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Year 2020, Volume: 49 Issue: 4, 1458 - 1470, 06.08.2020
https://doi.org/10.15672/hujms.464130

Abstract

References

  • [1] M. Ayad, Périodicité (mod q) des suites elliptiques et points S-entiers sur les courbes elliptiques, Ann. Inst. Fourier, 43 (3), 585–618, 1993.
  • [2] W. Bosma, J. Cannon, and C. Playoust, The Magma Algebra System I. The user language, J. Symbolic Comput. 24 (3-4), 235–265, 1997.
  • [3] A. Bremner and N. Tzanakis, Lucas sequences whose 12th or 9th term is a square, J. Number Theory, 107, 215–227, 2004.
  • [4] A. Bremner and N. Tzanakis, On squares in Lucas sequences , J. Number Theory, 124, 511–520, 2007.
  • [5] J. Cheon and S. Hahn, Explicit valuations of division polynomials of an elliptic curve, Manuscripta Math. 97, 319–328, 1998.
  • [6] G. Everest, A. van der Poorten, I. Shparlinski, and T. Ward, Recurrence Sequences, Math. Surveys Monogr. 104, AMS, Providence, RI, 2003.
  • [7] J. Gebel, A. Pethő, and H.G. Zimmer, Computing integral points on elliptic curves, Acta Arith. 68, 171–192, 1994.
  • [8] B. Gezer, Elliptic divisibility sequences, squares and cubes, Publ. Math. Debrecen, 83 (3), 481–515, 2013.
  • [9] B. Gezer, Sequences associated to elliptic curves, arXiv:1909.12654.
  • [10] B. Gezer and O. Bizim, Squares in elliptic divisibility sequences, Acta Arith. 144 (2), 125–134, 2010.
  • [11] B. Gezer and O. Bizim, Sequences generated by elliptic curves, Acta Arith. 188 (3), 253–268, 2019.
  • [12] D. Husemöller, Elliptic Curves, Springer Verlag, New York, 1987.
  • [13] D.S. Kubert, Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc. 33 (3), 193–237, 1976.
  • [14] http://magma.maths.usyd.edu.au/calc/
  • [15] B. Mazur, Modular curves and the Eisenstein ideal, Publ. Math. IHES, 47, 33–186, 1977.
  • [16] V. Mahé, Prime power terms in elliptic divisibility sequences, Math. Comp. 83 (288), 1951–1991, 2014.
  • [17] http://pari.maths.u-bordeaux.fr/
  • [18] J. Reynolds, Perfect powers in elliptic divisibility sequences, J. Number Theory, 132, 998–1015, 2012.
  • [19] P. Ribenboim, Pell numbers, squares and cubes, Publ. Math. Debrecen, 54, 131–152, 1999.
  • [20] P. Ribenboim and W. McDaniel, The square terms in Lucas sequences, J. Number Theory, 58, 104–123, 1996.
  • [21] P. Ribenboim and W. McDaniel, Squares in Lucas sequences having an even first parameter, Colloq. Math. 78, 29–34, 1998.
  • [22] R. Shipsey, Elliptic divisibility sequences, PhD thesis, Goldsmiths, University of London, 2000.
  • [23] J.H. Silverman, p-adic properties of division polynomials and elliptic divisibility sequences, Math. Ann. 332 (2), 443–471, 2005, addendum 473–474.
  • [24] J.H. Silverman, The Arithmetic of Elliptic Curves (2nd Edition), Graduate Texts in Mathematics, 106, Springer, Dordrecht, 2009.
  • [25] J.H. Silverman and N. Stephens, The sign of an elliptic divisibility sequence, J. Ramanujan Math. Soc. 21 (1), 1–17, 2006.
  • [26] J.H. Silverman and J. Tate, Rational Points on Elliptic Curves, Undergraduate Texts in Mathematics, Springer, 1992.
  • [27] K. Stange, Integral points on elliptic curves and explicit valuations of division polynomials, Canad. J. Math. 68 (5), 1120–1158, 2016.
  • [28] R.J. Stroeker, N. Tzanakis N, Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms, Acta Arith. 67, 177–196, 1994.
  • [29] C.S. Swart, Elliptic curves and related sequences, PhD Thesis, Royal Holloway, University of London, 2003.
  • [30] M. Ward, Memoir on elliptic divisibility sequences, Amer. J. Math. 70, 31–74, 1948.
  • [31] M. Ward, The law of repetition of primes in an elliptic divisibility sequences, Duke Math. J. 15, 941–946, 1948.

