Rational periodic sequences for the Lyness recurrence

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February 2012, 32(2): 587-604. doi: 10.3934/dcds.2012.32.587

Rational periodic sequences for the Lyness recurrence

A. Gasull ^1, , Víctor Mañosa ^2, and Xavier Xarles ^3,

1.	Dept. de Matemµatiques, Universitat Autónoma de Barcelona, Edifici C, 08193-Bellaterra, Barcelona, Spain
2.	Dept. de Matemàtica Aplicada III (MA3), Control, Dynamics and Applications Group (CoDALab), Universitat Politècnica de Catalunya (UPC), Colom 1, 08222 Terrassa, Spain
3.	Dept. de Matemàtiques, Universitat Autònoma de Barcelona, Ediﬁci C, 08193 Bellaterra, Barcelona, Spain

Received September 2010 Revised December 2010 Published September 2011

Abstract
Related Papers

Consider the celebrated Lyness recurrence $ x_{n+2}=(a+x_{n+1})/x_{n}$ with $a\in\mathbb{Q}$. First we prove that there exist initial conditions and values of $a$ for which it generates periodic sequences of rational numbers with prime periods $1,2,3,5,6,7,8,9,10$ or $12$ and that these are the only periods that rational sequences $\{x_n\}_n$ can have. It is known that if we restrict our attention to positive rational values of $a$ and positive rational initial conditions the only possible periods are $1,5$ and $9$. Moreover 1-periodic and 5-periodic sequences are easily obtained. We prove that for infinitely many positive values of $a,$ positive 9-period rational sequences occur. This last result is our main contribution and answers an open question left in previous works of Bastien & Rogalski and Zeeman. We also prove that the level sets of the invariant associated to the Lyness map is a two-parameter family of elliptic curves that is a universal family of the elliptic curves with a point of order $n, n\ge5,$ including $n$ infinity. This fact implies that the Lyness map is a universal normal form for most birational maps on elliptic curves.

Keywords: rational points over elliptic curves, Lyness difference equations, universal family of elliptic curves., periodic points.

Mathematics Subject Classification: Primary: 39A20; Secondary: 39A11, 14H5.

Citation: A. Gasull, Víctor Mañosa, Xavier Xarles. Rational periodic sequences for the Lyness recurrence. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 587-604. doi: 10.3934/dcds.2012.32.587

References:

[1]	A. O. L. Atkin and F. Morain, Finding suitable curves for the elliptic curve method of factorization, Math. Comp., 60 (1993), 399-405. doi: 10.1090/S0025-5718-1993-1140645-1.
[2]	E. Barbeau, B. Gelbord and S. Tanny, Periodicities of solutions of the generalized Lyness recursion, J. Difference Equations Appl., 1 (1995), 291-306. doi: 10.1080/10236199508808028.
[3]	G. Bastien and M. Rogalski, Global behavior of the solutions of Lyness' difference equation $u_{n+2}u_n=u_{n+1}+a$, J. Difference Equations Appl., 10 (2004), 977-1003. doi: 10.1080/10236190410001728104.
[4]	A. Beauville, Les familles stables de courbes elliptiques sur P1 admettant quatre fibres singulières, C. R. Acad. Sci. Paris, Série I Math., 294 (1982), 657-660.
[5]	F. Beukers and R. Cushman, Zeeman's monotonicity conjecture, J. Differential Equations, 143 (1998), 191-200. doi: 10.1006/jdeq.1997.3359.
[6]	W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125.
[7]	H. Cohen, "Number Theory. Volume I: Tools and Diophantine Equations," Graduate Texts in Mathematics, 239, Springer, New York, 2007.
[8]	J. E. Cremona, "Elliptic Curve Data," Web page maintained by W. Stein, University of Warwick., Available from: \url{http://www.warwick.ac.uk/staff/J.E.Cremona//ftp/data/}., ().
[9]	A. Dujella, "Elliptic Curve Tables," Web page maintained by author, University of Zagreb., Available from: \url{http://web.math.hr/~duje/}., ().
[10]	N. Elkies, Rational points near curves and small nonzero $\|x^3 - y^2\|$ via lattice reduction, in "Algorithmic Number Theory" (Leiden, 2000), 33-63. Lecutre Notes in Comput. Sci., 1838, Springer, Berlin, 2000.
[11]	J. Esch and T. D. Rogers, The screensaver map: Dynamics on elliptic curves arising from polygonal folding, Discrete Comput. Geom., 25 (2001), 477-502. doi: 10.1007/s004540010075.
[12]	D. Husemoller, "Elliptic Curves," With an appendix by Ruth Lawrence, Graduate Texts in Mathematics, 111, Springer-Verlag, New York, 1987.
[13]	D. Jogia, J. A. G. Roberts and F. Vivaldi, An algebraic geometric approach to integrable maps of the plane, J. Phys. A, 39 (2006), 1133-1149. doi: 10.1088/0305-4470/39/5/008.
[14]	I. Niven, H. S. Zukerman and H. L. Montgomery, "An Introduction to the Theory of Numbers,'' Fifth edition, John Wiley & Sons, Inc., New York, 1991.
[15]	F. P. Rabarison, "Torsion et Rang des Courbes Elliptiques Définies sur les Corps de Nombres Algébriques,'' Thèse de doctorat, Université de Caen, 2008.
[16]	W. Stein, et al., Sage: Open Source Mathematical Software (Version 4.0), The Sage Group, 2009., Available from: \url{http://www.sagemath.org/}., ().
[17]	J. Silverman, "Advanced Topics in the Arithmetic of Elliptic Curves,'' Graduate Texts in Mathematics, 151, Springer-Verlag, New York, 1994.
[18]	J. Silverman, "The Arithmetic of Elliptic Curves,'' Second edition, Graduate Texts in Mathematics, 106, Springer, Dordrecht, 2009.
[19]	J. Silverman and J. Tate., "Rational Points on Elliptic Curves,'' Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992.
[20]	J. M. H. Olmsted, Rational values of trigonometric functions, Amer. Math. Monthly, 52 (1945), 507-508. doi: 10.2307/2304540.
[21]	E. C. Zeeman, Geometric unfolding of a difference equation, Hertford College, Oxford, (1996), Unpublished paper, Reprinted as a Preprint of the Warwick Mathematics Institute, 2008. A video of the distinguished lecture, with the same title, at PIMS on March 21, 2000, can be downloaded from: http://www.pims.math.ca/resources/multimedia/video. The slides can be obtained at: http://zakuski.utsa.edu/~gokhman/ecz/gu.html.

