Rational periodic points for quadratic maps
Annales de l'Institut Fourier, Volume 60 (2010) no. 3, pp. 953-985.

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.

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.

DOI: 10.5802/aif.2544
Classification: 11G99,  14G05,  14L30
Keywords: Rational maps, moduli spaces, S-unit equations, reduction modulo 𝔭
Canci, Jung Kyu 1

1 Université Lille 1 Laboratoire Paul Painlevé, Mathématiques 59655 Villeneuve d’Ascq Cedex (France)
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Canci, Jung Kyu. Rational periodic points for quadratic maps. Annales de l'Institut Fourier, Volume 60 (2010) no. 3, pp. 953-985. doi : 10.5802/aif.2544. https://aif.centre-mersenne.org/articles/10.5802/aif.2544/

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

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

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

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

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

[6] DeMarco, Laura Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity, Math. Ann., Volume 326 (2003) no. 1, pp. 43-73 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

[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, Lecture Notes in Mathematics, 785, Springer, Berlin, 1980 | MR | Zbl

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

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

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

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

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