Rational periodic points for quadratic maps
[Points rationnels périodiques pour les endomorphismes quadratiques]
Annales de l'Institut Fourier, Tome 60 (2010) no. 3, pp. 953-985.

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.

DOI : 10.5802/aif.2544
Classification : 11G99, 14G05, 14L30
Keywords: Rational maps, moduli spaces, $S$-unit equations, reduction modulo $\mathfrak{p}$
Mot clés : applications rationnelles, espaces de modules, équations en $S$-unités, réduction modulo $\mathfrak{p}$
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, Tome 60 (2010) no. 3, pp. 953-985. doi : 10.5802/aif.2544. https://aif.centre-mersenne.org/articles/10.5802/aif.2544/

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