Rational periodic points for quadratic maps
Annales de l'Institut Fourier, Volume 60 (2010) no. 3, p. 953-985

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.

Soit K un corps de nombres. Soit S un ensemble fini de places de K contenant toutes les places archimédiennes. Soit R S l’anneau des S-entiers de K. Dans cet article on considère les endomorphismes de degré 2 de la droite projective, définie sur K, avec bonne réduction en dehors de S. On démontre qu’il n’existe qu’un nombre fini de tels endomorphismes, à conjugaison par l’action de PGL 2 (R S ) près, qui admettent un point périodique K-rationnel d’ordre >3. De plus, toutes les classes, sauf un nombre fini, ayant un point périodique K-rationnel d’ordre 3, sont paramétrées par une courbe irréductible.

DOI : https://doi.org/10.5802/aif.2544
Classification:  11G99,  14G05,  14L30
Keywords: Rational maps, moduli spaces, S-unit equations, reduction modulo 𝔭
@article{AIF_2010__60_3_953_0,
     author = {Canci, Jung Kyu},
     title = {Rational periodic points for quadratic maps},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {60},
     number = {3},
     year = {2010},
     pages = {953-985},
     doi = {10.5802/aif.2544},
     mrnumber = {2680821},
     zbl = {pre05763357},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2010__60_3_953_0}
}
Canci, Jung Kyu. Rational periodic points for quadratic maps. Annales de l'Institut Fourier, Volume 60 (2010) no. 3, pp. 953-985. doi : 10.5802/aif.2544. http://www.numdam.org/item/AIF_2010__60_3_953_0/

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