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