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 $\Vert y^{k}f(z){\Vert }_{\infty }\ll N^{-1/6+\epsilon }$ where the trivial bound is $\Vert y^{k}f(z){\Vert }_{\infty }\ll 1$. This result is an analog of a similar bound in Maaß form case.

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• Level Aspect Subconvexity For Rankin-Selberg $L$-functions