Advertisement

Cohen–Lenstra Heuristics of Quadratic Number Fields

  • Étienne Fouvry
  • Jürgen Klüners
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4076)

Abstract

We establish a link between some heuristic asymptotic formulas (due to Cohen and Lenstra) concerning the moments of the p–part of the class groups of quadratic fields and formulas giving the frequency of the values of the p–rank of these class groups.

Furthermore we report on new results for 4–ranks of class groups of quadratic number fields.

Keywords

Abelian Group Limit Point Class Group Vector Subspace Quadratic Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cohen, H., Lenstra, H.W.: Heuristics on class groups of number fields. In: Number theory, Noordwijkerhout 1983. Lecture Notes in Math., vol. 1068, pp. 33–62. Springer, Berlin (1984)CrossRefGoogle Scholar
  2. 2.
    Davenport, H., Heilbronn, H.: On the density of discriminants of cubic fields II. Proc. Roy. Soc. London Ser. A 322(1551), 405–420 (1971)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Fouvry, E., Klüners, J.: On the 4–rank of class groups of quadratic number fields (preprint, 2006)Google Scholar
  4. 4.
    Gerth III, F.: The 4-class ranks of quadratic fields. Invent. Math. 77(3), 489–515 (1984)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Gerth III, F.: Extension of conjectures of Cohen and Lenstra. Exposition. Exposition. Math. 5(2), 181–184 (1987)MATHMathSciNetGoogle Scholar
  6. 6.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers. Oxford University Press, Oxford (1975)Google Scholar
  7. 7.
    Heath–Brown, D.R.: The size of Selmer groups for the congruent number problem II. Inv. Math. 118, 331–370 (1994)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Redei, L.: Arithmetischer Beweis des Satzes über die Anzahl der durch vier teilbaren Invarianten der absoluten Klassengruppe im quadratischen Zahlkörper. J. Reine Angew. Math. 171, 55–60 (1934)CrossRefGoogle Scholar
  9. 9.
    Rudin, W.: Real and Complex Analysis, 2nd edn. McGraw–Hill Book Company, New York (1974)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Étienne Fouvry
    • 1
  • Jürgen Klüners
    • 2
  1. 1.Mathématique, Bât. 425Univ. Paris–SudORSAYFrance
  2. 2.Fachbereich Mathematik/InformatikUniversität KasselKasselGermany

Personalised recommendations