# Rational periodic points for quadratic maps

Jung Kyu Canci^{[1]}

- [1] Université Lille 1 Laboratoire Paul Painlevé, Mathématiques 59655 Villeneuve d’Ascq Cedex (France)

Annales de l’institut Fourier (2010)

- Volume: 60, Issue: 3, page 953-985
- ISSN: 0373-0956

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topCanci, Jung Kyu. "Rational periodic points for quadratic maps." Annales de l’institut Fourier 60.3 (2010): 953-985. <http://eudml.org/doc/116297>.

@article{Canci2010,

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

affiliation = {Université Lille 1 Laboratoire Paul Painlevé, Mathématiques 59655 Villeneuve d’Ascq Cedex (France)},

author = {Canci, Jung Kyu},

journal = {Annales de l’institut Fourier},

keywords = {Rational maps; moduli spaces; $S$-unit equations; reduction modulo $\mathfrak\{p\}$; rational maps; -unit equations; reduction modulo },

language = {eng},

number = {3},

pages = {953-985},

publisher = {Association des Annales de l’institut Fourier},

title = {Rational periodic points for quadratic maps},

url = {http://eudml.org/doc/116297},

volume = {60},

year = {2010},

}

TY - JOUR

AU - Canci, Jung Kyu

TI - Rational periodic points for quadratic maps

JO - Annales de l’institut Fourier

PY - 2010

PB - Association des Annales de l’institut Fourier

VL - 60

IS - 3

SP - 953

EP - 985

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

LA - eng

KW - Rational maps; moduli spaces; $S$-unit equations; reduction modulo $\mathfrak{p}$; rational maps; -unit equations; reduction modulo

UR - http://eudml.org/doc/116297

ER -

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