Rational periodic points for quadratic maps
[Points rationnels périodiques pour les endomorphismes quadratiques]
Canci, Jung Kyu
Annales de l'Institut Fourier, Tome 60 (2010), p. 953-985 / Harvested from Numdam

Soit K un corps de nombres. Soit S un ensemble fini de places de K contenant toutes les places archimédiennes. Soit R S l’anneau des S-entiers de K. Dans cet article on considère les endomorphismes de degré 2 de la droite projective, définie sur K, avec bonne réduction en dehors de S. On démontre qu’il n’existe qu’un nombre fini de tels endomorphismes, à conjugaison par l’action de PGL 2 (R S ) près, qui admettent un point périodique K-rationnel d’ordre >3. De plus, toutes les classes, sauf un nombre fini, ayant un point périodique K-rationnel d’ordre 3, sont paramétrées par une courbe irréductible.

Let K be a number field. Let S be a finite set of places of K containing all the archimedean ones. Let R S be the ring of S-integers of K. In the present paper we consider endomorphisms of 1 of degree 2, defined over K, with good reduction outside S. We prove that there exist only finitely many such endomorphisms, up to conjugation by PGL 2 (R S ), admitting a periodic point in 1 (K) of order >3. Also, all but finitely many classes with a periodic point in 1 (K) of order 3 are parametrized by an irreducible curve.

Publié le : 2010-01-01
DOI : https://doi.org/10.5802/aif.2544
Classification:  11G99,  14G05,  14L30
Mots clés: applications rationnelles, espaces de modules, équations en S-unités, réduction modulo 𝔭
@article{AIF_2010__60_3_953_0,
     author = {Canci, Jung Kyu},
     title = {Rational periodic points for quadratic maps},
     journal = {Annales de l'Institut Fourier},
     volume = {60},
     year = {2010},
     pages = {953-985},
     doi = {10.5802/aif.2544},
     zbl = {pre05763357},
     mrnumber = {2680821},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2010__60_3_953_0}
}
Canci, Jung Kyu. Rational periodic points for quadratic maps. Annales de l'Institut Fourier, Tome 60 (2010) pp. 953-985. doi : 10.5802/aif.2544. http://gdmltest.u-ga.fr/item/AIF_2010__60_3_953_0/

[1] Benedetto, Robert L. Reduction, dynamics, and Julia sets of rational functions, J. Number Theory, Tome 86 (2001) no. 2, pp. 175-195 | Article | MR 1813109 | Zbl 0978.37039

[2] Birch, B. J.; Merriman, J. R. Finiteness theorems for binary forms with given discriminant, Proc. London Math. Soc. (3), Tome 24 (1972), pp. 385-394 | Article | MR 306119 | Zbl 0248.12002

[3] Bombieri, Enrico; Gubler, Walter Heights in Diophantine Geometry, Cambridge University Press, Cambridge, New Mathematical Monographs (2006) no. 4 | MR 2216774 | Zbl 1115.11034

[4] Canci, Jung Kyu Cycles for rational maps with good reduction outside a prescribed set, Monatsh. Math., Tome 149 (2007) no. 4, pp. 265-287 | Article | MR 2284648 | Zbl 1171.11041

[5] Corvaja, Pietro; Zannier, Umberto A lower bound for the height of a rational function at S-unit points, Monatsh. Math., Tome 144 (2005) no. 3, pp. 203-224 | Article | MR 2130274 | Zbl 1086.11035

[6] Demarco, Laura Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity, Math. Ann., Tome 326 (2003) no. 1, pp. 43-73 | Article | MR 1981611 | Zbl 1032.37029

[7] Evertse, J.-H.; Győry, K. Effective finiteness results for binary forms with given discriminant, Compositio Math., Tome 79 (1991) no. 2, pp. 169-204 | Numdam | MR 1117339 | Zbl 0746.11020

[8] Evertse, Jan-Hendrik On sums of S-units and linear recurrences, Compositio Math., Tome 53 (1984) no. 2, pp. 225-244 | Numdam | MR 766298 | Zbl 0547.10008

[9] Hindry, Marc; Silverman, Joseph H. Diophantine Geometry, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 201 (2000) (An introduction) | MR 1745599 | Zbl 0948.11023

[10] Lang, Serge Algebra, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 211 (2002) | MR 1878556 | Zbl 0984.00001

[11] Marcus, Daniel A. Number fields, Springer-Verlag, New York (1977) (Universitext) | MR 457396 | Zbl 0383.12001

[12] Milnor, John Geometry and dynamics of quadratic rational maps, Experiment. Math., Tome 2 (1993) no. 1, pp. 37-83 (With an appendix by the author and Lei Tan) | MR 1246482 | Zbl 0922.58062

[13] Morton, Patrick; Silverman, Joseph H. Rational periodic points of rational functions, Internat. Math. Res. Notices (1994) no. 2, pp. 97-110 | Article | MR 1264933 | Zbl 0819.11045

[14] Morton, Patrick; Silverman, Joseph H. Periodic points, multiplicities, and dynamical units, J. Reine Angew. Math., Tome 461 (1995), pp. 81-122 | Article | MR 1324210 | Zbl 0813.11059

[15] Van Der Poorten, A. J.; Schlickewei, H. P. The growth condition for recurrence sequences (1982) (Rep. No. 82-0041)

[16] Schmidt, Wolfgang Diophantine Approximation, Springer, Berlin, Lecture Notes in Mathematics, Tome 785 (1980) | MR 568710 | Zbl 0421.10019

[17] Schmidt, Wolfgang M. Diophantine approximations and Diophantine equations, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1467 (1991) | MR 1176315 | Zbl 0754.11020

[18] Serre, Jean-Pierre Lectures on the Mordell-Weil Theorem, Friedr. Vieweg & Sohn, Braunschweig, Aspects of Mathematics (1997) (Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt, with a foreword by Brown and Serre) | MR 1757192 | Zbl 0676.14005

[19] Silverman, Joseph H. The space of rational maps on 1 , Duke Math. J., Tome 94 (1998) no. 1, pp. 41-77 | Article | MR 1635900 | Zbl 0966.14031

[20] Silverman, Joseph H. The arithmetic of dynamical systems, Springer, New York, Graduate Texts in Mathematics, Tome 241 (2007) | MR 2316407 | Zbl 1130.37001