Sequences associated to elliptic curves with non-cyclic torsion subgroup

Year 2020, Volume: 49 Issue: 4, 1458 - 1470, 06.08.2020
https://doi.org/10.15672/hujms.464130

Abstract

Let $E$ be an elliptic curve defined over $K$ given by a Weierstrass equation and let $P=(x,y)\in E(K)$ be a point. Then for each $n$ $\geq 1$ we can write the $x$- and $y$-coordinates of the point $[n]P$ as
\[ [n]P=\left( \frac{G_{n}(P)}{F_{n}^{2}(P)},\frac{H_{n}(P)}{F_{n}^{3}(P)}\right)\]
where $F_{n}$, $G_{n}$, and $H_{n}\in K[x,y]$ are division polynomials of $E$. In this work we give explicit formulas for sequences
\[(F_{n}(P))_{n\geq 0},\,(G_{n}(P))_{n\geq 0},\,\text{and}\,(H_{n}(P))_{n\geq 0}\]
associated to an elliptic curve $E$ defined over $\mathbb{Q}$ with non-cyclic torsion subgroup. As applications we give similar formulas for elliptic divisibility sequences associated to elliptic curves with non-cyclic torsion subgroup and determine square terms in these sequences.

References

  • [1] M. Ayad, Périodicité (mod q) des suites elliptiques et points S-entiers sur les courbes elliptiques, Ann. Inst. Fourier, 43 (3), 585–618, 1993.
  • [2] W. Bosma, J. Cannon, and C. Playoust, The Magma Algebra System I. The user language, J. Symbolic Comput. 24 (3-4), 235–265, 1997.
  • [3] A. Bremner and N. Tzanakis, Lucas sequences whose 12th or 9th term is a square, J. Number Theory, 107, 215–227, 2004.
  • [4] A. Bremner and N. Tzanakis, On squares in Lucas sequences , J. Number Theory, 124, 511–520, 2007.
  • [5] J. Cheon and S. Hahn, Explicit valuations of division polynomials of an elliptic curve, Manuscripta Math. 97, 319–328, 1998.
  • [6] G. Everest, A. van der Poorten, I. Shparlinski, and T. Ward, Recurrence Sequences, Math. Surveys Monogr. 104, AMS, Providence, RI, 2003.
  • [7] J. Gebel, A. Pethő, and H.G. Zimmer, Computing integral points on elliptic curves, Acta Arith. 68, 171–192, 1994.
  • [8] B. Gezer, Elliptic divisibility sequences, squares and cubes, Publ. Math. Debrecen, 83 (3), 481–515, 2013.
  • [9] B. Gezer, Sequences associated to elliptic curves, arXiv:1909.12654.
  • [10] B. Gezer and O. Bizim, Squares in elliptic divisibility sequences, Acta Arith. 144 (2), 125–134, 2010.
  • [11] B. Gezer and O. Bizim, Sequences generated by elliptic curves, Acta Arith. 188 (3), 253–268, 2019.
  • [12] D. Husemöller, Elliptic Curves, Springer Verlag, New York, 1987.
  • [13] D.S. Kubert, Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc. 33 (3), 193–237, 1976.
  • [14] http://magma.maths.usyd.edu.au/calc/
  • [15] B. Mazur, Modular curves and the Eisenstein ideal, Publ. Math. IHES, 47, 33–186, 1977.
  • [16] V. Mahé, Prime power terms in elliptic divisibility sequences, Math. Comp. 83 (288), 1951–1991, 2014.
  • [17] http://pari.maths.u-bordeaux.fr/
  • [18] J. Reynolds, Perfect powers in elliptic divisibility sequences, J. Number Theory, 132, 998–1015, 2012.
  • [19] P. Ribenboim, Pell numbers, squares and cubes, Publ. Math. Debrecen, 54, 131–152, 1999.
  • [20] P. Ribenboim and W. McDaniel, The square terms in Lucas sequences, J. Number Theory, 58, 104–123, 1996.
  • [21] P. Ribenboim and W. McDaniel, Squares in Lucas sequences having an even first parameter, Colloq. Math. 78, 29–34, 1998.
  • [22] R. Shipsey, Elliptic divisibility sequences, PhD thesis, Goldsmiths, University of London, 2000.
  • [23] J.H. Silverman, p-adic properties of division polynomials and elliptic divisibility sequences, Math. Ann. 332 (2), 443–471, 2005, addendum 473–474.
  • [24] J.H. Silverman, The Arithmetic of Elliptic Curves (2nd Edition), Graduate Texts in Mathematics, 106, Springer, Dordrecht, 2009.
  • [25] J.H. Silverman and N. Stephens, The sign of an elliptic divisibility sequence, J. Ramanujan Math. Soc. 21 (1), 1–17, 2006.
  • [26] J.H. Silverman and J. Tate, Rational Points on Elliptic Curves, Undergraduate Texts in Mathematics, Springer, 1992.
  • [27] K. Stange, Integral points on elliptic curves and explicit valuations of division polynomials, Canad. J. Math. 68 (5), 1120–1158, 2016.
  • [28] R.J. Stroeker, N. Tzanakis N, Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms, Acta Arith. 67, 177–196, 1994.
  • [29] C.S. Swart, Elliptic curves and related sequences, PhD Thesis, Royal Holloway, University of London, 2003.
  • [30] M. Ward, Memoir on elliptic divisibility sequences, Amer. J. Math. 70, 31–74, 1948.
  • [31] M. Ward, The law of repetition of primes in an elliptic divisibility sequences, Duke Math. J. 15, 941–946, 1948.
There are 31 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Betül Gezer 0000-0001-9133-1734

