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.

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:
Accepted:
Published online:
DOI: 10.5802/jtnb.1068
Classification: 37P35, 37P05
Keywords: rational functions, preperiodic points, preperiodicity graphs, moduli curves
Vishkautsan, Solomon 1

1 Tel-Hai Academic College, Israel
@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
SP  - 49
EP  - 79
VL  - 31
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - http://archive.numdam.org/articles/10.5802/jtnb.1068/
DO  - 10.5802/jtnb.1068
LA  - en
ID  - JTNB_2019__31_1_49_0
ER  - 
%0 Journal Article
%A Vishkautsan, Solomon
%T Quadratic rational functions with a rational periodic critical point of period $3$
%J Journal de théorie des nombres de Bordeaux
%D 2019
%P 49-79
%V 31
%N 1
%I Société Arithmétique de Bordeaux
%U http://archive.numdam.org/articles/10.5802/jtnb.1068/
%R 10.5802/jtnb.1068
%G en
%F 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. http://archive.numdam.org/articles/10.5802/jtnb.1068/

[1] Bosma, Wieb; Cannon, John; Playoust, Catherine The Magma algebra system. I. The user language, J. Symb. Comput., Volume 24 (1997) no. 3-4, pp. 235-265 | DOI | MR | Zbl

[2] Bruin, Nils; Stoll, Michael The Mordell–Weil sieve: proving non-existence of rational points on curves, LMS J. Comput. Math., Volume 13 (2010), pp. 272-306 | DOI | MR | Zbl

[3] Canci, Jung Kyu; Vishkautsan, Solomon Quadratic maps with a periodic critical point of period 2, Int. J. Number Theory, Volume 13 (2017) no. 6, pp. 1393-1417 | DOI | MR | Zbl

[4] Cremona, John The elliptic curve database for conductors to 130000, Algorithmic number theory (Lecture Notes in Computer Science), Volume 4076, Springer, 2006, pp. 11-29 | DOI | MR | Zbl

[5] Doyle, John R.; Faber, Xander; Krumm, David Preperiodic points for quadratic polynomials over quadratic fields, New York J. Math., Volume 20 (2014), pp. 507-605 | MR | Zbl

[6] Fakhruddin, Najmuddin Questions on self maps of algebraic varieties, J. Ramanujan Math. Soc., Volume 18 (2003) no. 2, pp. 109-122 | MR | Zbl

[7] Flynn, Eugene V.; Poonen, Bjorn; Schaefer, Edward F. Cycles of quadratic polynomials and rational points on a genus-2 curve, Duke Math. J., Volume 90 (1997) no. 3, pp. 435-463 | DOI | MR | Zbl

[8] Hutz, Benjamin; Ingram, Patrick On Poonen’s conjecture concerning rational preperiodic points of quadratic maps, Rocky Mt. J. Math., Volume 43 (2013) no. 1, pp. 193-204 | DOI | MR | Zbl

[9] Lukas, David; Manes, Michelle; Yap, Diane A census of quadratic post-critically finite rational functions defined over , LMS J. Comput. Math., Volume 17A (2014), pp. 314-329 | DOI | MR | Zbl

[10] Manes, Michelle -rational cycles for degree-2 rational maps having an automorphism, Proc. Lond. Math. Soc., Volume 96 (2008) no. 3, pp. 669-696 | DOI | MR | Zbl

[11] Milnor, John Geometry and dynamics of quadratic rational maps, Exp. Math., Volume 2 (1993) no. 1, pp. 37-83 | MR | Zbl

[12] Morton, Patrick Rational periodic points of rational functions, Int. Math. Res. Not., Volume 1994 (1994) no. 2, pp. 97-110 | MR | Zbl

[13] Morton, Patrick Periodic points, multiplicities, and dynamical units, J. Reine Angew. Math., Volume 461 (1995), pp. 81-122 | MR | Zbl

[14] Morton, Patrick Arithmetic properties of periodic points of quadratic maps. II, Acta Arith., Volume 87 (1998) no. 2, pp. 89-102 | MR | Zbl

[15] Northcott, D. G. Periodic points on an algebraic variety, Ann. Math., Volume 51 (1950), pp. 167-177 | DOI | MR | Zbl

[16] Poonen, Bjorn The classification of rational preperiodic points of quadratic polynomials over : a refined conjecture, Math. Z., Volume 228 (1998) no. 1, pp. 11-29 | DOI | MR | Zbl

[17] Silverman, Joseph H. The arithmetic of dynamical systems, Graduate Texts in Mathematics, 241, European Mathematical Society, 2007 | MR | Zbl

[18] Stoll, Michael Rational 6-cycles under iteration of quadratic polynomials, LMS J. Comput. Math., Volume 11 (2008), pp. 367-380 | DOI | MR | Zbl

[19] Bosma, Wieb; Cannon, John; Playoust, Catherine The Magma algebra system. I. The user language, J. Symb. Comput., Volume 24 (1997) no. 3-4, pp. 235-265 | DOI | MR | Zbl

[20] Stoll, Michael Independence of rational points on twists of a given curve, Compos. Math., Volume 142 (2006) no. 5, pp. 1201-1214 | DOI | MR | Zbl

[21] Stoll, Michael Rational 6-cycles under iteration of quadratic polynomials, LMS J. Comput. Math., Volume 11 (2008), pp. 367-380 | DOI | MR | Zbl

Cited by Sources: