From pseudodifferential analysis to modular form theory

André Unterberger

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A spectral analysis of automorphic distributions and Poisson summation formulas

André Unterberger (2004)

Annales de l’institut Fourier

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Automorphic distributions are distributions on $ℝ^{d}$ , invariant under the linear action of the group $S L (d, ℤ)$ . Combs are characterized by the additional requirement of being measures supported in $ℤ^{d}$ : their decomposition into homogeneous components involves the family ${(𝔈_{i λ}^{d})}_{λ \in ℝ}$ , of Eisenstein distributions, and the coefficients of the decomposition are given as Dirichlet series $𝒟 (s)$ . Functional equations of the usual (Hecke) kind relative to $𝒟 (s)$ turn out to be equivalent to the invariance of the comb under some modification...