Journal d’Analyse Mathématique

, Volume 68, Issue 1, pp 121–143 | Cite as

Algebras of symbols and modular forms

  • André Unterberger
  • Julianne Unterberger


A formula of H. Cohen permits building a sequence of modular forms of weightsk 1+k 2+2j, j≥0, from two modular forms of weights,k 1 andk 2. We show that these bilinear products, can be interpreted as arising from the composition formula associated with a symbolic calculus of operators linked to the principal series of representations of SL(2, ℝ)


Modular Form Principal Series Composition Rule Hypergeometric Equation Symbolic Calculus 


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  1. [1]
    F. A. Berezin,Quantization, Math. USSR Izvestija38 (1974), 1109–1165.CrossRefMathSciNetGoogle Scholar
  2. [2]
    F. A. Berezin,Quantization in complex symmetric spaces, Math. USSR Izvestija39 (1975), 341–379.CrossRefMathSciNetGoogle Scholar
  3. [3]
    H. Cohen,Sums involving the values at negative integers of L-functions of quadratic characters, Math. Ann.217 (1975), 271–295.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    J. Faraut,Distributions sphériques sur les espaces hyperboliques, J. Math. Pures Appl.58 (1979), 369–444.MATHMathSciNetGoogle Scholar
  5. [5]
    I. M. Gelfand, M. I. Graev, and N. Ya. Vilenkin,Generalized Functions 5, Academic Press, New York, 1966.Google Scholar
  6. [6]
    A. W. Knapp,Representation Theory of Semi-Simple Groups, Princeton Univ. Press, Princeton, 1986.Google Scholar
  7. [7]
    W. Magnus, F. Oberhettinger and R. P. Soni,Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd edition, Springer-Verlag, Berlin, 1966.MATHGoogle Scholar
  8. [8]
    V. F. Molchanov,Quantization on the imaginary Lobachevskii plane, Funksional'nyi Analiz, Ego Prilozheniya14 (1980), 73–74.MATHMathSciNetGoogle Scholar
  9. [9]
    R. S. Strichartz,Harmonic analysis of hyperboloids, J. Funct. Anal.12 (1973), 341–383.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    A. Unterberger,Analyse harmonique et analyse pseudo-différentielle du cône de lumière, Astérisque156, Soc. Math. de France, Paris, 1987.Google Scholar
  11. [11]
    A. Unterberger,Relativity, spherical functions and the hypergeometric, equation, Ann. Inst. H. Poincaré, Phys. Théorique62 (1995), 103–144.MATHMathSciNetGoogle Scholar
  12. [12]
    A. Unterberger and J. Unterberger,Quantification et analyse pseudodifférentielle, Ann. Sci. Ec. Norm. Sup.21 (1988), 133–158.MATHMathSciNetGoogle Scholar
  13. [13]
    A. Unterberger and J. Unterberger,A quantization of the Cartan domain BDI (q=2) and operators on the light-cone, J. Funct. Anal.72 (1987), 279–319.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    A. Unterberger and J. Unterberger,Representations of SL (2, ℝ)and symbolic calculi, Integr. Equat. Oper. Th.18 (1994), 303–334.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    D. Zagier,Introduction to modular forms, inFrom Number Theory to Physics (M. Waldschmidt, P. Moussa, J.-M. Luck and C. Itzykson, eds.), Springer-Verlag, Berlin, 1992, pp. 238–291.Google Scholar

Copyright information

© The Magnes Press, The Hebrew University 1996

Authors and Affiliations

  • André Unterberger
    • 1
  • Julianne Unterberger
    • 1
  1. 1.Department of MathematicsUniversity of ReimsReims Cedex 2France

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