Rational periodic points for quadratic maps

Details Rational periodic points for quadratic maps Rational periodic points for quadratic maps

2008-01-30

arxiv.org/abs/0801.4636v3

Mathematics

Number Theory

32 pages. To appear on Annales de l'Insitut Fourier. Corrected some mistakes in the proofs of Lemma 6 and Lemma 8. Thanks to t

Scientific paper

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 $\pro$ 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 $\po$ of order $>3$. Also, all but finitely many classes with a periodic point in $\po$ of order 3 are parametrized by an irreducible curve.

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