Unterberger, André A spectral analysis of automorphic distributions and Poisson summation formulas. (English) Zbl 1066.11040 Ann. Inst. Fourier 54, No. 5, 1151-1196 (2004). Define \(\mathcal{F}\) to be the Fourier transform on \(\mathbb{R}^{d}\) and \({\mathcal E}\) to be the Euler operator, \[ \mathcal{E}= \frac{1}{2\pi i} \left( \sum x_{j}\frac{\partial}{\partial x_{j}}+\frac{d}{2 }\right). \] If \(h\) is a Schwartz function on \(\mathbb{R}^{d}\), set \( t^{2\pi i\mathcal{E}}h(x)=t^{d/2}h(tx)\), for \(t>0.\) If \(\mathcal{S}\) is a tempered distribution, set \(\left\langle t^{2\pi i\mathcal{E}}\mathcal{S} ,h\right\rangle =\left\langle \mathcal{S},t^{-2\pi i\mathcal{E} }h\right\rangle .\) For a sequence of complex numbers \(\mathbf{a} =(a_{n})_{n\geq 1}\) such that the series \(D_{\mathbf{a}}(s)=\sum_{n\geq 1}\frac{a_{n}}{n^{s}}\) is nice, set \(D_{\mathbf{b}}(a)(s)=\frac{D_{ \mathbf{a}}(s)}{\zeta (s)},\) where \(\zeta (s)\) is Riemann’s zeta function. Define the Eisenstein distribution \(\mathfrak{E}_{\nu }^{d}\) for Schwartz functions \(h\) by \[ \left\langle \mathfrak{E}_{\nu }^{d},h\right\rangle =\sum_{m\in \mathbb{Z}^{d}-\{0\}}\int_{0}^{\infty }t^{-\nu +\frac{d}{2}}h(tm) \frac{dt}{t}. \] This converges for \(\text{Re}\nu <-d/2\) where it is an SL\((d,\mathbb{Z})\) -invariant, even, tempered distribution. For \(h(x)=e^{-\parallel x\parallel ^{2}}\) this is an Eisenstein series for SL\((d,\mathbb{Z}).\)Theorem 2.5 of the paper under review says \(\mathfrak{E}_{\nu }^{d}\) extends as a tempered distribution to a meromorphic function of \(\nu \) whose only poles are at \(\nu =\pm d/2.\) One has \(\mathcal{F}\mathfrak{E}_{\nu }^{d}=\mathfrak{E}_{-\nu }^{d},\nu \neq \pm d/2.\) Proposition 2.6 says if \(r(m)\) is the greatest common divisor of the entries of \(m\in \mathbb{Z}^{d},\) and \(h\) is a Schwartz function on \(\mathbb{R}^{d}, \) then assuming the sequence \(\mathbf{a}=(a_{n})_{n\geq 1}\) has at most polynomial increase and the Dirichlet series \(D_{\mathbf{a}}(s)\) has at most polynomial increase on vertical strips and that the function \(D_{ \mathbf{b}}(a)(s)=\frac{D_{a}(s)}{\zeta (s)}\) extends as a meromorphic function in the plane, with no pole with real part \(\geq d/2,\) except possibly for a pole at \(s=d\), one has \[ 2\pi \sum_{m\in \mathbb{Z}^{d}-\{0\}}a_{r(m)}h(m)=\int_{- \infty }^{\infty }D_{\mathbf{b}}(\frac{d}{2}-i\lambda )\left\langle \mathfrak{E}_{i\lambda }^{d},h\right\rangle d\lambda +2\pi \text{Re} s_{s=d}\left( D_{\mathbf{b}}(s)\left\langle \mathfrak{E}_{\frac{d}{2} -s}^{d},h\right\rangle \right). \] The right hand side makes sense when \(h\) is slightly more general than a Schwartz function and can then serve to define the left hand side. Section 3 applies the theory to sequences \(a=(a_{n})_{n\geq 1}\) coming from Fourier coefficients of holomorphic modular forms \(f(z)\), \(z\) in the Poincaré upper half plane; i.e., \( f(z)=f(z+2)\) and \(f(-1/z)=\kappa \left( \frac{z}{i}\right) ^{wd}f(z),\kappa =\pm 1.\) Suppose \[ f(z)=f_{0}+\sum_{n\geq 1}f_{n}e^{\pi inz}. \] Set \(c_{0}=-f_{0}\) and for \(m\neq 0,\) set \(c_{m}=\sum\limits_{\underset{ n\geq 1} { n^w| m_j,all j}}f_{n}.