$L^4$-norms of Hecke newforms of large level

TL;DR

This paper establishes a new upper bound for the $L^4$-norm of holomorphic Hecke newforms with large prime level, utilizing a sharp mean value estimate for a related GL(6) $L$-function.

Contribution

It introduces a novel upper bound for the $L^4$-norm of Hecke newforms at large prime levels, based on advanced mean value estimates for higher-rank $L$-functions.

Findings

Established a new upper bound for the $L^4$-norm of Hecke newforms.

Proved a sharp mean value estimate for a related $L$-function on GL(6).

Enhanced understanding of the behavior of automorphic forms at large levels.

Abstract

We prove a new upper bound for the $L^4$-norm of a holomorphic Hecke newform of large fixed weight and prime level $q\to \infty$. This is achieved by proving a sharp mean value estimate for a related $L$-function on GL(6)