The cubic moment of Hecke--Maass cusp forms and moments of $L$-functions

TL;DR

This paper proves the vanishing of smooth cubic moments for Hecke--Maass cusp forms, supporting the random wave conjecture, and establishes new subconvexity bounds for certain $L$-functions, advancing understanding of their moments.

Contribution

It introduces new bounds and decay rates for moments of $L$-functions associated with Hecke--Maass forms, including the first proof of cubic moment vanishing and subconvexity bounds.

Findings

Smooth cubic moments vanish for Hecke--Maass cusp forms.

Established polynomial decay for smooth cubic moments.

Proved new subconvexity bounds for $ m GL(3) imes GL(2)$ $L$-functions.

Abstract

In this paper, we prove the smooth cubic moments vanish for the Hecke--Maass cusp forms, which gives a new case of the random wave conjecture. In fact, we can prove a polynomial decay for the smooth cubic moments, while for the smooth second moment (i.e. QUE) no rate of decay is known unconditionally for general Hecke--Maass cusp forms. The proof bases on various estimates of moments of central $L$-values. We prove the Lindel\"of on average bound for the first moment of $\rm GL(3)\times GL(2)$ $L$-functions in short intervals of the subconvexity strength length, and the convexity strength upper bound for the mixed moment of $\rm GL(2)$ and the triple product $L$-functions. In particular, we prove new subconvexity bounds of certain $\rm GL(3)\times GL(2)$ $L$-functions.