Hybrid sup-norm bounds for Hecke–Maass cusp forms

Abstract

Let $f$ be a Hecke–Maass cusp form of eigenvalue $\lambda$ and square-free level $N$. Normalize the hyperbolic measure such that $\mathrm{vol}(Y_0(N))=1$ and the form $f$ such that $\Vert f{\Vert }_2=1$. It is shown that $\Vert f{\Vert }_{\infty }{\ll }_{ϵ}{\lambda }^{\frac{5}{24}+ϵ}{N}^{\frac{1}{3}+ϵ}$ for all $ϵ>0$. This generalizes simultaneously the current best bounds in the eigenvalue and level aspects.