Automorphic distributions are distributions on , invariant under the linear action of the group . Combs are characterized by the additional requirement of being measures supported in : their decomposition into homogeneous components involves the family , of Eisenstein distributions, and the coefficients of the decomposition are given as Dirichlet series . Functional equations of the usual (Hecke) kind relative to 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 sont celles invariantes par l’action linéaire du groupe . Un cas particulier est constitué par les peignes, qui sont en outre des mesures à support dans : la décomposition d’un peigne en ses composantes homogènes se fait suivant la famille , des distributions d’Eisenstein, les coefficients étant donnés par une série de Dirichlet . Les équations fonctionnelles du genre usuel (Hecke) relatives à , 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
@article{AIF_2004__54_5_1151_0, author = {Unterberger, Andr\'e}, title = {A spectral analysis of automorphic distributions and {Poisson} summation formulas}, journal = {Annales de l'Institut Fourier}, pages = {1151--1196}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {54}, number = {5}, year = {2004}, doi = {10.5802/aif.2048}, mrnumber = {2127847}, zbl = {1066.11040}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2048/} }
TY - JOUR AU - Unterberger, André TI - A spectral analysis of automorphic distributions and Poisson summation formulas JO - Annales de l'Institut Fourier PY - 2004 SP - 1151 EP - 1196 VL - 54 IS - 5 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2048/ DO - 10.5802/aif.2048 LA - en ID - AIF_2004__54_5_1151_0 ER -
%0 Journal Article %A Unterberger, André %T A spectral analysis of automorphic distributions and Poisson summation formulas %J Annales de l'Institut Fourier %D 2004 %P 1151-1196 %V 54 %N 5 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2048/ %R 10.5802/aif.2048 %G en %F AIF_2004__54_5_1151_0
Unterberger, André. 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://archive.numdam.org/articles/10.5802/aif.2048/
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