Skip to main content
Log in

Icosahedral representations and elliptic curves

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

In this paper we show a connection between icosahedral Artin representations of the rationals and elliptic curves. More specifically, we prove for each suitable elliptic curve defined over there is an associated icosahedral Artin representation defined over the rationals.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Buhler, J.P.: Icosahedral Galois Representations, volume 654 of Lecture Notes in Mathematics. Springer-Verlag, 1978

  2. Buzzard, K., Dickinson, M., Shepherd-Barron, N., Taylor, R.: On icosahedral Artin representations. Duke Math Journal 109 (2), 283–318 (2001)

    Google Scholar 

  3. Buzzard, K., Stein, W.: A mod five approach to modularity of icosahedral Galois representations. Pacific J. Math. 203 (2), 265–282 (2002)

    Google Scholar 

  4. Buzzard, K., Taylor, R.: Companion forms and weight one forms. Annals of Mathematics, Second Series 149 (3), 905–919 (1999)

    Google Scholar 

  5. Carter, R.W.: Simple Groups of Lie Type. John Wiley & Sons, 1972

  6. Cox, D.A.: Primes of the Form x2 + ny2: Fermat, Class Field Theory, and Complex Multiplication. John Wiley & Sons, 1989

  7. Girardi, T.: Icosahedral Extensions of and Elliptic Curves. Rutgers University, 1999

  8. Goins, E.: Icosahedral -Curve Extensions. Mathematical Research Letters 10 (2–3), 205–217 (2003)

  9. Isaacs, I.M.: Character Theory of Finite Groups. Dover Publications, Inc., 1976

  10. Kiming, I., Wang, X.: Examples of 2-dimensional odd Galois representations of A5-type over satisfying Artin’s conjecture. In: G. Frey (ed) On Artin’s Conjecture for Odd 2-dimensional Representations, Volume 1585. Springer-Verlag, 1994

  11. Kinzelbach, M.L.: Konstruktion von zweidimensionalen Galoisdarstellungen mit Hilfe von Kurven vom Geschlecht ≤ 2. Universität Gesamthochschule Essen, 1993

  12. Klute, A.: Icosahedral Galois Extensions and Elliptic Curves. manuscripta mathematica 93, 301–324 (1997)

    Google Scholar 

  13. Roberts, B.B.: -Curves over Quadratic Fields. University of Maryland, 1995

  14. Robinson, D.J.S.: A Course in the Theory of Groups. Springer-Verlag, 1993

  15. Serre, J.-P.: Propriètès galoisiennes des points d’order fini des courbes elliptiques. Inventiones mathematicae 15, 259–331 (1972)

    Google Scholar 

  16. Shih, K.-y.: p-Division Points on Certain Elliptic Curves. Composito Mathematica 36 (2), 113–129 (1978)

    Google Scholar 

  17. Shimura, G.: A reciprocity law in non-solvable extensions. Journal für die Reine und Angewandte Mathematik 221, 209–220 (1966)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Theresa Girardi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Girardi, T. Icosahedral representations and elliptic curves. manuscripta math. 117, 239–263 (2005). https://doi.org/10.1007/s00229-005-0558-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-005-0558-1

Keywords

Navigation