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.
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.
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, Tome 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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