A spectral analysis of automorphic distributions and Poisson summation formulas
Annales de l'Institut Fourier, Volume 54 (2004) no. 5, p. 1151-1196

Automorphic distributions are distributions on d , invariant under the linear action of the group SL(d,). Combs are characterized by the additional requirement of being measures supported in d : their decomposition into homogeneous components involves the family (𝔈 iλ d ) λ , 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 of the Fourier transformation. This leads to an automatic way to associate Poisson-like (or Voronoï-like) summation formulas to (holomorphic or non-holomorphic) modular forms

Les distributions automorphes sur d sont celles invariantes par l’action linéaire du groupe SL(d,). Un cas particulier est constitué par les peignes, qui sont en outre des mesures à support dans d : la décomposition d’un peigne en ses composantes homogènes se fait suivant la famille (𝔈 iλ d ) λ , des distributions d’Eisenstein, les coefficients étant donnés par une série de Dirichlet 𝒟(s). Les équations fonctionnelles du genre usuel (Hecke) relatives à 𝒟(s), peuvent se traduire en termes de l’invariance du peigne considéré par la transformation de Fourier, légèrement modifiée. Ceci conduit à une façon automatique d’associer des formules du genre de la formule de Poisson, ou de celle de Voronoï, aux formes modulaires, holomorphes ou non-holomorphes

DOI : https://doi.org/10.5802/aif.2048
Classification:  11E45,  11M36,  46F99
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     author = {Unterberger, Andr\'e},
     title = {A spectral analysis of automorphic distributions and Poisson summation formulas},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {54},
     number = {5},
     year = {2004},
     pages = {1151-1196},
     doi = {10.5802/aif.2048},
     zbl = {1066.11040},
     mrnumber = {2127847},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2004__54_5_1151_0}
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A spectral analysis of automorphic distributions and Poisson summation formulas. Annales de l'Institut Fourier, Volume 54 (2004) no. 5, pp. 1151-1196. doi : 10.5802/aif.2048. http://www.numdam.org/item/AIF_2004__54_5_1151_0/

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