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

Résumé

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 ${\rm PGL}_2(R_S)$ 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.

Abstract

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 $\mathbb{P}_1$ 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 ${\rm PGL}_2(R_S)$, admitting a periodic point in $\mathbb{P}_1(K)$ of order $>3$. Also, all but finitely many classes with a periodic point in $\mathbb{P}_1(K)$ of order $3$ are parametrized by an irreducible curve.

Détail

Publié le : 2010-01-01

MR 2680821 | Zbl pre05763357

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 $\mathfrak{p}$

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