The average of $L^4$-norms of holomorphic cusp forms

TL;DR

This paper computes the average $L^4$-norms of holomorphic cusp forms of varying weights, providing precise asymptotic formulas with power-saving error terms.

Contribution

It introduces a method to evaluate the average $L^4$-norms of holomorphic cusp forms with improved error bounds compared to previous results.

Findings

Derived an asymptotic formula for the average $L^4$-norms

Achieved power-saving error estimates

Enhanced understanding of the distribution of cusp form norms

Abstract

We find the average value of the $L^4$-norm of holomorphic Hecke cusp forms of weight within a dyadic interval, up to an error which saves a power of the weight.