Let be a number field. Let be a finite set of places of containing all the archimedean ones. Let be the ring of -integers of . In the present paper we consider endomorphisms of of degree , defined over , with good reduction outside . We prove that there exist only finitely many such endomorphisms, up to conjugation by , admitting a periodic point in of order . Also, all but finitely many classes with a periodic point in of order are parametrized by an irreducible curve.
Soit un corps de nombres. Soit un ensemble fini de places de contenant toutes les places archimédiennes. Soit l’anneau des -entiers de . Dans cet article on considère les endomorphismes de degré de la droite projective, définie sur , avec bonne réduction en dehors de . On démontre qu’il n’existe qu’un nombre fini de tels endomorphismes, à conjugaison par l’action de près, qui admettent un point périodique -rationnel d’ordre . De plus, toutes les classes, sauf un nombre fini, ayant un point périodique -rationnel d’ordre , sont paramétrées par une courbe irréductible.
Keywords: Rational maps, moduli spaces, $S$-unit equations, reduction modulo $\mathfrak{p}$
Mot clés : applications rationnelles, espaces de modules, équations en $S$-unités, réduction modulo $\mathfrak{p}$
@article{AIF_2010__60_3_953_0, author = {Canci, Jung Kyu}, title = {Rational periodic points for quadratic maps}, journal = {Annales de l'Institut Fourier}, pages = {953--985}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {60}, number = {3}, year = {2010}, doi = {10.5802/aif.2544}, mrnumber = {2680821}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2544/} }
TY - JOUR AU - Canci, Jung Kyu TI - Rational periodic points for quadratic maps JO - Annales de l'Institut Fourier PY - 2010 SP - 953 EP - 985 VL - 60 IS - 3 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2544/ DO - 10.5802/aif.2544 LA - en ID - AIF_2010__60_3_953_0 ER -
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/articles/10.5802/aif.2544/
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