Glasnik Matematicki, Vol. 56, No. 1 (2021), 47-61.

EXPLICIT CHARACTERIZATION OF THE TORSION GROWTH OF RATIONAL ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION OVER QUADRATIC FIELDS

Enrique González-Jiménez

Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, Spain
e-mail: enrique.gonzalez.jimenez@uam.es

Abstract.   In a series of papers we classify the possible torsion structures of rational elliptic curves base-extended to number fields of a fixed degree. In this paper we turn our attention to the question of how the torsion of an elliptic curve with complex multiplication defined over the rationals grows over quadratic fields. We go further and we give an explicit characterization of the quadratic fields where the torsion grows in terms of some invariants attached to the curve.

2010 Mathematics Subject Classification.   11G05, 11G15

Key words and phrases.   Elliptic curves, complex multiplication, torsion subgroup, rationals, quadratic fields

Full text (PDF) (access from subscribing institutions only)

https://doi.org/10.3336/gm.56.1.04

References:

  1. W. Bosma, J. Cannon, C. Fieker, and A. Steel (eds.), Handbook of Magma functions, Edition 2.23, http://magma.maths.usyd.edu.au/magma, 2019.

  2. A. Bourdon, P. L. Clark, amd J. Stankewicz, Torsion points on CM elliptic curves over real number fields, Trans. Amer. Math. Soc. 369 (2017), 8457-8496.
    MathSciNet     CrossRef

  3. A. Bourdon and P. Pollack, Torsion subgroups of CM elliptic curves over odd degree number fields, Int. Math. Res. Not. IMRN 2017 (2017), 4923-4961.
    MathSciNet     CrossRef

  4. M. Chou, Torsion of rational elliptic curves over quartic Galois number fields, J. Number Theory 160 (2016), 603-628.
    MathSciNet     CrossRef

  5. P. L. Clark, Bounds for torsion on abelian varieties with integral moduli,
    https://arxiv.org/abs/math/0407264v2.

  6. P. L. Clark, P. Corn, A. Rice and J. Stankewicz, Computation on elliptic curves with complex multiplication, LMS J. Comput. Math. 17 (2014), 509-535.
    MathSciNet     CrossRef

  7. H. B. Daniels and E. González-Jiménez, On the torsion of rational elliptic curves over sextic fields, Math. Comp. 89 (2020), 411-435.
    MathSciNet     CrossRef

  8. M. Derickx, A. Etropolski, M. van Hoeij, J. Morrow and D. Zureick-Brown, Sporadic cubic torsion, to appear in Algebra Number Theory.

  9. P. K. Dey, Torsion groups of a family of elliptic curves over number fields, Czechoslovak Math. J. 69(144) (2019), 161-171.
    MathSciNet     CrossRef

  10. L. Dieulefait, E. González-Jiménez and J. Jiménez Urroz, On fields of definition of torsion points of elliptic curves with complex multiplication Proc. Amer. Math. Soc. 139 (2011), 1961-1969.
    MathSciNet     CrossRef

  11. G. Fung, H. Ströher, H. Williams, H. Zimmer, Torsion groups of elliptic curves with integral j-invariant over pure cubic fields, J. Number Theory 36 (1990) 12-45.
    MathSciNet     CrossRef

  12. R. Fueter, Ueber kubische diophantische Gleichungen, Comment. Math. Helv. 2 (1930), 69-89.
    MathSciNet     CrossRef

  13. E. González-Jiménez, Complete classification of the torsion structures of rational elliptic curves over quintic number fields, J. Algebra 478 (2017), 484-505.
    MathSciNet     CrossRef

  14. E. González-Jiménez, Magma scripts and electronic transcript of computations for the paper "Explicit characterization of the torsion growth of rational elliptic curves with complex multiplication over quadratic fields"".
    http://matematicas.uam.es/ enrique.gonzalez.jimenez

  15. E. González-Jiménez, Torsion growth over cubic fields of rational elliptic curves with complex multiplication, Publ. Math. Debrecen 97 (2020), 63-76.
    MathSciNet     CrossRef

  16. E. González-Jiménez, Torsion of rational elliptic curves with complex multiplication over number fields of low degree, in preparation.

  17. E. González-Jiménez and Á. Lozano-Robledo, On the torsion of rational elliptic curves over quartic fields, Math. Comp. 87 (2018), 1457-1478.
    MathSciNet     CrossRef

  18. E. González-Jiménez and F. Najman, Growth of torsion groups of elliptic curves upon base change, Math. Comp. 89 (2020), 1457-1485.
    MathSciNet     CrossRef

  19. E. González-Jiménez and F. Najman, An algorithm for determining torsion growth of elliptic curves, to appear in Exp. Math.

  20. E. González-Jiménez, F. Najman and J.M. Tornero, Torsion of rational elliptic curves over cubic fields, Rocky Mountain J. Math. 46 (2016), 1899-1917.
    MathSciNet     CrossRef

  21. E. González-Jiménez and J.M. Tornero, Torsion of rational elliptic curves over quadratic fields, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 108 (2014), 923-934.
    MathSciNet     CrossRef

  22. E. González-Jiménez and J.M. Tornero, Torsion of rational elliptic curves over quadratic fields II, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 110 (2016), 121-143.
    MathSciNet     CrossRef

  23. T. Gužvić, Torsion growth of rational elliptic curves in sextic number fields, J. Number Theory 220 (2021), 330-345.
    MathSciNet     CrossRef

  24. S. Kamienny, Torsion points on elliptic curves and q-coefficients of modular forms, Invent. Math. 109 (1992), 221-229.
    MathSciNet     CrossRef

  25. M. A. Kenku and F. Momose, Torsion points on elliptic curves defined over quadratic fields, Nagoya Math. J. 109 (1988), 125-149.
    MathSciNet     CrossRef

  26. B. Mazur, Rational isogenies of prime degree, Invent. Math.44 (1978), 129-162.
    MathSciNet     CrossRef

  27. L. Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124 (1996), 437-449.
    MathSciNet     CrossRef

  28. H. Müller, H. Ströher and H. Zimmer, Torsion groups of elliptic curves with integral j-invariant over quadratic fields, J. Reine Angew. Math. 397 (1989), 100-161.
    MathSciNet    

  29. F. Najman, Torsion of rational elliptic curves over cubic fields and sporadic points on X1(n), Math. Res. Lett. 23 (2016), 245-272.
    MathSciNet     CrossRef

  30. L. Olson, Points of finite order on elliptic curves with complex multiplication, Manuscripta Math. 14 (1974), 195-205.
    MathSciNet     CrossRef

  31. A. Pethő, T. Weis and H. Zimmer. Torsion groups of elliptic curves with integral j-invariant over general cubic number fields, Internat. J. Algebra Comput. 7 (1997) 353-413.
    MathSciNet     CrossRef

  32. J-H. Silverman, The arithmetic of elliptic curves, Springer-Verlag, New York, 2009.
    MathSciNet     CrossRef

  33. J-H. Silverman, Advanced topics in the arithmetic of elliptic curves, Springer-Verlag, New York, 1994.
    MathSciNet     CrossRef

  34. L. C. Washington, Elliptic curves. Number theory and cryptography, Chapman & Hall, Boca Ratón, 2008.
    MathSciNet     CrossRef
Glasnik Matematicki Home Page