Quantization and Non-holomorphic Modular Forms

André Unterberger (auth.)9783540678618, 3-540-67861-1

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).

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Table of contents :

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 –