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References:

[1]	A. O. L. Atkin and F. Morain, Finding suitable curves for the elliptic curve method of factorization, Math. Comp., 60 (1993), 399-405. doi: 10.1090/S0025-5718-1993-1140645-1.
[2]	E. Barbeau, B. Gelbord and S. Tanny, Periodicities of solutions of the generalized Lyness recursion, J. Difference Equations Appl., 1 (1995), 291-306. doi: 10.1080/10236199508808028.
[3]	G. Bastien and M. Rogalski, Global behavior of the solutions of Lyness' difference equation $u_{n+2}u_n=u_{n+1}+a$, J. Difference Equations Appl., 10 (2004), 977-1003. doi: 10.1080/10236190410001728104.
[4]	A. Beauville, Les familles stables de courbes elliptiques sur P1 admettant quatre fibres singulières, C. R. Acad. Sci. Paris, Série I Math., 294 (1982), 657-660.
[5]	F. Beukers and R. Cushman, Zeeman's monotonicity conjecture, J. Differential Equations, 143 (1998), 191-200. doi: 10.1006/jdeq.1997.3359.
[6]	W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125.
[7]	H. Cohen, "Number Theory. Volume I: Tools and Diophantine Equations," Graduate Texts in Mathematics, 239, Springer, New York, 2007.
[8]	J. E. Cremona, "Elliptic Curve Data," Web page maintained by W. Stein, University of Warwick., Available from: \url{http://www.warwick.ac.uk/staff/J.E.Cremona//ftp/data/}., ().
[9]	A. Dujella, "Elliptic Curve Tables," Web page maintained by author, University of Zagreb., Available from: \url{http://web.math.hr/~duje/}., ().
[10]	N. Elkies, Rational points near curves and small nonzero $\|x^3 - y^2\|$ via lattice reduction, in "Algorithmic Number Theory" (Leiden, 2000), 33-63. Lecutre Notes in Comput. Sci., 1838, Springer, Berlin, 2000.
[11]	J. Esch and T. D. Rogers, The screensaver map: Dynamics on elliptic curves arising from polygonal folding, Discrete Comput. Geom., 25 (2001), 477-502. doi: 10.1007/s004540010075.
[12]	D. Husemoller, "Elliptic Curves," With an appendix by Ruth Lawrence, Graduate Texts in Mathematics, 111, Springer-Verlag, New York, 1987.
[13]	D. Jogia, J. A. G. Roberts and F. Vivaldi, An algebraic geometric approach to integrable maps of the plane, J. Phys. A, 39 (2006), 1133-1149. doi: 10.1088/0305-4470/39/5/008.
[14]	I. Niven, H. S. Zukerman and H. L. Montgomery, "An Introduction to the Theory of Numbers,'' Fifth edition, John Wiley & Sons, Inc., New York, 1991.
[15]	F. P. Rabarison, "Torsion et Rang des Courbes Elliptiques Définies sur les Corps de Nombres Algébriques,'' Thèse de doctorat, Université de Caen, 2008.
[16]	W. Stein, et al., Sage: Open Source Mathematical Software (Version 4.0), The Sage Group, 2009., Available from: \url{http://www.sagemath.org/}., ().
[17]	J. Silverman, "Advanced Topics in the Arithmetic of Elliptic Curves,'' Graduate Texts in Mathematics, 151, Springer-Verlag, New York, 1994.
[18]	J. Silverman, "The Arithmetic of Elliptic Curves,'' Second edition, Graduate Texts in Mathematics, 106, Springer, Dordrecht, 2009.
[19]	J. Silverman and J. Tate., "Rational Points on Elliptic Curves,'' Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992.
[20]	J. M. H. Olmsted, Rational values of trigonometric functions, Amer. Math. Monthly, 52 (1945), 507-508. doi: 10.2307/2304540.
[21]	E. C. Zeeman, Geometric unfolding of a difference equation, Hertford College, Oxford, (1996), Unpublished paper, Reprinted as a Preprint of the Warwick Mathematics Institute, 2008. A video of the distinguished lecture, with the same title, at PIMS on March 21, 2000, can be downloaded from: http://www.pims.math.ca/resources/multimedia/video. The slides can be obtained at: http://zakuski.utsa.edu/~gokhman/ecz/gu.html.

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