\(L^ \infty\)-norms of eigenfunctions for arithmetic hyperbolic 3-manifolds
Let \(\Gamma\) be an arithmetic subgroup of \(SL_2 (\mathbb{R})\). Then \(\Gamma\) acts on the hyperbolic 2-space \({\mathfrak h}\) with quotient \(X_\Gamma= \Gamma \backslash {\mathfrak h}\) of finite volume. For \(\lambda\geq 0\) let \(\text{Eig} (\lambda)\) denote the \(\lambda \)-eigenspace of the Laplace-Beltrami-operator on \(X_\Gamma\). Let \[ M_\infty (\lambda): =\sup \left\{\left. {|\Phi |_\infty \over|\Phi|_2} \right|\Phi\in \text{Eig} (\lambda) \right\}. \] \textit{H. Iwaniec} and \textit{P. Sarnak} showed in [\(L^\infty\)-norms of eigenfunctions of arithmetic surfaces, Ann. Math., II. Ser. 141, No. 2, 301-320 (1995; Zbl 0833.11019)] that for \(\Gamma\) being the full norm-1-group of a maximal order in a quaternion algebra over \(\mathbb{Q}\) which splits at \(\infty\), we have \[ M_\infty (\lambda)= O\bigl(\lambda^{5/24+ \varepsilon} \sqrt{\text{dim Eig} (\lambda)}\bigr) \] for any \(\varepsilon >0\). Since this result is much sharper than what holds for more general manifolds, it is clear that one will need the arithmetic structure for the proof. In this case a careful analysis of the action of the Hecke-algebra yields the result. In the present paper these methods are transferred to the three dimensional case. For \(\Gamma\) being as above for a quaternion algebra over \(\mathbb{Q}(i)\) one gets: \[ M_\infty(\lambda) =O\bigl(\lambda^{37/76+ \varepsilon} \sqrt{\text{dim Eig} (\lambda)} \bigr). \]
eigenspace of the Laplace-Beltrami-operator
scientific article; zbMATH DE number 784721
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