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On the fourth moment of Hecke-Maass forms and the random wave conjecture. (English) Zbl 1398.11072

The paper under review studies the behavior of the fourth moment of a Hecke-Maass cusp form for \(\Gamma=\mathrm{SL}_2(\mathbb{Z})\) with large spectral parameter. The main theorem (Theorem 1.1) is as follows:
Assume the generalized Lindelöf hypothesis (GLH). Let \(f\) be a Hecke-Maass cusp form for \(\Gamma\) with Laplacian eigenvalue \(1/4+T^2\) \((T>0)\), normalized such that \[ \int_{\Gamma\backslash\mathbb{H}}f(z)^2\frac{dxdy}{y^2}=\mathrm{Vol}(\Gamma\backslash\mathbb{H}). \] Then there exists a constant \(\delta>0\) such that \[ \mathrm{Vol}(\Gamma\backslash\mathbb{H})^{-1}\int_{\Gamma\backslash\mathbb{H}}f(z)^4\frac{dxdy}{y^2}=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}t^4e^{-t^2/2}dt+O(T^{-\delta}). \]
By the spectral expansion and Parseval, the authors reduce the fourth moment to the estimate of several terms, among which the most interesting is \[ \sum_{j\geq1}|\langle f^2,u_j\rangle|^2. \] This sum from the discrete spectrum is, by Watson’s formula, related to the triple product \(L\)-function \(L(s,f\times f\times \bar{u_j})\). Using this as a starting point, and with sophisticated analysis (using Kuznetsov’s formula for Kloosterman sums, spectral large sieve, Voronoi summation, etc.) the authors prove Proposition 2.2 that under GLH there exists some \(\delta>0\) such that \(\sum_{j\geq1}|\langle f^2,u_j\rangle|^2=\frac{2}{3}\pi+O(T^{-\delta})\).
Based on the widely believed GLH, the main result of this paper adds another piece of evidence for the random wave conjecture to hold.

MSC:

11F12 Automorphic forms, one variable
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
81Q50 Quantum chaos

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References:

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