Advertisement

manuscripta mathematica

, Volume 54, Issue 3, pp 279–322 | Cite as

Jacobiformen und Thetareihen

  • Jürg Kramer
Article

Abstract

We give a characterisation of Jacobi forms by classical modular forms from which we obtain dimension formulas for the spaces of Jacobi forms in certain cases. Then we consider the ordinary theta series to the quaternary quadratic forms of discriminant q2 (q an odd prime) representing 2; these possess a ‘natural’ continuation to Jacobi forms for which we give a sufficient condition of linear independence. If this condition is fulfilled and if there is no cusp form of weight 4 with respect to Γo(q) which vanishes at the cusp 0 with a certain order then the classical theta series are also linear independent.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literaturverzeichnis

  1. [E1]
    M.Eichler; Quadratische Formen und orthogonale Gruppen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen Band 63, Springer-Verlag Berlin-Heidelberg-New York, 1974 (2.Auflage)Google Scholar
  2. [E2]
    M.Eichler; Einführung in die Theorie der algebraischen Zahlen und Funktionen, Birkhäuser-Verlag Basel und Stuttgart, 1963Google Scholar
  3. [E3]
    M.Eichler; Zur Zahlentheorie der Quaternionen-Algebren, J.reine angew.Math. 195 (1955), 127–151Google Scholar
  4. [E4]
    M.Eichler; Ueber die Darstellbarkeit von Modulformen durch Thetareihen, J.reine angew.Math. 195 (1955), 156–171Google Scholar
  5. [E5]
    M.Eichler; The basis problem for modular forms and the traces of the Hecke operators, in: Modular Functions of One Variable I, Lecture Notes in Math. 320, Springer-Verlag Berlin-Heidelberg-New York 1973, 75–151Google Scholar
  6. [E6]
    M.Eichler; Eine neue Klasse von Modulformen und Modulfunktionen, erscheint in Abh.Hamb.Google Scholar
  7. [EZ]
    M.Eichler and D.Zagier; On the theory of Jacobi forms I, to appear in Progress in Math., Birkhäuser-Verlag Boston-Basel-Stuttgart, 1984Google Scholar
  8. [G]
    B.Gross; Heights and the special values of L-series, to appear in Sém.Math.Sup., Presses Univ.MontréalGoogle Scholar
  9. [K1]
    W.Kohnen; Modular forms of half-integral weight on Γo(4), Math. Ann. 248 (1980), 249–266Google Scholar
  10. [K2]
    W.Kohnen; Newforms of half-integral weight, J.reine angew.Math. 333 (1982), 32–72Google Scholar
  11. [P1]
    P.Ponomarev; A correspondence between quaternary quadratic forms, Nagoya Math.J. 62 (1976), 125–140Google Scholar
  12. [P2]
    P.Ponomarev; Ternary quadratic forms and an explicit quaternary correspondence, Proceedings of the Conference on Quadratic Forms, Queen's Papers in Pure and Applied Mathematics, No.46 (1977), 582–594Google Scholar
  13. [P3]
    P.Ponomarev; Ternary quadratic forms and Shimura's correspondence, Nagoya Math.J. 81 (1981), 123–151Google Scholar
  14. [SS]
    J.-P.Serre and H. M.Stark; Modular forms of weight 1/2, in: Modular Functions of One Variable VI, Lecture Notes in Math. 627, Springer-Verlag Berlin-Heidelberg-New York 1977, 27–67Google Scholar
  15. [S1]
    G.Shimura; Introduction to the arithmetic theory of automorphic functions, Princeton University Press, 1971Google Scholar
  16. [S2]
    G.Shimura; On modular forms of half-integral weight, Ann. of Math. 97 (1973), 440–481Google Scholar
  17. [V]
    M.-F.Vignéras, Arithmétique des Algèbres de Quaternions, Lecture Notes in Math. 800, Springer-Verlag Berlin-Heidelberg-New York, 1980Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Jürg Kramer
    • 1
  1. 1.Mathematisches Institut der Universität BaselBasel

Personalised recommendations