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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 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.

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