Have a personal or library account? Click to login
Torsion subgroups of rational Mordell curves over some families of number fields Cover

Torsion subgroups of rational Mordell curves over some families of number fields

Open Access
|Jun 2022

References

  1. [1] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235265.10.1006/jsco.1996.0125
  2. [2] A. Bourdon and H. P. Chaos, Torsion for CM elliptic curves defined over number fields of degree 2p, https://arxiv.org/abs/2110.07819v1.
  3. [3] A. Bourdon and P. L. Clark, Torsion points and Galois representations on CM elliptic curves, Pacific J. Math. 305 (2020), no. 1, 43-88.
  4. [4] A. Bourdon and P. L. Clark, Torsion points and isogenies on CM elliptic curves, J. London Math. Soc. 102 (2020), no. 2, 580-622.
  5. [5] A. Bourdon and P. Pollack, Torsion subgroups of CM elliptic curves over odd degree number fields, Math. Res. Not. IMRN (2017), no. 16, 4923-4961.
  6. [6] P. L. Clark, Bounds for torsion on abelian varieties with integral moduli, arXiv:math/0407264.
  7. [7] P. L. Clark, P. Corn, A. Rice and J. Stankewicz, Computation on elliptic curves with complex multiplication, LMS J. Comput. Math. 17 (1) (2014), 509-535.10.1112/S1461157014000072
  8. [8] H. B. Daniels and E. González-Jiménez, On the torsion of rational elliptic curves over sextic fields, Math. Comp. 89 (321) (2020), 411-435.10.1090/mcom/3440
  9. [9] M. Derickx, A. Etropolski, M. V. Hoeij, J. S. Morrow and D. Zureick-Brown, Sporadic Cubic Torsion, Algebra & Number Theory 15 (7), 1837-1864.10.2140/ant.2021.15.1837
  10. [10] P. K. Dey, Torsion groups of a family of elliptic curves over number fields, Czechoslovak Math. J. 69 (144) (1) (2019), 161-171.10.21136/CMJ.2018.0214-17
  11. [11] P. K. Dey and B. Roy, Torsion groups of Mordell curves over cubic and sextic fields, Publicationes Mathematicae Debrecen 99/3-4 (13) (2021).10.5486/PMD.2021.8771
  12. [12] G. Fung, H. Ströher, H. Williams, and H. Zimmer. Torsion groups of elliptic curves with integral j-invariant over pure cubic fields, J. Number Theory 36 (1) (1990), 12-45.10.1016/0022-314X(90)90003-A
  13. [13] R. Fueter, Ueber kubische diophantische Gleichungen, Comment. Math. Helv. 2 (1930), no. 1, 6989.
  14. [14] E. González-Jiménez, Complete classification of the torsion structures of rational elliptic curves over quintic number fields, J. Algebra. 478 (2017), 484-505.10.1016/j.jalgebra.2017.01.012
  15. [15] E. González-Jiménez, Torsion growth over cubic fields of rational elliptic curves with complex multiplication, Publicationes Mathematicae Debrecen 97/1-2, 63-76 (2020).10.5486/PMD.2020.8697
  16. [16] E. González-Jiménez, Explicit description of the growth of the torsion subgroup of rational elliptic curves with complex multiplication over quadratic fields, Glas. Mat. Ser. III, 56(76)(2021), 47-61.10.3336/gm.56.1.04
  17. [17] E. González-Jiménez and F. Najman, Growth of torsion groups of elliptic curves upon base change, Math. Comp. 89 (323) (2020), 1457-1485.10.1090/mcom/3478
  18. [18] T. Guvi, Torsion growth of rational elliptic curves in sextic number fields, J. Number Theory 220 (2021), 330-345.10.1016/j.jnt.2020.09.010
  19. [19] T. Guvi, Torsion of elliptic curves with rational j-invariant defined over number fields of prime degree, Proc. Amer. Math. Soc. 149 (2021), 3261-3275.10.1090/proc/15500
  20. [20] S. Kamienny, Torsion points on elliptic curves and q-coefficients of modular forms, Invent. Math. 109 (2) (1992), 221-229.10.1007/BF01232025
  21. [21] M. A. Kenku and F. Momose, Torsion points on elliptic curves defined over quadratic fields, Nagoya Math. J. 109 (1988), 125-149.10.1017/S0027763000002816
  22. [22] A. W. Knapp, Elliptic Curves, Mathematical Notes, Vol. 40, Princeton Univ. Press, Princeton, 1992.
  23. [23] A. Lozano-Robledo, Galois representations attached to elliptic curves with complex multiplications, arXiv: 1809.02584, Algebra & Number Theory (to appear).
  24. [24] B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33-186.10.1007/BF02684339
  25. [25] H. H. Müller, H. Ströher, and H. G. Zimmer, Torsion groups of elliptic curves with integral j-invariant over quadratic fields, J. Reine Angew. Math. 397 (1989), 100161.10.1515/crll.1989.397.100
  26. [26] F. Najman, Torsion of rational elliptic curves over cubic fields and sporadic points on X1(n), Math. Res. Letters. 23 (1) (2016), 245-272.10.4310/MRL.2016.v23.n1.a12
  27. [27] L. D. Olson, Points of finite order on elliptic curves with complex multiplication, Manuscripta math. 14 (1974), 195-205.10.1007/BF01171442
  28. [28] A. Pethő, T. Weis, and H. Zimmer, Torsion groups of elliptic curves with integral j-invariant over general cubic number fields, Int. J. Algebra Comput. 7 (1997) 353-413.10.1142/S0218196797000174
  29. [29] J. H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, 106. Springer-Verlag, New York, 1986.10.1007/978-1-4757-1920-8
  30. [30] D. J. Zywina, On the possible images of the mod ℓ representations associated to elliptic curves over ℚ, https://arxiv.org/abs/1508.07660.
DOI: https://doi.org/10.2478/auom-2022-0022 | Journal eISSN: 1844-0835 | Journal ISSN: 1224-1784
Language: English
Page range: 125 - 132
Submitted on: May 12, 2021
Accepted on: Jan 24, 2022
Published on: Jun 2, 2022
Published by: Ovidius University of Constanta
In partnership with: Paradigm Publishing Services
Publication frequency: 3 issues per year

© 2022 Tomislav Gužvić, Bidisha Roy, published by Ovidius University of Constanta
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.