On the asymptotic formula of Kuznetsov and existence of generic cusp forms
type: Item
instance of: scholarly article
title: On the asymptotic formula of Kuznetsov and existence of generic cusp forms
zbMATH Open document ID: 0810.11029
DOI: 10.1007/BF03025738
author: Andre Reznikov
published in: Mathematische Zeitschrift
publication date: 1994-05-19T00:00:00Z
full work available at URL: https://eudml.org/doc/174546
review text: The author generalizes an asymptotic formula of Kuznetsov for the mean value of squares of Fourier coefficients of cuspidal Maass forms; see Theorem 6 in \textit{N. V. Kuznetsov} [Mat. Sb., Nov. Ser. 111(153), 334-383 (1980; Zbl 0427.10016)]. The main result of this paper is \[ \sum_ j {{| a_ \psi (\varphi_ j)|^ 2} \over {\cos\pi t_ j}} e^{-\delta t_ j}+ \sum_ l \int_{-\infty}^ \infty | Z^ l (ir\psi)|^ 2 e^{-\delta r} dr= {1\over {\pi^ 2 \delta^ 2}}+ O(1/ \delta+| \psi|^ K) \] as \(\delta\downarrow 0\), for some \(K>0\). The paper gives three layers of interpretation of this formula. In the first layer \(\psi\in \mathbb Z\setminus \{0\}\). The \(\varphi_ j\) run through a basis of cusp forms of weight zero for a cofinite discrete subgroup \(\Gamma \subset \text{PSL}_ 2 (\mathbb R)\). The eigenvalue is \({1\over 4}+ t_ j^ 2\), and \(a_ \psi (\varphi_ j)\) is the \(\psi\)- th Fourier coefficient. The \(Z^ l (ir,\psi)\) are Fourier coefficients of Eisenstein series; \(l\) runs through the cusps of \(\Gamma\). In the second layer, \(\Gamma\) is a discrete subgroup of \(\text{SU} (1,n)\), the \(\varphi_ j\) are eigenfunctions of the Laplacian with eigenvalue \(4n^ 2+ t_ j^ 2\), and again the \(a_ \psi\) and \(Z^ i (r,\psi)\) are Fourier coefficients of the \(\varphi_ j\), respectively of Eisenstein series. The third layer concerns the case of discrete subgroups of an arbitrary Lie group of real rank one. Proofs are given for the first two layers of interpretation. Kuznetsov's original result concerns layer one, with \(\Gamma= \text{PSL}_ 2 (\mathbb R)\). For that case estimates of \(Z(ir,\psi)\) are known that allow the Eisenstein terms to be absorbed into the error term. Analogous estimates are given for the case that \(\Gamma\subset \text{SU}(1,2)\) is a congruence subgroup. This leads to the corollary that in this case there are infinitely many \(\varphi_ j\) with non-zero Fourier coefficients. Basis of the proof is Kuznetsov's sum formula, generalized for groups of r…
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 11F37
Mathematics Subject Classification ID: 11F55
Mathematics Subject Classification ID: 11F72
zbMATH DE Number: 561307
zbMATH Keywords: generic cusp forms
zbMATH Keywords: Kloosterman sums
zbMATH Keywords: asymptotic formula
zbMATH Keywords: mean value of squares of Fourier coefficients
zbMATH Keywords: cuspidal Maass forms
zbMATH Keywords: Kuznetsov's sum formula
reviewed by: Roelof Bruggeman
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 561307