Quantization and Non-holomorphic Modular Forms [Paperback]

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).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 L2 of the fundamental domain 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 R2 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 P1(R) 20. Quadratic orbits: a dual problem Index of notations: page 1 Index of notations: page 2 Subject index ReferencesSpringer Book Archives