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).
Author(s): André Unterberger (auth.)
Series: Lecture Notes in Mathematics 1742
Edition: 1
Publisher: Springer Berlin Heidelberg
Year: 2000
Language: English
Pages: X, 258 p.
City: Berlin; New York
Content:
Front Matter....Pages -
Introduction....Pages 1-9
Distributions associated with the non-unitary principal series....Pages 11-15
Modular distributions....Pages 17-23
The principal series of SL(2, ℝ) and the Radon transform....Pages 25-31
Another look at the composition of Weyl symbols....Pages 33-44
The Roelcke-Selberg decomposition and the Radon transform....Pages 45-59
Recovering the Roelcke-Selberg coefficients of a function in L 2(Γ∖� )....Pages 61-68
The “product” of two Eisenstein distributions....Pages 69-75
The roelcke-selberg expansion of the product of two eisenstein series: the continuous part....Pages 77-90
A digression on kloosterman sums....Pages 91-96
The roelcke-selberg expansion of the product of two eisenstein series: the discrete part....Pages 97-109
The expansion of the poisson bracket of two eisenstein series....Pages 111-117
Automorphic distributions on ℝ2 ....Pages 119-130
The Hecke decomposition of products or Poisson brackets of two Eisenstein series....Pages 131-147
A generating series of sorts for Maass cusp-forms....Pages 149-161
Some arithmetic distributions....Pages 163-176
Quantization, products and Poisson brackets....Pages 177-190
Moving to the forward light-cone: the Lax-Phillips theory revisited....Pages 191-212
Automorphic functions associated with quadratic PSL(2, ℤ)-orbits in P 1(ℝ)....Pages 213-230
Quadratic orbits: a dual problem....Pages 231-246
Back Matter....Pages -
