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

Abstract

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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.

How to cite

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Canci, 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 $&gt;3$. 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 $&gt;3$. 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 -

References

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