Mixed fourth moments of automorphic forms and the shifted moments of $L$-functions

TL;DR

This paper derives asymptotic formulas for mixed fourth moments of automorphic forms under GRH and GRC, revealing non-equidistribution phenomena and proposing conjectures on Eisenstein series' joint value distribution.

Contribution

It provides new asymptotic formulas for mixed moments of automorphic forms assuming GRH and GRC, and explores their distribution properties and related conjectures.

Findings

Asymptotic formulas for mixed fourth moments under GRH and GRC

Evidence of non-equidistribution over the full fundamental domain

Proposed conjecture on joint value distribution of Eisenstein series

Abstract

In this article, we study the mixed fourth moments of Hecke--Maass cusp forms and Eisenstein series with type $(2, 2)$. Under the assumptions of the Generalized Riemann Hypothesis (GRH) and the Generalized Ramanujan Conjecture (GRC), we establish asymptotic formulas for these moments. Our results give an interesting non-equidistribution phenomenon over the full fundamental domain. In fact, this independent equidistribution should be true in a compact set. We further investigate this behaviour by examining a truncated version involving truncated Eisenstein series. Additionally, we propose a conjecture on the joint value distribution of Eisenstein series. The proofs are based on the bounds of the shifted mixed moments of $L$-functions.