Rational periodic points for quadratic maps

Canci, Jung Kyu

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

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