| | | |

## Sequences associated to elliptic curves with non-cyclic torsion subgroup

#### Betül GEZER [1]

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.
Elliptic curves, division polynomials, elliptic divisibility sequences
• [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.
Primary Language en Mathematics Mathematics Orcid: 0000-0001-9133-1734Author: Betül GEZER (Primary Author)Institution: BURSA ULUDAĞ ÜNİVERSİTESİCountry: Turkey Publication Date : August 6, 2020
 Bibtex @research article { hujms464130, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2020}, volume = {49}, pages = {1458 - 1470}, doi = {10.15672/hujms.464130}, title = {Sequences associated to elliptic curves with non-cyclic torsion subgroup}, key = {cite}, author = {Gezer, Betül} } APA Gezer, B . (2020). Sequences associated to elliptic curves with non-cyclic torsion subgroup . Hacettepe Journal of Mathematics and Statistics , 49 (4) , 1458-1470 . DOI: 10.15672/hujms.464130 MLA Gezer, B . "Sequences associated to elliptic curves with non-cyclic torsion subgroup" . Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1458-1470 Chicago Gezer, B . "Sequences associated to elliptic curves with non-cyclic torsion subgroup". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1458-1470 RIS TY - JOUR T1 - Sequences associated to elliptic curves with non-cyclic torsion subgroup AU - Betül Gezer Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.464130 DO - 10.15672/hujms.464130 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1458 EP - 1470 VL - 49 IS - 4 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.464130 UR - https://doi.org/10.15672/hujms.464130 Y2 - 2019 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics Sequences associated to elliptic curves with non-cyclic torsion subgroup %A Betül Gezer %T Sequences associated to elliptic curves with non-cyclic torsion subgroup %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 4 %R doi: 10.15672/hujms.464130 %U 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 AMA Gezer B . Sequences associated to elliptic curves with non-cyclic torsion subgroup. Hacettepe Journal of Mathematics and Statistics. 2020; 49(4): 1458-1470. Vancouver Gezer B . Sequences associated to elliptic curves with non-cyclic torsion subgroup. Hacettepe Journal of Mathematics and Statistics. 2020; 49(4): 1458-1470.

Authors of the Article