On the Structure of Operators with Automorphic Symbols

  • André Unterberger
Part of the Operator Theory: Advances and Applications book series (OT, volume 210)


We consider automorphic distributions on ℝ2, a concept slightly more precise than that of automorphic functions on the hyperbolic half-plane: these objects introduce themselves in a natural way in symbolic calculi of operators such as Weyl’s or the horocyclic calculus, linked to the projective discrete series of SL(2, ℝ). We interpret in operator- theoretic terms various constructions of number-theoretic interest, some of which raise quite deep questions.

Mathematics Subject Classification (2000)

11F11 11F27 47G30 


Automorphic distributions horocyclic calculus Berezin calculus 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    D. Bump, Automorphic Forms and Representations, Cambridge Series in Adv.Math. 55, Cambridge, 1996.Google Scholar
  2. [2]
    H. Iwaniec, Topics in Classical Automorphic Forms, Graduate Studies in Math. 17, A.M.S., Providence, 1997.Google Scholar
  3. [3]
    P.D. Lax, R.S. Phillips, Scattering Theory for Automorphic Functions, Ann.Math. Studies 87, Princeton Univ. Press, 1976.Google Scholar
  4. [4]
    W. Magnus, F. Oberhettinger, R.P. Soni, Formulas and theorems for the special functions of mathematical physics, 3rd edition, Springer-Verlag, Berlin, 1966.zbMATHGoogle Scholar
  5. [5]
    G. Shimura, Modular Forms of half-integral weight, Lecture Notes in Math. 320, Springer-Verlag, Berlin-Heidelberg-New York, 1973.Google Scholar
  6. [6]
    A. Unterberger, Quantization and non-holomorphic modular forms, Lecture Notes in Math. 1742, Springer-Verlag, Berlin-Heidelberg, 2000.Google Scholar
  7. [7]
    A. Unterberger, Automorphic pseudodifferential analysis and higher-level Weyl calculi, Progress in Math., Birkhäuser, Basel-Boston-Berlin, 2002.Google Scholar
  8. [8]
    A. Unterberger, Quantization and Arithmetic, Progress in Math., Birkhäuser, Basel-Boston-Berlin, 2008.Google Scholar
  9. [9]
    N. Vasilevski, Bergman space structure, commutative algebras of Toeplitz operators, and hyperbolic geometry, Integral Equations and Operator Theory 46,2 (2003), 235–251.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • André Unterberger
    • 1
  1. 1.Mathématiques (FRE 3111)Université de ReimsREIMS Cedex 2France

Personalised recommendations