\) Let \(\Phi \) be a radial Schwartz function (or the slightly weakened definition mentioned in Prop. 2.6 above). Define \(\Psi =\mathcal{K}_{d,w}\mathcal{F}\Phi ,\) where (with \(\mathcal{F}=\) Fourier transform and \(\mathcal{E}=\) Euler’s operator) and \(\mathcal{K}_{d,w}\) for Schwartz functions \(h\), given by \[ \mathcal{K}_{d,w}h(x)=2\pi \int_{0}^{\infty }t^{-wd}h(t^{-2w}x)J_{wd-1}(2wt)\,dt. \] Theorem 3.1 says then that \[ \sum_{m\in \mathbb{Z}^{d}}c_{m}\Phi (tm)=\kappa t^{-d}\sum_{m\in \mathbb{Z}^{d}}c_{m}\Psi \left( \frac{1}{t}m\right) . \] Section 4 extends the results to Maass wave forms. Section 5 finds the analog of the Gaussian for the transformation \(\mathcal{K}_{d,w}\mathcal{F},\) also analogs of Hermite functions. Section 6 gives general remarks on automorphic forms for \(GL(d,\mathbb{Z})\). In particular, it is noted that the present generalizations of Poisson summation are not related to non-Euclidean Poisson summation considered by the reviewer in her book “Harmonic analysis on symmetric spaces and applications. I, II”, Springer-Verlag (1985; Zbl 0574.10029), (1988; Zbl 0668.10033). Reviewer: Audrey A. Terras (La Jolla) Cited in 2 Documents MSC: 11M41 Other Dirichlet series and zeta functions 11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas) 46F99 Distributions, generalized functions, distribution spaces Keywords:Poisson–like summation formulas Citations:Zbl 0574.10029; Zbl 0668.10033 × Cite Format Result Cite Review PDF Full Text: DOI Numdam Numdam EuDML References: [1] [1] , Automorphic Forms and Representations, Cambridge Series in Adv. Math55 (1996) · Zbl 0868.11022 [2] [2] , Riemann’s zeta function, Aca. Press, 1974 · Zbl 1113.11303 [3] [3] & , An Introduction to the Theory of Numbers, fourth edition, Oxford Univ. Press, 1962 · Zbl 0423.10001 [4] [4] , The Selberg trace formula and the Riemann zeta function, Duke Math. J43 (1976) no.3 p. 441-482 · Zbl 0346.10010 [5] [5] , Introduction to the spectral theory of automorphic forms, Revista Matemática Iberoamericana, Madrid (1995) · Zbl 0847.11028 [6] [6] , Topics in Classical Automorphic Forms, Graduate Studies in Math 17, A.M.S., 1997 · Zbl 0905.11023 [7] [7] , Elementary Theory of Eisenstein Series, Kodansha Ltd, Tokyo, Halsted Press, 1973 · Zbl 0268.10012 [8] [8] & , Scattering Theory for Automorphic Functions, Ann. Math. Studies 87, Princeton Univ.Press, 1976 · Zbl 0362.10022 [9] [9] , & , Formulas and theorems for the special functions of mathematical physics, 3rd edition, Springer-Verlag, 1966 · Zbl 0143.08502 [10] [10] , Modular Forms and Dirichlet Series, Benjamin Inc., 1969 · Zbl 0191.38101 [11] [11] , On the Estimation of Fourier Coefficients of Modular Forms, Proc. Symp. Pure Math8 (1963) p. 1-15 · Zbl 0142.33903 [12] [12] , Old and new conjectures and results about a class of Dirichlet series, 1992 · Zbl 0787.11037 [13] [13] , Cours d’Arithmétique, Presses Univ. de France, 1970 · Zbl 0225.12002 [14] [14] , Introduction à la théorie analytique et probabiliste des nombres, Cours spécialisés, Soc. Math. France, 1995 · Zbl 0880.11001 [15] [15] , Harmonic analysis on symmetric spaces and applications. I., Springer-Verlag, 1985 · Zbl 0574.10029 [16] [16] , Harmonic analysis on symmetric spaces and applications. II., Springer-Verlag, 1988 · Zbl 0668.10033 [17] [17] , Quantization and non-holomorphic modular forms, Lecture Notes in Math 1742, Springer-Verlag, · Zbl 0970.11014 [18] [18] , Automorphic pseudodifferential analysis and higher-level Weyl calculi, Progress in Math 209, Birkhäuser, 2002 · Zbl 1018.11018 [19] [19] , Sur le développement, à l’aide des fonctions cylindriques, des sommes doubles \(\sum f(pm^2,qmn+rn^2), 1904\), p. 241-245 · JFM 36.0516.02 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.