On the asymptotic formula of Kuznetsov and existence of generic cusp forms
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 real rank one by \textit{R. J. Miatello} and \textit{N. R. Wallach} [J. Funct. Anal. 93, No. 1, 171-206 (1990; Zbl 0711.11023); see formula (1.11)]. For the case \(\Gamma\subset \text{PSL}_ 2 (\mathbb R)\) (first layer of interpretation) the author chooses explicit test functions, and studies the integral transform occurring in the Kloosterman term of the sum formula. In \S3 of the paper he uses this to obtain the main result. Special care is given to the Kloosterman term, as for general \(\Gamma\) only trivial estimates of Kloosterman sums can be used. (I am not convinced by some of the steps in this part of the proof.) With some modifications, indicated in \S4, the same method is used in the case \(\Gamma\subset \text{SU}(1,n)\).
mean value of squares of Fourier coefficients
scientific article; zbMATH DE number 561307
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