
DO  - 10.1016/j.crma.2015.01.007
LA  - en
ID  - CRMATH_2015__353_4_283_0
ER  -

%0 Journal Article
%A Chang, Mei-Chu
%T On periods modulo p in arithmetic dynamics
%J Comptes Rendus. Mathématique
%D 2015
%P 283-285
%V 353
%N 4
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.crma.2015.01.007/
%R 10.1016/j.crma.2015.01.007
%G en
%F CRMATH_2015__353_4_283_0

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

Bibliographie
Cité par

[1] Akbary, A.; Ghioca, D. Periods of orbits modulo primes, J. Number Theory, Volume 129 (2009), pp. 2831-2842

[2] Baker, M.; DeMarco, L. Preperiodic points and unlikely intersections, Duke Math. J., Volume 159 (2011), pp. 1-29

[3] Baker, M.; DeMarco, L. Special curves and postcritically-finite polynomials, Forum Math. Pi, Volume 1 (2013) e3 (35 p.)

[4] Berenstein, C.; Yger, A. Effective Bezout identities in $Q [Z 1, \dots, Z n]$ , Acta Math., Volume 166 (1991), pp. 69-120

[5] Chang, M.-C. Elements of large order in prime finite fields, Bull. Aust. Math. Soc., Volume 88 (2013), pp. 169-176

[6] D. Ghioca, H. Krieger, K. Nguyen, A case of the dynamical Andre–Oort conjecture, Preprint.

[7] Ghioca, D.; Hsia, L.-C.; Tucker, T.J. Preperiodic points for families of polynomials, Algebra Number Theory, Volume 7 (2012), pp. 701-732

[8] Ghioca, D.; Hsia, L.-C.; Tucker, T.J. Preperiodic points for families of rational maps, Proc. Lond. Math. Soc. (2015) (in press) | arXiv

[9] Silverman, J.J. Variation of periods modulo p in arithmetic dynamics, N.Y. J. Math., Volume 14 (2008), pp. 601-616

Cité par Sources :