A spectral analysis of automorphic distributions and Poisson summation formulas

André Unterberger[1]

  • [1] Université de Reims, Mathématiques (UMR 6056), Moulin de la Housse, B.P.1039, 51687 REIMS Cedex 2 (France)

Annales de l’institut Fourier (2004)

  • Volume: 54, Issue: 5, page 1151-1196
  • ISSN: 0373-0956

Abstract

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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 ∈ 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 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

How to cite

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Unterberger, André. "A spectral analysis of automorphic distributions and Poisson summation formulas." Annales de l’institut Fourier 54.5 (2004): 1151-1196. <http://eudml.org/doc/116141>.

@article{Unterberger2004,
abstract = {Automorphic distributions are distributions on $\{\mathbb \{R\}\}^d$, invariant under the linear action of the group $SL(d,\{\mathbb \{Z\}\})$. Combs are characterized by the additional requirement of being measures supported in $\{\mathbb \{Z\}\}^d$: their decomposition into homogeneous components involves the family $(\{\mathfrak \{E\}\}^d_\{i\lambda \})_\{\lambda \in \{\mathbb \{R\}\}\}$, of Eisenstein distributions, and the coefficients of the decomposition are given as Dirichlet series $\{\mathcal \{D\}\}(s)$. Functional equations of the usual (Hecke) kind relative to $\{\mathcal \{D\}\}(s)$ 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},
affiliation = {Université de Reims, Mathématiques (UMR 6056), Moulin de la Housse, B.P.1039, 51687 REIMS Cedex 2 (France)},
author = {Unterberger, André},
journal = {Annales de l’institut Fourier},
keywords = {automorphic distributions; summation formulas; Voronoï's formula; Poisson–like summation formulas},
language = {eng},
number = {5},
pages = {1151-1196},
publisher = {Association des Annales de l'Institut Fourier},
title = {A spectral analysis of automorphic distributions and Poisson summation formulas},
url = {http://eudml.org/doc/116141},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Unterberger, André
TI - A spectral analysis of automorphic distributions and Poisson summation formulas
JO - Annales de l’institut Fourier
PY - 2004
PB - Association des Annales de l'Institut Fourier
VL - 54
IS - 5
SP - 1151
EP - 1196
AB - Automorphic distributions are distributions on ${\mathbb {R}}^d$, invariant under the linear action of the group $SL(d,{\mathbb {Z}})$. Combs are characterized by the additional requirement of being measures supported in ${\mathbb {Z}}^d$: their decomposition into homogeneous components involves the family $({\mathfrak {E}}^d_{i\lambda })_{\lambda \in {\mathbb {R}}}$, of Eisenstein distributions, and the coefficients of the decomposition are given as Dirichlet series ${\mathcal {D}}(s)$. Functional equations of the usual (Hecke) kind relative to ${\mathcal {D}}(s)$ 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
LA - eng
KW - automorphic distributions; summation formulas; Voronoï's formula; Poisson–like summation formulas
UR - http://eudml.org/doc/116141
ER -

References

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  1. D. Bump, Automorphic Forms and Representations, Cambridge Series in Adv. Math 55 (1996) Zbl0868.11022MR1431508
  2. H.M. Edwards, Riemann's zeta function, (1974), Aca. Press, New York-London Zbl1113.11303
  3. G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, (1962), Oxford Univ. Press, London Zbl0423.10001MR568909
  4. D.A. Hejhal, The Selberg trace formula and the Riemann zeta function, Duke Math. J 43 (1976), 441-482 Zbl0346.10010MR414490
  5. H. Iwaniec, Introduction to the spectral theory of automorphic forms, Revista Matemática Iberoamericana, Madrid (1995) Zbl0847.11028MR1325466
  6. H. Iwaniec, Topics in Classical Automorphic Forms, 17 (1997), A.M.S., Providence Zbl0905.11023MR1474964
  7. T. Kubota, Elementary Theory of Eisenstein Series, (1973), Halsted Press, New York Zbl0268.10012MR429749
  8. P.D. Lax, R.S. Phillips, Scattering Theory for Automorphic Functions, 87 (1976), Princeton Univ.Press Zbl0362.10022MR562288
  9. W. Magnus, F. Oberhettinger, R.P. Soni, Formulas and theorems for the special functions of mathematical physics, (1966), Springer-Verlag, Berlin Zbl0143.08502MR232968
  10. A. Ogg, Modular Forms and Dirichlet Series, (1969), Benjamin Inc., New York-Amsterdam Zbl0191.38101MR256993
  11. A. Selberg, On the Estimation of Fourier Coefficients of Modular Forms, Proc. Symp. Pure Math 8 (1963), 1-15 Zbl0142.33903MR182610
  12. A. Selberg, Old and new conjectures and results about a class of Dirichlet series, Proc. of the Amalfi Conf. 1989, Univ. of Salerno (1992) Zbl0787.11037
  13. J.P. Serre, Cours d'Arithmétique, (1970), Presses Univ. de France, Paris Zbl0225.12002MR255476
  14. G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, (1995), Soc. Math. France, Paris Zbl0880.11001MR1366197
  15. A. Terras, Harmonic analysis on symmetric spaces and applications. I., (1985), Springer-Verlag, New York-Berlin-Heidelberg-Tokyo Zbl0574.10029MR791406
  16. A. Terras, Harmonic analysis on symmetric spaces and applications. II., (1988), Springer-Verlag, New York-Berlin-Heidelberg-Tokyo Zbl0668.10033MR955271
  17. A. Unterberger, Quantization and non-holomorphic modular forms, 1742, Springer-Verlag, Berlin-Heidelberg Zbl0970.11014MR1783191
  18. A. Unterberger, Automorphic pseudodifferential analysis and higher-level Weyl calculi, 209 (2002), Birkhäuser, Basel-Boston-Berlin Zbl1018.11018MR1956320
  19. G. VoronoÏ, Sur le développement, à l’aide des fonctions cylindriques, des sommes doubles ∑ f ( p m 2 , q m n + r n 2 ) , Ver. Math. Kongr. Heidelberg (1904), 241-245 

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