Publication Date August 6, 2020
Published in Issue Year 2020 Volume: 49 Issue: 4

Cite

APA Gezer, B. (2020). Sequences associated to elliptic curves with non-cyclic torsion subgroup. Hacettepe Journal of Mathematics and Statistics, 49(4), 1458-1470. https://doi.org/10.15672/hujms.464130
AMA Gezer B. Sequences associated to elliptic curves with non-cyclic torsion subgroup. Hacettepe Journal of Mathematics and Statistics. August 2020;49(4):1458-1470. doi:10.15672/hujms.464130
Chicago Gezer, Betül. “Sequences Associated to Elliptic Curves With Non-Cyclic Torsion Subgroup”. Hacettepe Journal of Mathematics and Statistics 49, no. 4 (August 2020): 1458-70. https://doi.org/10.15672/hujms.464130.
EndNote Gezer B (August 1, 2020) Sequences associated to elliptic curves with non-cyclic torsion subgroup. Hacettepe Journal of Mathematics and Statistics 49 4 1458–1470.
IEEE B. Gezer, “Sequences associated to elliptic curves with non-cyclic torsion subgroup”, Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 4, pp. 1458–1470, 2020, doi: 10.15672/hujms.464130.
ISNAD Gezer, Betül. “Sequences Associated to Elliptic Curves With Non-Cyclic Torsion Subgroup”. Hacettepe Journal of Mathematics and Statistics 49/4 (August 2020), 1458-1470. https://doi.org/10.15672/hujms.464130.
JAMA Gezer B. Sequences associated to elliptic curves with non-cyclic torsion subgroup. Hacettepe Journal of Mathematics and Statistics. 2020;49:1458–1470.
MLA Gezer, Betül. “Sequences Associated to Elliptic Curves With Non-Cyclic Torsion Subgroup”. Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 4, 2020, pp. 1458-70, doi:10.15672/hujms.464130.
Vancouver Gezer B. Sequences associated to elliptic curves with non-cyclic torsion subgroup. Hacettepe Journal of Mathematics and Statistics. 2020;49(4):1458-70.