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Geodesic restrictions of arithmetic eigenfunctions. (English) Zbl 1377.11059
Let \(X\) be a compact congruence arithmetic hyperbolic surface arising from a quaternion division algebra over \({\mathbb Q}\), let \(\psi\) be a Hecke-Maass form on \(X\) whose \(L^2\)-norm is \(1\). Let \(\lambda\) be the spectral parameter of \(\psi\). Let \(\ell\) be a geodesic segment on \(X\). This paper studies the bound of the \(L^2\)-norm of \(\psi\) restricted to \(\ell\).
Previously, a special case of the paper [N. Burq et al., Duke Math. J. 138, No. 3, 445–486 (2007; Zbl 1131.35053)] shows the above \(L^2\) norm is bounded by \(\lambda^{1/4}\). The author uses the amplification method of H. Iwaniec and P. Sarnak [Ann. Math. (2) 141, No. 2, 301–320 (1995; Zbl 0833.11019)] to improve the bound to \(\lambda^{3/14+\varepsilon}\). The author also gives better bounds on Fourier coefficients of \(\psi\) along the geodesic \(\ell\).

MSC:
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F25 Hecke-Petersson operators, differential operators (one variable)
35P20 Asymptotic distributions of eigenvalues in context of PDEs
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