Fast Computation of Relative Class Numbers of CM-Fields

  • Stéphane Louboutin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1838)

Abstract

Let χ be a nontrivial Hecke character on a (strict) ray class group of a totally real number field L of discriminant \(d_{\textbf{L}}\). Then, L(0, χ) is an algebraic number of some cyclotomic number field. We develop an efficient technique for computing the exact values at s = 0 of such Abelian Hecke L-functions over totally real number fields L. Let f χ denote the norm of the finite part of the conductor of χ. Then, roughly speaking, we can compute L(0, χ in \(O((d_{\textbf{L}}f_{x})^{0.5+\epsilon})\) elementary operations. We then explain how the computation of relative class numbers of CM-fields boils down to the computation of exact values at s=0 of such Abelian Hecke L-functions over totally real number fields L. Finally, we give examples of relative class number computations for CM-fields of large degrees based on computations of L(0, χ) over totally real number fields of degree 2 and 6. This paper being an abridged version of [Lou4], the reader will find there all the details glossed over here.

Keywords and phrases

CM-field relative class number Hecke L-function 

1991 Mathematics Subject Classification

Primary 11R29 11R21 11Y35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Cas.
    Cassou-Nogués, P.: Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta p-adiques. Invent. Math. 51, 29–59 (1979)MATHCrossRefMathSciNetGoogle Scholar
  2. CS.
    Coates, J., Sinnott, W.: Integrality properties of the values of partial zeta functions. Proc. London Math. Soc. 34, 365–384 (1977)MATHCrossRefMathSciNetGoogle Scholar
  3. FQ.
    Fröhlich, A., Queyrut, J.: On the functional equation of the Artin L-function for characters of real representations. Invent. Math. 20, 125–138 (1973)MATHCrossRefMathSciNetGoogle Scholar
  4. Fro.
    Fröhlich, A.: Artin-root numbers and normal integral bases for quaternion fields. Invent. Math. 17, 143–166 (1972)MATHCrossRefMathSciNetGoogle Scholar
  5. Hid.
    Hida, H.: Elementary theory of L-functions and Eisenstein series. London Mathematical Society, Student Texts 26. Cambridge University Press (1993)Google Scholar
  6. Lan.
    Lang, S.: Algebraic Number Theory, Grad. Texts Math., 2nd edn., vol. 110. Springer, Heidelberg (1994)Google Scholar
  7. LLO.
    Lemmermeyer, F., Louboutin, S., Okazaki, R.: The class number one problem for some non-Abelian normal CM-fields of degree 24. Sem. Th. Nb. Bordeaux (to appear)Google Scholar
  8. LOO.
    Louboutin, S., Okazaki, R., Olivier, M.: The class number one problem for some non-Abelian normal CM-fields. Trans. Amer. Math. Soc. 349, 3657–3678 (1997)MATHCrossRefMathSciNetGoogle Scholar
  9. Lou1.
    Louboutin, S.: Upper bounds on |L(1,\( \rm \mathcal{X}\))|and applications. Canad. Math. J. 4(50), 794–815 (1998)CrossRefMathSciNetGoogle Scholar
  10. Lou2.
    Louboutin, S.: Computation of relative class numbers of CM-fields by using Hecke L-functions. Math. Comp. 69, 371–393 (2000)MATHCrossRefMathSciNetGoogle Scholar
  11. Lou3.
    Louboutin, S.: Formulae for some Artin root numbers. Tatra Mountains Math. Publ (to appear)Google Scholar
  12. Lou4.
    Louboutin, S.: Computation of L(1,X) and of relative class numbers of CM-fields. Nagoya Math. J (to appear)Google Scholar
  13. LPCK.
    Louboutin, S., Par, Y.-H., Chang, K.-Y., Kwon, S.-H.: The class number one problem for the non Abelian normal CM-fields of degree 2pq. (1999) (preprint)Google Scholar
  14. LPL.
    Louboutin, S., Park, Y.-H., Lefeuvre, Y.: Construction of the real dihedral number fields of degree 2p Applications. Acta Arith. 89, 201–215 (1999)MATHMathSciNetGoogle Scholar
  15. Mar.
    Martinet, J.: Sur l’arithmétique des extensions à groupe de Galois diédral d’ordre 2p. Ann. Inst. Fourier (Grenoble) 19Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Stéphane Louboutin
    • 1
  1. 1.Université de CaenCaen cedexFrance

Personalised recommendations