The fourth moment of truncated Eisenstein series

TL;DR

This paper proves that the fourth moment of truncated Eisenstein series behaves like a Gaussian, confirming the Random Wave Conjecture for Eisenstein series through novel averaging techniques.

Contribution

It provides the first asymptotic for the fourth moment of truncated Eisenstein series, linking it to Gaussian behavior and advancing understanding of the Random Wave Conjecture.

Findings

Main term matches Gaussian random behavior

Introduces averaging over truncation parameter

First asymptotic for fourth moment of Eisenstein series

Abstract

We obtain an asymptotic for the fourth moment of truncated Eisenstein series of large Laplacian eigenvalue, verifying for the first time that the main term corresponds to Gaussian random behavior. This is a manifestation of the Random Wave Conjecture, which for Eisenstein series was formulated by Hejhal and Rackner over thirty years ago. Our innovation is to tackle the problem after introducing, at no cost, an extra averaging over the truncation parameter.