Advertisement

Generating class fields using Shimura reciprocity

  • Alice Gee
  • Peter Stevenhagen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1423)

Abstract

The abelian extensions of an imaginary quadratic field can theoretically be generated by the values of the modular j-function, but these values are too large to be useful in practice. We show how Shimura's reciprocity law can be applied to find small generators for these extensions, and to compute the corresponding irreducible polynomials.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Birch B.: Weber's class invariants. Mathematika 16 (1969) 283–294MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    Cox D.A.: Primes of the form x 2 + ny 2. Wiley-Interscience (1989)Google Scholar
  3. [3]
    Gee, A.C.P.: thesis Universiteit van Amsterdam (in preparation)Google Scholar
  4. [4]
    Lang, S.: Elliptic functions, 2nd edition. Springer Graduate Text in Mathematics 112 (1987)Google Scholar
  5. [5]
    Schertz, R.: Die singulÄren Werte der Weberschen Funktionen f, f1, f223. J. Reine Angew. Math. 286/287 (1976) 46–74MathSciNetGoogle Scholar
  6. [6]
    Schertz, R.: Problèmes de construction en multiplication complexe. Sém. Théor. Nombres Bordeaux, (2) 4 (1992) no. 2, 239–262MATHMathSciNetGoogle Scholar
  7. [7]
    Shimura G.: Introduction to the Arithmetic Theory of Automorphic Functions. Iwanami Shoten and Princeton University Press (1971)Google Scholar
  8. [8]
    Shimura, G.: Complex Multiplication. Modular functions of One Variable I Springer Lecture Notes in Mathematics 320 (1973)Google Scholar
  9. [9]
    Stark, H.M. On a ”gap” in a theorem of Heegner. J. Number Theory 1 (1969) 16–27MATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    Washington, L.C.: Introduction to Cyclotomic Fields, 2nd edition. Springer Graduate Text in Mathematics 83 (1997)Google Scholar
  11. [11]
    Weber, H.: Lehrbuch der Algebra, Vol. III. Chelsea reprint original edition 1908Google Scholar
  12. [12]
    Yui, N. and Zagier, D.: On the singular values of Weber modular functions. Math. Comp. 66 (1997) no. 20, 1645–1662MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Alice Gee
    • 1
  • Peter Stevenhagen
    • 1
  1. 1.Universiteit van AmsterdamTV AmsterdamThe Netherlands

Personalised recommendations