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.