Hostname: page-component-5d59c44645-jb2ch Total loading time: 0 Render date: 2024-02-21T21:33:46.232Z Has data issue: false hasContentIssue false

Rational 6-Cycles Under Iteration of Quadratic Polynomials

Published online by Cambridge University Press:  01 February 2010

Michael Stoll
Department of Mathematics, University of Bayreuth, 95440 Bayreuth, Germany,


Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We present a proof, which is conditional on the Birch and Swinnerton-Dyer Conjecture for a specific abelian variety, that there do not exist rational numbers x and c such that x has exact period N = 6 under the iteration xx2 + c. This extends earlier results by Morton for N = 4 and by Flynn, Poonen and Schaefer for N = 5.

Research Article
Copyright © London Mathematical Society 2008


1.Bosch, S., Lütkebohmert, W. and Raynaud, M., Néron models, Ergeb. Math. Grenzgeb. (3) 21 (Springer, Berlin-Heidelberg, 1990).Google Scholar
2.Bousch, T., ‘Sur quelques problèmes de dynamique holomorphe’, PhD thesis, Université de Paris-Sud, Centre d'Orsay, 1992.Google Scholar
3.Chabauty, C., ‘Sur les points rationnels des courbes algébriques de genre supérieur à l'unité’, C R. Acad. Sci. Paris 212 (1941) 882885.Google Scholar
4.Coleman, R. F., ‘Effective Chabauty’, Duke Math. J. 52 (1985) 765770.Google Scholar
5.Conway, J. H., Curtis, R.T., Norton, S.P., Parker, R. A. and Wilson, R. A., Atlas of finite groups (Oxford University Press, Eynsham, 1985).Google Scholar
6.Dokchitser, T., ‘Computing special values of motivic L-functions’, Experiment. Math. 13 (2004) 137149.Google Scholar
7.Flynn, E. V., Leprévost, F., Schaefer, E. F., Stein, W. A., Stoll, M. and Wetherell, J. L., ‘Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves‘, Math. Comp. 70 (2001) 16751697.Google Scholar
8.Flynn, E. V., ‘An explicit theory of heights’, Trans. Amer. Math. Soc. 347 (1995) 30033015.Google Scholar
9.Flynn, E. V., Poonen, B. and Schaefer, E. F., ‘Cycles of quadratic polynomials and rational points on a genus-2 curve’, Duke Math. J. 90 (1997) 435463.Google Scholar
10.Hulsbergen, W. W. J., Conjectures in arithmetic algebraic geometry, 2nd revised edition (Vieweg & Sohn, Braunschweig-Wiesbaden, 1994).Google Scholar
11.Khovanskii, A. G., ‘Newton polyhedra and the genus of complete intersections’, Fund. Anal. Appl. 12 (1978) 3846. (Translated from Russian.)Google Scholar
12.Lau, E. and Schleicher, D., ‘Internal addresses in the Mandelbrot set and irreducibility of polynomials’, SUNY Stony Brook Institute for Mathematical Sciences, Preprint 1994–19; arXiv:math/9411238v2 [math.DS]Google Scholar
13.MAGMA is described in Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system I: The user language’, J. Symb. Comp. 24 (1997) 235265. (See also the Magma home page at Scholar
14.McCallum, W. and Poonen, B., ‘The method of Chabauty and Coleman’, Preprint, 2007; Scholar
15.Morton, P., ‘On certain algebraic curves related to polynomial maps’, Compositio Math. 103 (1996) 319350.Google Scholar
16.Morton, P., ‘Arithmetic properties of periodic points of quadratic maps, II’, Acta Arith. 87 (1998) 89102.Google Scholar
17.Narkiewicz, W., ‘Polynomial cycles in algebraic number fields’, Colloq. Math. 58 (1989) 151155.Google Scholar
18.Poonen, B., ‘The classification of rational preperiodic points of quadratic polynomials over ℚ: a refined conjecture’, Math. Z. 228 (1998) 1129.Google Scholar
19.Poonen, B., Schaffer, E. F. and Stoll, M., ‘Twists of X(7) and primitive solutions to x 2 + y 3 = z 7’, Duke Math. J. 137 (2007) 103158.Google Scholar
20.Poonen, B. and Stoll, M., ‘The Cassels-Tate pairing on polarized abelian varieties’, Ann. of Math. (2) 150 (1999) 11091149.Google Scholar
21.Silverman, J. H., The arithmetic of dynamical systems, Springer Graduate Texts in Mathematics 241 (Springer, New York, 2007).Google Scholar
22.Stoll, M., ‘Two simple 2-dimensional abelian varieties defined over Q with Mordell-Weil group of rank at least 19’, C. R. Acad. Sci. Paris 321 (1995) 13411345.Google Scholar
23.Stoll, M., ‘On the height constant for curves of genus two’, Acta Arith. 90 (1999) 183201.Google Scholar
24.Stoll, M., ‘On the height constant for curves of genus two, II’, Acta Arith. 104 (2002) 165182.Google Scholar
25.Stoll, M., ‘Independence of rational points on twists of a given curve’, Compositio Math. 142 (2006) 12011214.Google Scholar
26.Stoll, M., MAGMA script accompanying this paper, available at Scholar
27.Tate, J., ‘On the conjectures of Birch and Swinnerton-Dyer and a geometric analog’, Séminaire Bourbaki, vol. 9, exp. no. 306 (Soc. Math. France, Paris, 1995) 415440.Google Scholar
28.Waterhouse, W. C. and Milne, J. S., ‘Abelian varieties over finite fields’, 1969 Number Theory Institute, Stony Brook, NY, 1969, Proc. Sympos. Pure Math. 20 (Amer. Math. Soc, Providence, 1971) 5364.Google Scholar
Supplementary material: File

JCM 11 Stoll Appendix A

Stoll Appendix A

Download JCM 11 Stoll Appendix A(File)
File 24 KB