Czechoslovak Mathematical Journal, Vol. 69, No. 1, pp. 161-171, 2019


Torsion groups of a family of elliptic curves over number fields

Pallab Kanti Dey

Received May 8, 2017.   Published online July 24, 2018.

Abstract:  We compute the torsion group explicitly over quadratic fields and number fields of degree coprime to 6 for a family of elliptic curves of the form $E y^2 = x^3 +c$, where $c$ is an integer.
Keywords:  torsion group; elliptic curve; number field
Classification MSC:  14H52, 11R04


References:
[1] R. Ayoub: An Introduction to the Analytic Theory of Numbers. Mathematical Surveys 10, American Mathematical Society, Providence (1963). DOI 10.1090/surv/010 | MR 0160743 | Zbl 0128.04303
[2] A. Bourdon, P. L. Clark, J. Stankewicz: Torsion points on CM elliptic curves over real number fields. Trans. Am. Math. Soc. 369 (1996), 8457-8496. DOI 10.1090/tran/6905 | MR 3710632 | Zbl 06790352
[3] P. K. Dey: Elliptic curves with rank $0$ over number fields. Funct. Approximatio, Comment. Math. 56 (2017), 25-37. DOI 10.7169/facm/1585 | MR 3629008 | Zbl 06864143
[4] E. González-Jiménez: Complete classification of the torsion structures of rational elliptic curves over quintic number fields. J. Algebra 478 (2017), 484-505. DOI 10.1016/j.jalgebra.2017.01.012 | MR 3621686 | Zbl 1369.11040
[5] D. Jeon, C. H. Kim, E. Park: On the torsion of elliptic curves over quartic number fields. J. Lond. Math. Soc., II. Ser. 74 (2006), 1-12. DOI 10.1112/S0024610706022940 | MR 2254548 | Zbl 1165.11054
[6] S. Kamienny: Torsion points on elliptic curves and $q$-coefficients of modular forms. Invent. Math. 109 (1992), 221-229. DOI 10.1007/BF01232025 | MR 1172689 | Zbl 0773.14016
[7] M. A. Kenku, F. Momose: Torsion points on elliptic curves defined over quadratic fields. Nagoya Math. J. 109 (1988), 125-149. DOI 10.1017/S0027763000002816 | MR 0931956 | Zbl 0647.14020
[8] A. W. Knapp: Elliptic Curves. Mathematical Notes (Princeton) 40, Princeton University Press, Princeton (1992). MR 1193029 | Zbl 0804.14013
[9] B. Mazur: Modular curves and the Eisenstein ideal. Publ. Math., Inst. Hautes Étud. Sci. 47 (1977), 33-186. DOI 10.1007/BF02684339 | MR 0488287 | Zbl 0394.14008
[10] F. Najman: Complete classification of torsion of elliptic curves over quadratic cyclotomic fields. J. Number Theory 130 (2010), 1964-1968. DOI 10.1016/j.jnt.2009.12.008 | MR 2653208 | Zbl 1200.11039
[11] F. Najman: Torsion of elliptic curves over quadratic cyclotomic fields. Math. J. Okayama Univ. 53 (2011), 75-82. MR 2778886 | Zbl 1222.11076
[12] F. Najman: Torsion of rational elliptic curves over cubic fields and sporadic points on $X_1(n)$. Math. Res. Lett. 23 (2016), 245-272. DOI 10.4310/MRL.2016.v23.n1.a12 | MR 3512885 | Zbl 06609434
[13] L. D. Olson: Points of finite order on elliptic curves with complex multiplication. Manuscr. Math. 14 (1974), 195-205. DOI 10.1007/BF01171442 | MR 0352104 | Zbl 0292.14015
[14] L. C. Washington: Elliptic Curves. Number Theory and Cryptography. Chapman and Hall/CRC, Boca Raton (2008). DOI 10.4324/9780203484029 | MR 2404461 | Zbl 1200.11043

Affiliations:   Pallab Kanti Dey, Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad - 211019, India, e-mail: pallabkantidey@gmail.com


 
PDF available at: