The Sup-norm of Holomorphic Cusp Forms

Abstract

Let $f$ be a normalized holomorphic cusp form with a square-free level $N$ and weight $k$. Using a pre-trace formula, we establish a sup-norm bound of $f$ such that $\|y^kf(z)\|_{\infty} \ll N^{-1/6+\epsilon}$ where the trivial bound is $\|y^kf(z)\|_{\infty} \ll 1$. This result is an analog of a similar bound in Maa{\ss} form case.