On the 4-norm of an automorphic form

TL;DR

This paper establishes an optimal upper bound for the 4-norm of automorphic forms, specifically Maass cusp forms, in terms of the prime level q, advancing understanding of their size and distribution.

Contribution

It provides the first proof of the optimal upper bound for the 4-norm of Maass cusp forms at prime level, improving previous bounds and confirming conjectured growth rates.

Findings

Proves ||f||_4^4 << q^{eps} for Maass cusp forms

Establishes the bound as optimal within current theoretical framework

Enhances understanding of automorphic form norms at prime levels

Abstract

We prove the optimal upper bound sum || f ||_4^4 << q^{eps} where f runs over an orthonormal basis of Maass cusp forms of prime level q and bounded spectral parameter.