Comptes Rendus
Number theory/Dynamical systems
On periods modulo p in arithmetic dynamics
Comptes Rendus. Mathématique, Volume 353 (2015) no. 4, pp. 283-285.

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

Let d2 be an integer, K/Q a number field, and N=NK/Q(P) the norm of an ideal POK. Let h(z)K[z] be non-constant and not of the form h(z)=ξz, ξd1=1. Denote ft(z)=zd+t, gt(z)=zd+h(t), and F() the -th iteration of F. There are constants c1, c2 depending on d and h such that the following holds.

For almost all prime ideals POK, there is a finite subset TF¯P, |T|c1 such that if tF¯PT at least one of the sets

{ft()(0):=1,2,,[c2logN]},{gt()(0):=1,2,,[c2logN]}(1)
consists of distinct elements.

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

Soit d2 un entier, K/Q un corps de nombres, et N=NK/Q(P) la norme d'un idéal POK. Soit h(z)K[z] un polynôme non constant qui n'est pas de la forme h(z)=ξz, ξd1=1. Posons ft(z)=zd+t, gt(z)=zd+h(t) et F() les itérés de F. Il existe des constantes c1, c2, dépendant de d et h, possédant la propriété suivante : pour presque tout idéal premier POK, il y a un sous-ensemble TF¯P, |T|c1 tel que si tF¯PT, au moins un des ensembles

{ft()(0):=1,2,,[c2logN]},{gt()(0):=1,2,,[c2logN]}
se compose d'éléments distincts.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2015.01.007

Mei-Chu Chang 1

1 Department of Mathematics, University of California, Riverside, CA 92521, USA
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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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.01.007/

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

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

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

[4] C. Berenstein; A. Yger Effective Bezout identities in Q[Z1,,Zn], Acta Math., Volume 166 (1991), pp. 69-120

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

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

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

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