Sharp bound for the fourth moment of holomorphic Hecke cusp forms

TL;DR

This paper establishes a conditional sharp bound for the fourth moment of holomorphic Hecke cusp forms, linking it to the Riemann Hypothesis for a specific degree 8 L-function, and employs advanced techniques from L-function moment theory.

Contribution

It provides a novel conditional bound for the fourth moment of holomorphic Hecke cusp forms using Watson's formula and recent advances in L-function moment bounds.

Findings

Conditional bound for the fourth moment established

Uses Watson's formula to connect cusp forms and L-functions

Applies Soundararajan and Harper's techniques for sharp L-function moment bounds

Abstract

We prove that the fourth moment of holomorphic Hecke cusp forms is bounded provided that the Riemann Hypothesis holds for an appropriate degree 8 L-function. We accomplish this using Watson's formula, which translates the question in hand into a moment problem for L-functions which is amenable to the techniques of Soundararajan and Harper on obtaining sharp bounds for moments of the Riemann zeta function.