Large values of eigenfunctions on arithmetic hyperbolic surfaces
Let \(K\) be a totally real number field of narrow class number one, and let \({\mathfrak o}=R_k\) be its ring of integers. Let \(A\) be a quaternion algebra over \(K\) ramified at all infinite places except one, let \({\mathcal O}\) be an Eichler \({\mathfrak o}\)-order in \(A\), and let \({\mathcal O}^1\) be the group of elements of norm one. Let \(\rho\) be a \(K\)-embedding of \(A\) into \(M_2({\mathbb R})\), and let \(\Gamma=P\rho({\mathcal O}^1)<\text{PSL}_2({\mathbb R}) \) be the corresponding arithmetic Fuchsian group, acting on the upper half-space \(\mathfrak h\). The Hilbert space \(L_0^2(\Gamma /{\mathfrak h} )=L^2((\Gamma /{\mathfrak h}))\), if \(\Gamma\) is cocompact, \(L_0^2(\Gamma /{\mathfrak h} )=(L^2_{\text{Eis}})^\bot\), if \(\Gamma\) is non-cocompact of finite volume, has an orthonormal basis decomposition \(L_0^2(\Gamma /{\mathfrak h} )=\overline{\oplus_{j\geq 0}{\mathbb C}\phi_j}\), where \( \phi_j\) are Hecke-Maass eigenforms (for \( j\geq j_0\)), with \((\Delta+\lambda_j)\phi_j=0\), \(\Delta\) the Laplacian. The author proves that for every fixed CM-point \(z\in \mathfrak h\), we have, as \(j\to\infty\), \[ |\phi_j(z)|=\Omega\left(\exp\left(\sqrt{\frac{\log\lambda_j}{\log\log\lambda_j}}\left(1+O\left(\frac{\log\log\log \lambda_j}{\log\log\lambda_j}\right)\right)\right)\right). \]
scientific article; zbMATH DE number 5834471
scientific article