Supnorm of an eigenfunction of finitely many Hecke operators

Abstract

Let $\phi$ be a Laplace eigenfunction on a compact hyperbolic surface attached to an order in a quaternion algebra. Assuming that $\phi$ is an eigenfunction of Hecke operators at a \emph{fixed finite} collection of primes, we prove an $L^\infty$-norm bound for $\phi$ that improves upon the trivial estimate by a power of the logarithm of the eigenvalue. We have constructed an amplifier whose length depends on the support of the amplifier on Hecke trees. We have used a method of B\'erard in \cite{Be} to improve the archimedean amplification.