Supnorm of an eigenfunction of finitely many Hecke operators

Abstract

Let $\varphi$ be a Laplace eigenfunction on a compact hyperbolic surface attached to an order in a quaternion algebra. Assuming that $\varphi$ is an eigenfunction of Hecke operators at a \emph{fixed finite} collection of primes, we prove an $L^{\infty }$-norm bound for $\varphi$ 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 Bérard in \cite{Be} to improve the archimedean amplification.

14 pages, Almost accepted version

References in corpus (1)

• Bounds for eigenforms on arithmetic hyperbolic 3-manifolds