The eighth moment of the family of $\Gamma_1(q)$-automorphic $L$-functions

TL;DR

This paper establishes a groundbreaking average bound for the eighth moment of a family of automorphic $L$-functions on $GL(2)$, overcoming previous limitations and introducing a novel proof technique.

Contribution

It provides the first proven bound for the eighth moment of these $L$-functions, using a new approach that addresses orthogonality challenges.

Findings

First bound for the eighth moment of the family

New method overcoming orthogonality issues

Extends understanding of automorphic $L$-functions

Abstract

We prove a Lindel\"of on average bound for the eighth moment of a family of $L$-functions attached to automorphic forms on $GL(2)$, the first time this has been accomplished. Previously, such a bound had been proven for the sixth moment for our family by Djankovi\'c and for a similar family by Young. Our proof rests on a new approach which overcomes the lack of perfect orthogonality in the family initially observed by Iwaniec and Xiaoqing Li.