Supnorm of an eigenfunction of finitely many Hecke operators

TL;DR

This paper establishes a logarithmically improved $L^$-norm bound for Laplace eigenfunctions on hyperbolic surfaces that are eigenfunctions of Hecke operators at finitely many primes, using an amplifier method.

Contribution

It introduces a new amplifier construction tailored to finitely many Hecke operators and applies Be9rard's method to enhance archimedean amplification.

Findings

Achieved a power of logarithm improvement over trivial bounds.

Constructed an amplifier with support on Hecke trees.

Applied Be9rard's method to improve archimedean amplification.

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.