Nous établissons une classification complète des graphes des points rationnels prépériodiques des fonctions rationnelles de degré ayant un point critique rationnel de période sous les hypothèses suivantes : ces fonctions n’admettent pas de points de période supérieure à et une certaine conjecture sur le nombre de points rationnels sur une courbe affine plane de genre est vraie. Nous montrons qu’il y a exactement six graphes possibles et que les fonctions associées possèdent au plus onze points prépériodiques.
We provide a complete classification of possible graphs of rational preperiodic points of quadratic rational functions defined over the rationals with a rational periodic critical point of period 3, under two assumptions: that these functions have no periodic points of period at least 5 and the conjectured enumeration of rational points on a certain genus 6 affine plane curve. We show that there are exactly six such possible graphs, and that rational functions satisfying the conditions above have at most eleven rational preperiodic points.
Accepté le :
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DOI : https://doi.org/10.5802/jtnb.1068
Classification : 37P35, 37P05
Mots clés : rational functions, preperiodic points, preperiodicity graphs, moduli curves
@article{JTNB_2019__31_1_49_0,
author = {Vishkautsan, Solomon},
title = {Quadratic rational functions with a rational periodic critical point of period $3$},
journal = {Journal de Th\'eorie des Nombres de Bordeaux},
pages = {49--79},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {31},
number = {1},
year = {2019},
doi = {10.5802/jtnb.1068},
mrnumber = {3994719},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/jtnb.1068/}
}
TY - JOUR AU - Vishkautsan, Solomon TI - Quadratic rational functions with a rational periodic critical point of period $3$ JO - Journal de Théorie des Nombres de Bordeaux PY - 2019 DA - 2019/// SP - 49 EP - 79 VL - 31 IS - 1 PB - Société Arithmétique de Bordeaux UR - http://archive.numdam.org/articles/10.5802/jtnb.1068/ UR - https://www.ams.org/mathscinet-getitem?mr=3994719 UR - https://doi.org/10.5802/jtnb.1068 DO - 10.5802/jtnb.1068 LA - en ID - JTNB_2019__31_1_49_0 ER -
Vishkautsan, Solomon. Quadratic rational functions with a rational periodic critical point of period $3$. Journal de Théorie des Nombres de Bordeaux, Tome 31 (2019) no. 1, pp. 49-79. doi : 10.5802/jtnb.1068. http://archive.numdam.org/articles/10.5802/jtnb.1068/
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