Integrals of products of eigenfunctions. (English) Zbl 0833.11020

Sarnak, Peter

Int. Math. Res. Not. 1994, No. 6, 251-260 (1994).

Using the notation of the preceding review (Zbl 0833.11019), the object here is to consider \(X= \Gamma\setminus H^n\), a compact hyperbolic \(n\)-manifold. Throughout much of the paper \(n=3\). Take \(\varphi_j\) to be an orthonormal basis of \(L^2 (X)\) consisting of eigenfunctions of the Laplacian \(\Delta \varphi_j=- \lambda_j \varphi_j\). Suppose \(P= P(\varphi_1, \dots, \varphi_k)\) is a polynomial in the \(\varphi_j\). Then it is proved that there are constants \(A\) and \(B\), with \(B\) depending on \(n\) and the degree of \(P\) such that for all \(j\), \[ |\langle P, \varphi_j \rangle|\leq A(\lambda_j+ 1)^B \exp \biggl( {{-\pi \sqrt {\lambda_j}} \over 2} \biggr). \] The constant \(A\) depends on \(P\) but not on \(j\).
Some comments about hyperbolic 2-manifolds are made at the end. Here fractional integrals appear in the computations. It is stated without giving the details that one can also apply the proof to Eisenstein series \(E(z, s)\) and obtain for \(n=2\): \[ \int_T^{T+1} \Bigl|\langle \varphi^2, E( \cdot ,{\textstyle {1\over 2}}+ it)\rangle \Bigr|^2 dt\;\ll\;(T\log T)^2 e^{-\pi T}. \] This can then be used to bound Fourier coefficients \(a_n\) of Maass cusp forms \(\varphi\) on \(\Gamma \setminus H^2\), obtaining \[ |a_n |\;\ll_{\varepsilon, \varphi} |n|^{(5/12)+ \varepsilon}, \qquad \forall \varepsilon>0. \] For general \(\Gamma\) and \(\varphi\), this is the first estimate to go beyond Hecke’s. It also applies to nonarithmetic Hecke groups. For special groups much better bounds are known. See D. Bump, W. Duke, J. Hoffstein and H. Iwaniec [Int. Math. Res. Not. 1992, 75-81 (1992; Zbl 0760.11017)]. The author concludes: “One might take the new estimate above as well as the numerical calculations of [Hejhal, Rackner and Wang, Minnesota Supercomputer Research Report, 1994]as a hint that the Ramanujan conjecture, \(|a_n|\ll_\varepsilon |n|^\varepsilon\), may be a general feature having little to do with arithmetic.”
The proof for the case \(H^3\) again starts with the spectral expansion using a function \(h_T (\tau)= \exp(- (\tau- T)^2)+ \exp(- (\tau+ T)^2)\). Classical spherical harmonics and properties of the Gauss hypergeometric function are involved.

Reviewer: A.A.Terras (La Jolla)

MSC:

11F72	Spectral theory; trace formulas (e.g., that of Selberg)
35P10	Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
30F99	Riemann surfaces

Keywords:

Maass wave form; Ramanujan conjecture; eigenfunctions; Laplacian; Fourier coefficients; Maass cusp forms

Citations:

Zbl 0760.11017; Zbl 0833.11019