Title: Quantization and non-holomorphic modular forms /
Authors: Unterberger, André.
Publisher: Springer,
Year: 2000
Subjects: Geometric quantization.; Forms, Modular.; Mathematics.; Number theory.
Abstract: 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).
Contents: 1. Introduction -- 2. Distributions associated with the non-unitary principal series -- 3. Modular distributions -- 4. The principal series of SL(2,R) and the Radon transform -- 5. Another look at the composition of Weyl symbols -- 6. The Roelcke-Selberg decomposition and the Radon transform -- 7. Recovering the Roelcke-Selberg coefficients of a function in L[superscript 2](Gamma\II) -- 8. The "product" of two Eisenstein distributions -- 9. The Roelcke-Selberg expansion of the product of two Eisenstein series: the continuous part -- 10. A digression on Kloosterman sums -- 11. The Roelcke-Selberg expansion of the product of two Eisenstein series: the discrete part -- 12. The expansion of the Poisson bracket of two Eisenstein series -- 13. Automorphic distributions on R[superscript 2] -- 14. The Hecke decomposition of products or Poisson brackets of two Eisenstein series -- 15. A generating series of sorts for Maass cusp-forms -- 16. Some arithmetic distributions -- 17. Quantization, products and Poisson brackets -- 18. Moving to the forward light-cone: the Lax-Phillips theory revisited -- 19. Automorphic functions associated with quadratic PSL(2, Z)- orbits in P[subscript 1](R) -- 20. Quadratic orbits: a dual problem -- Index of notations: page 1 -- Index of notations: page 2.