Sup-norms of Eigenfunctions on Arithmetic Ellipsoids

Let B be a positive quaternion algebra, and let $\mathscr O\subseteq{\mathrm B}$ be an Eichler order. There is associated, in a natural way, a variety $X=X(\mathscr O)$ the connected components of which are indexed by the ideal classes of $\mathscr O$ and are isomorphic to spheres. This variety is naturally equipped with a Laplace operator and a large family of Hecke operators. For a joint eigenfunction φ of the Hecke algebra and of the Laplace operator with eigenvalue λ, the hybrid sup norm bound $\parallel\varphi\parallel_{\infty}\ll(tV)^{-\delta}t^{1/2}\parallel\varphi\parallel_2$ for any $\delta < \frac{1}{60}$ is shown, where t=(1+λ)^1/2 and $V={\mathrm vol}(X(\mathscr O))$.