Quantization and Non-holomorphic Modular Forms

1. Front Matter

   Pages I-VIII

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2. Introduction

   • André Unterberger

   Pages 1-9

3. Distributions associated with the non-unitary principal series

   • André Unterberger

   Pages 11-15

4. Modular distributions

   • André Unterberger

   Pages 17-23

5. The principal series of SL(2, ℝ) and the Radon transform

   • André Unterberger

   Pages 25-31

6. Another look at the composition of Weyl symbols

   • André Unterberger

   Pages 33-44

7. The Roelcke-Selberg decomposition and the Radon transform

   • André Unterberger

   Pages 45-59

8. Recovering the Roelcke-Selberg coefficients of a function in L ²(Γ∖Π)

   • André Unterberger

   Pages 61-68

9. The “product” of two Eisenstein distributions

   • André Unterberger

   Pages 69-75

10. The roelcke-selberg expansion of the product of two eisenstein series: the continuous part

    • André Unterberger

    Pages 77-90

11. A digression on kloosterman sums

    • André Unterberger

    Pages 91-96

12. The roelcke-selberg expansion of the product of two eisenstein series: the discrete part

    • André Unterberger

    Pages 97-109

13. The expansion of the poisson bracket of two eisenstein series

    • André Unterberger

    Pages 111-117

14. Automorphic distributions on ℝ2

    • André Unterberger

    Pages 119-130

15. The Hecke decomposition of products or Poisson brackets of two Eisenstein series

    • André Unterberger

    Pages 131-147

16. A generating series of sorts for Maass cusp-forms

    • André Unterberger

    Pages 149-161

17. Some arithmetic distributions

    • André Unterberger

    Pages 163-176

18. Quantization, products and Poisson brackets

    • André Unterberger

    Pages 177-190

19. Moving to the forward light-cone: the Lax-Phillips theory revisited

    • André Unterberger

    Pages 191-212

20. Automorphic functions associated with quadratic PSL(2, ℤ)-orbits in P ₁(ℝ)

    • André Unterberger

    Pages 213-230

This is a new approach to the theory of non-holomorphic modular forms, based on ideas from quantization theory or pseudodifferential analysis. Extending the Rankin-Selberg method so as to apply it to the calculation of the Roelcke-Selberg decomposition of the product of two Eisenstein series, one lets Maass cusp-forms appear as residues of simple, Eisenstein-like, series. Other results, based on quantization theory, include a reinterpretation of the Lax-Phillips scattering theory for the automorphic wave equation, in terms of distributions on R2 automorphic with respect to the linear action of SL(2,Z).

• Boundary value problem
• Kloosterman series
• Lax-Phillips theory
• Non-holomorphic modular forms
• Rankin-Cohen products
• automorphic distributions
• scattering theory
• wave equation

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