Advertisement

Visibility of 4-covers of elliptic curves

  • Nils Bruin
  • Tom Fisher
Research
  • 77 Downloads

Abstract

Let C be a 4-cover of an elliptic curve E, written as a quadric intersection in \({\mathbb P}^3\). Let \(E'\) be another elliptic curve with 4-torsion isomorphic to that of E. We show how to write down the 4-cover \(C'\) of \(E'\) with the property that C and \(C'\) are represented by the same cohomology class on the 4-torsion. In fact we give equations for \(C'\) as a curve of degree 8 in \({\mathbb P}^5\). We also study the K3-surfaces fibred by the curves \(C'\) as we vary \(E'\). In particular we show how to write down models for these surfaces as complete intersections of quadrics in \({\mathbb P}^5\) with exactly 16 singular points. This allows us to give examples of elliptic curves over \({\mathbb Q}\) that have elements of order 4 in their Tate–Shafarevich group that are not visible in a principally polarized abelian surface.

Keywords

Elliptic curves Tate–Shafarevich groups Mazur visibility Descent K3 surfaces Local-global obstructions 

Mathematics Subject Classification

11G05 11G35 14H10 

Notes

References

  1. 1.
    Barth, W.: Projective models of Shioda modular surfaces. Manuscripta Math. 50(1), 73–132 (1985)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bosma, W., Cannon, J., Cannon, C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24(3), 235–265 (1997)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bruin, N., Doerksen, K.: The arithmetic of genus two curves with \((4,4)\)-split Jacobians. Canad. J. Math 63(5), 992–1024 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bruin, N.: Visualising Sha[2] in abelian surfaces. Math. Comp. 73(247), 1459–1476 (2004)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bruin, N., Dahmen, S.R.: Visualizing elements of Sha[3] in genus 2 jacobians. In: Hanrot, G., Morain, F., Thomé, E. (eds.) Algorithmic Number Theory. ANTS 2010. Lecture Notes in Computer Science, vol. 6197. Springer, Berlin (2010).  https://doi.org/10.1007/978-3-642-14518-6_12
  6. 6.
    Cremona, J. E.: Algorithms for modular elliptic curves, 2nd edn. Cambridge University Press, Cambridge. http://www.warwick.ac.uk/~masgaj/ftp/data/ (1997)
  7. 7.
    Cremona, J.E.: Classical invariants and 2-descent on elliptic curves. J. Symbolic Comput. 31(1–2), 71–87 (2001)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cremona, J.E., Fisher, T.A., O’Neil, C., Simon, D., Stoll, M.: Explicit \(n\)-descent on elliptic curves. I. Algebra. J. Reine Angew. Math. 615(121–155), 0075–4102 (2008)MathSciNetMATHGoogle Scholar
  9. 9.
    Cremona, J.E., Fisher, T.A., Stoll, M.: Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves. Algebra Number Theory 4(6), 763–820 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Cremona, J.E., Mazur, B.: Visualizing elements in the Shafarevich–Tate group. Exp. Math 9(1), 13–28 (2000)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Fisher, T.: Some improvements to 4-descent on an elliptic curve. In: van der Poorten, A.J., Stein, A. (eds.) Algorithmic Number Theory. ANTS 2008. Lecture Notes in Computer Science, vol. 5011. Springer, Berlin (2008).  https://doi.org/10.1007/978-3-540-79456-1_8
  12. 12.
    Fisher, T.: The Hessian of a genus one curve. Proc. Lond. Math. Soc 104(3), 613–648 (2012)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Fisher, T.: Invariant theory for the elliptic normal quintic I. Twists of X(5). Math. Ann. 356(2), 589–616 (2013)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Fisher, T.: Invisibility of Tate–Shafarevich groups in abelian surfaces. IMRN 15, 4085–4099 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Klenke, T.A.: Visualizing elements of order two in the Weil–Châtelet group. J. Number Theory 110(2), 387–395 (2005)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Mazur, B.: Visualizing elements of order three in the Shafarevich–Tate group. Asian J. Math. 3(1), 221–232 (1999)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Merriman, J.R., Siksek, S., Smart, N.P.: Explicit \(4\)-descents on an elliptic curve. Acta Arith 77(4), 385–404 (1996)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Milne, J.S.: Abelian varieties, Arithmetic geometry (Storrs, Conn., 1984), pp. 103–150. Springer, New York (1986)Google Scholar
  19. 19.
    Silverberg, A.: Explicit families of elliptic curves with prescribed mod N representations. In: Modular forms and Fermat’s last theorem (Boston, MA, 1995), pp. 447–461. Springer, New York (1997)Google Scholar
  20. 20.
    Stamminger, S.K.M.: Explicit 8-descent on elliptic curves, International University Bremen, http://nbn-resolving.de/urn:nbn:de:101:1-201305171186, (PhD thesis) (2005)
  21. 21.
    Womack, T.: Explicit descent on elliptic curves, University of Nottingham, (PhD thesis) (2003)Google Scholar

Copyright information

© SpringerNature 2018

Authors and Affiliations

  1. 1.Department of MathematicsSimon Fraser UniversityBurnabyCanada
  2. 2.DPMMS, Centre for Mathematical SciencesUniversity of CambridgeCambridgeUK

Personalised recommendations