Sup-norms of Eigenfunctions on Arithmetic Ellipsoids

Let B be a positive quaternion algebra, and let O subset of B be an Eichler order. There is associated, in a natural way, a variety X = X(O) the connected components of which are indexed by the ideal classes of 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 phi of the Hecke algebra and of the Laplace operator with eigenvalue lambda, the hybrid sup norm bound parallel to phi parallel to(infinity) < (tV)(-delta)t(1/2)parallel to phi parallel to(2) for any delta < 1/60 is shown, where t = (1 + lambda)(1/2) and V = vol(X(O)).