Bounding sup-norms of cusp forms of large level

TL;DR

This paper establishes new bounds on the maximum size of cusp forms of large square-free level, improving understanding of their sup-norms and deriving hybrid bounds involving eigenvalues and level.

Contribution

It provides the first non-trivial bound on the sup-norm of cusp forms of large level, extending previous results to a broader class of forms and deriving hybrid bounds.

Findings

Sup-norm bound:  f _{ abla} ll N^{-1/37}

Hybrid bound involving eigenvalue  abla ll  abla^{1/4} (N abla)^{-}

Applicable to holomorphic cusp forms with explicit dependence on weight k

Abstract

Let f be an $L^2$-normalized weight zero Hecke-Maass cusp form of square-free level N, character $\chi$ and Laplacian eigenvalue $\lambda\geq 1/4$. It is shown that $\| f \|_{\infty} \ll_{\lambda} N^{-1/37}$, from which the hybrid bound $\|f \|_{\infty} \ll \lambda^{1/4} (N\lambda)^{-\delta}$ (for some $\delta > 0$) is derived. The first bound holds also for $f = y^{k/2}F$ where F is a holomorphic cusp form of weight k with the implied constant now depending on k.