Quantization and Non-holomorphic Modular Forms

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).

Distributions associated with the non-unitary principal series. - Modular distributions. - The principal series of SL(2, ?) and the Radon transform. - Another look at the composition of Weyl symbols. - The Roelcke-Selberg decomposition and the Radon transform. - Recovering the Roelcke-Selberg coefficients of a function in L 2(? ? ?). - The product of two Eisenstein distributions. - The roelcke-selberg expansion of the product of two eisenstein series: the continuous part. - A digression on kloosterman sums. - The roelcke-selberg expansion of the product of two eisenstein series: the discrete part. - The expansion of the poisson bracket of two eisenstein series. - Automorphic distributions on ? 2. - The Hecke decomposition of products or Poisson brackets of two Eisenstein series. - A generating series of sorts for Maass cusp-forms. - Some arithmetic distributions. - Quantization, products and Poisson brackets. - Moving to the forward light-cone: the Lax-Phillips theory revisited. - Automorphic functions associated with quadratic PSL(2, ?)-orbits in P 1(?). - Quadratic orbits: a dual problem.