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Rational periodic points for quadratic maps

Jung Kyu Canci — 2010

Annales de l’institut Fourier

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 $ℙ_{1}$ 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 ${PGL}_{2} (R_{S})$ , admitting a periodic point in $ℙ_{1} (K)$ of order $> 3$ . Also, all but finitely many classes with a periodic point in $ℙ_{1} (K)$ of order $3$ are parametrized by an irreducible curve.

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