Rational periodic points for quadratic maps

TL;DR

This paper proves finiteness results for quadratic endomorphisms over number fields with good reduction outside a set S, focusing on the existence and classification of periodic points of order greater than 3.

Contribution

It establishes the finiteness of quadratic maps with certain periodic points over number fields, and parametrizes most with points of order 3 via an irreducible curve.

Findings

Finitely many quadratic maps have periodic points of order >3.

Most maps with a 3-cycle are parametrized by an irreducible curve.

Results depend on good reduction outside a finite set S.

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 $\pro$ 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.