Quadratic rational functions with a rational periodic critical point of period 3
Solomon Vishkautsan
Journal de Théorie des Nombres de Bordeaux, Volume 31 (2019) no. 1, p. 49-79

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.

Nous établissons une classification complète des graphes des points rationnels prépériodiques des fonctions rationnelles de degré 2 ayant un point critique rationnel de période 3 sous les hypothèses suivantes : ces fonctions n’admettent pas de points de période supérieure à 5 et une certaine conjecture sur le nombre de points rationnels sur une courbe affine plane de genre 6 est vraie. Nous montrons qu’il y a exactement six graphes possibles et que les fonctions associées possèdent au plus onze points prépériodiques.

Received : 2018-01-12
Accepted : 2018-06-06
Published online : 2019-07-29
DOI : https://doi.org/10.5802/jtnb.1068
Classification:  37P35,  37P05
Keywords: rational functions, preperiodic points, preperiodicity graphs, moduli curves
@article{JTNB_2019__31_1_49_0,
     author = {Solomon Vishkautsan},
     title = {Quadratic rational functions with a rational periodic critical point of period $3$},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {1},
     year = {2019},
     pages = {49-79},
     doi = {10.5802/jtnb.1068},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_2019__31_1_49_0}
}
Vishkautsan, Solomon. Quadratic rational functions with a rational periodic critical point of period $3$. Journal de Théorie des Nombres de Bordeaux, Volume 31 (2019) no. 1, pp. 49-79. doi : 10.5802/jtnb.1068. jtnb.centre-mersenne.org/item/JTNB_2019__31_1_49_0/

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