
Previous Article
On strange attractors in a class of pinched skew products
 DCDS Home
 This Issue

Next Article
Genus and braid index associated to sequences of renormalizable Lorenz maps
Rational periodic sequences for the Lyness recurrence
1.  Dept. de Matemµatiques, Universitat Autónoma de Barcelona, Edifici C, 08193Bellaterra, 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 
References:
[1] 
A. O. L. Atkin and F. Morain, Finding suitable curves for the elliptic curve method of factorization, Math. Comp., 60 (1993), 399405. doi: 10.1090/S00255718199311406451. 
[2] 
E. Barbeau, B. Gelbord and S. Tanny, Periodicities of solutions of the generalized Lyness recursion, J. Difference Equations Appl., 1 (1995), 291306. 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), 9771003. 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), 657660. 
[5] 
F. Beukers and R. Cushman, Zeeman's monotonicity conjecture, J. Differential Equations, 143 (1998), 191200. 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), 235265. 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), 3363. 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), 477502. doi: 10.1007/s004540010075. 
[12] 
D. Husemoller, "Elliptic Curves," With an appendix by Ruth Lawrence, Graduate Texts in Mathematics, 111, SpringerVerlag, 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), 11331149. doi: 10.1088/03054470/39/5/008. 
[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, SpringerVerlag, 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, SpringerVerlag, New York, 1992. 
[20] 
J. M. H. Olmsted, Rational values of trigonometric functions, Amer. Math. Monthly, 52 (1945), 507508. 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. 
show all references
References:
[1] 
A. O. L. Atkin and F. Morain, Finding suitable curves for the elliptic curve method of factorization, Math. Comp., 60 (1993), 399405. doi: 10.1090/S00255718199311406451. 
[2] 
E. Barbeau, B. Gelbord and S. Tanny, Periodicities of solutions of the generalized Lyness recursion, J. Difference Equations Appl., 1 (1995), 291306. 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), 9771003. 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), 657660. 
[5] 
F. Beukers and R. Cushman, Zeeman's monotonicity conjecture, J. Differential Equations, 143 (1998), 191200. 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), 235265. 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), 3363. 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), 477502. doi: 10.1007/s004540010075. 
[12] 
D. Husemoller, "Elliptic Curves," With an appendix by Ruth Lawrence, Graduate Texts in Mathematics, 111, SpringerVerlag, 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), 11331149. doi: 10.1088/03054470/39/5/008. 
[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, SpringerVerlag, 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, SpringerVerlag, New York, 1992. 
[20] 
J. M. H. Olmsted, Rational values of trigonometric functions, Amer. Math. Monthly, 52 (1945), 507508. 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. 
[1] 
Jaime Gutierrez. Reconstructing points of superelliptic curves over a prime finite field. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022022 
[2] 
David L. Finn. Convexity of level curves for solutions to semilinear elliptic equations. Communications on Pure and Applied Analysis, 2008, 7 (6) : 13351343. doi: 10.3934/cpaa.2008.7.1335 
[3] 
Joseph H. Silverman. Localglobal aspects of (hyper)elliptic curves over (in)finite fields. Advances in Mathematics of Communications, 2010, 4 (2) : 101114. doi: 10.3934/amc.2010.4.101 
[4] 
Ferruh Özbudak, Burcu Gülmez Temür, Oǧuz Yayla. Further results on fibre products of Kummer covers and curves with many points over finite fields. Advances in Mathematics of Communications, 2016, 10 (1) : 151162. doi: 10.3934/amc.2016.10.151 
[5] 
Massimo Grossi. On the number of critical points of solutions of semilinear elliptic equations. Electronic Research Archive, 2021, 29 (6) : 42154228. doi: 10.3934/era.2021080 
[6] 
Philip N. J. Eagle, Steven D. Galbraith, John B. Ong. Point compression for Koblitz elliptic curves. Advances in Mathematics of Communications, 2011, 5 (1) : 110. doi: 10.3934/amc.2011.5.1 
[7] 
Eszter Fehér, Gábor Domokos, Bernd Krauskopf. Tracking the critical points of curves evolving under planar curvature flows. Journal of Computational Dynamics, 2021, 8 (4) : 447494. doi: 10.3934/jcd.2021017 
[8] 
Kensuke Yoshizawa. The critical points of the elastic energy among curves pinned at endpoints. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 403423. doi: 10.3934/dcds.2021122 
[9] 
Alice Silverberg. Some remarks on primality proving and elliptic curves. Advances in Mathematics of Communications, 2014, 8 (4) : 427436. doi: 10.3934/amc.2014.8.427 
[10] 
Jaime Arango, Adriana Gómez. Critical points of solutions to elliptic problems in planar domains. Communications on Pure and Applied Analysis, 2011, 10 (1) : 327338. doi: 10.3934/cpaa.2011.10.327 
[11] 
Motoko Qiu Kawakita. Certain sextics with many rational points. Advances in Mathematics of Communications, 2017, 11 (2) : 289292. doi: 10.3934/amc.2017020 
[12] 
Daniele Bartoli, Adnen Sboui, Leo Storme. Bounds on the number of rational points of algebraic hypersurfaces over finite fields, with applications to projective ReedMuller codes. Advances in Mathematics of Communications, 2016, 10 (2) : 355365. doi: 10.3934/amc.2016010 
[13] 
Ravi Vakil and Aleksey Zinger. A natural smooth compactification of the space of elliptic curves in projective space. Electronic Research Announcements, 2007, 13: 5359. 
[14] 
Guilin Ji, Changjian Liu. The cyclicity of a class of quadratic reversible centers defining elliptic curves. Discrete and Continuous Dynamical Systems  B, 2021 doi: 10.3934/dcdsb.2021299 
[15] 
K. H. Kim, F. W. Roush and J. B. Wagoner. Inert actions on periodic points. Electronic Research Announcements, 1997, 3: 5562. 
[16] 
Charles Pugh, Michael Shub. Periodic points on the $2$sphere. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 11711182. doi: 10.3934/dcds.2014.34.1171 
[17] 
Nazar Arakelian, Saeed Tafazolian, Fernando Torres. On the spectrum for the genera of maximal curves over small fields. Advances in Mathematics of Communications, 2018, 12 (1) : 143149. doi: 10.3934/amc.2018009 
[18] 
Stefania Fanali, Massimo Giulietti, Irene Platoni. On maximal curves over finite fields of small order. Advances in Mathematics of Communications, 2012, 6 (1) : 107120. doi: 10.3934/amc.2012.6.107 
[19] 
Anton Stolbunov. Constructing publickey cryptographic schemes based on class group action on a set of isogenous elliptic curves. Advances in Mathematics of Communications, 2010, 4 (2) : 215235. doi: 10.3934/amc.2010.4.215 
[20] 
Ananta Acharya, R. Shivaji, Nalin Fonseka. $ \Sigma $shaped bifurcation curves for classes of elliptic systems. Discrete and Continuous Dynamical Systems  S, 2022 doi: 10.3934/dcdss.2022067 
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]