Advertisement

Periodica Mathematica Hungarica

, Volume 68, Issue 2, pp 222–230 | Cite as

High rank elliptic curves with prescribed torsion group over quadratic fields

  • Julián Aguirre
  • Andrej Dujella
  • Mirela Jukić Bokun
  • Juan Carlos Peral
Article

Abstract

There are 26 possibilities for the torsion groups of elliptic curves defined over quadratic number fields. We present examples of high rank elliptic curves with a given torsion group which set the current rank records for most of the torsion groups. In particular, we show that for each possible torsion group, except maybe for \(\mathbb {Z}/15\mathbb {Z}\), there exists an elliptic curve over some quadratic field with this torsion group and with rank \(\ge 2\).

Keywords

Quadratic fields Elliptic curves Rank Torsion group 

Mathematics Subject Classification

11G05 14H52 11R11 

Notes

Acknowledgments

The authors would like to thank Filip Najman and the referee for very useful comments on the previous version of this paper.

References

  1. 1.
    J. Aguirre, F. Castañeda, J.C. Peral, High rank elliptic curves with torsion group \({\mathbb{Z}}/2{\mathbb{Z}}\). Math. Comp. 73, 323–331 (2004)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    B.J. Birch, in Elliptic Curves and Modular Functions, Symposia Mathematica, vol. IV, (Academic Press, London, 1970), pp. 27–32Google Scholar
  3. 3.
    W. Bosma, J. Cannon, C. Playoust, The magma algebra system i: the user language. J. Symb. Comp. 24, 235–265 (1997)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    J. Bosman, P. Bruin, A. Dujella, F. Najman, Ranks of elliptic curves with prescribed torsion over number fields. Int. Math. Res. Notices (2013). doi: 10.1093/imrn/rnt013
  5. 5.
    J. Cremona, Algorithms for Modular Elliptic Curves (Cambridge University Press, Cambridge, 1997)MATHGoogle Scholar
  6. 6.
    A. Dujella, High rank elliptic curves with prescribed torsion. http://web.math.hr/duje/tors/tors.html
  7. 7.
    A. Dujella, M. Jukić Bokun, on the rank of elliptic curves over \({\mathbb{Q}}(i)\) with torsion group \({\mathbb{Z}}_4\times {\mathbb{Z}}_4\). Proc. Jpn. Acad. Ser. A Math. Sci. 86, 93–96 (2010)Google Scholar
  8. 8.
    N. D. Elkies, Three lectures on elliptic surfaces and curves of high rank. Lecture notes, Oberwolfach (2007) arXiv:0709.2908
  9. 9.
    M. Jukić Bokun, On the rank of elliptic curves over \({\mathbb{Q}}(\sqrt{-3})\) with torsion group \({\mathbb{Z}}_3\times {\mathbb{Z}}_3\) and \({\mathbb{Z}}_3\times {\mathbb{Z}}_6\). Proc. Jpn. Acad. Ser. A Math. Sci. 87, 61–64 (2011)Google Scholar
  10. 10.
    S. Kamienny, Torsion points on elliptic curves and \(q\)-coefficients of modular forms. Invent. Math. 109, 221–229 (1992)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    M.A. Kenku, F. Momose, Torsion points on elliptic curves defined over quadratic fields. Nagoya Math. J. 109, 125–149 (1988)MATHMathSciNetGoogle Scholar
  12. 12.
    L. Kulesz, C. Stahlke, Elliptic curves of high rank with nontrivial torsion group over \({\mathbb{Q}}\). Exp. Math. 10, 475–480 (2001)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    J.-F. Mestre, Rang des courbes elliptiques d’ invariant donné. C. R. Acad. Sci. Paris 314, 919–922 (1992)MATHMathSciNetGoogle Scholar
  14. 14.
    J.-F. Mestre, Rang de certaines familles de courbes elliptiques d’ invariant donné. C. R. Acad. Sci. Paris 327, 763–764 (1998)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    K. Nagao, An example of elliptic curve over q with rank \(\ge 20\). Proc. Jpn. Acad. Ser. A Math. Sci. 69, 291–293 (1993)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    PARI/GP, version 2.4.0, Bordeaux (2008). http://pari.math.u-bordeaux.fr
  17. 17.
    F.P. Rabarison, Structure de torsion des courbes elliptiques sur les corps quadratiques. Acta Arith. 144, 17–52 (2010)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    K. Rubin, A. Silverberg, Rank frequencies for quadratic twists of elliptic curves. Exp. Math. 10, 559–569 (2001)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    K. Rubin, A. Silverberg, Twists of elliptic curves of rank at least four, in Ranks of Elliptic Curves and Random Matrix Theory, ed. by J.B. Conrey, D.W. Farmer, F. Mezzadri, N.C. Snaith (Cambridge University Press, Cambridge, 2007), pp. 177–188Google Scholar
  20. 20.
    U. Schneiders, H.G. Zimmer, The rank of elliptic curves upon quadratic extension, in Computational Number Theory, ed. by A. Pethő, M.E. Pohst, H.C. Williams, H.G. Zimmer (W. de Gruyter, Berlin, 1991), pp. 239–260Google Scholar
  21. 21.
    J. Silverman, The Arithmetic of Elliptic curves (Springer, New York, 2009)CrossRefMATHGoogle Scholar
  22. 22.
    C.L. Stewart, J. Top, On ranks of twists of elliptic curves and power-free values of binary forms. J. Am. Math. Soc. 8, 943–973 (1995)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    T. Womack, Curves with moderate rank and interesting torsion group. http://tom.womack.net/maths/torsion.htm

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2014

Authors and Affiliations

  • Julián Aguirre
    • 1
  • Andrej Dujella
    • 2
  • Mirela Jukić Bokun
    • 3
  • Juan Carlos Peral
    • 1
  1. 1.Departamento de MatemáticasUniversidad del País VascoBilbaoSpain
  2. 2.Department of MathematicsUniversity of ZagrebZagrebCroatia
  3. 3.Department of MathematicsUniversity of OsijekOsijekCroatia

Personalised recommendations