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.

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.

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.

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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 = {Solomon Vishkautsan},
     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 = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1068/}
}
Solomon Vishkautsan. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1068/

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