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

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 Lindelöf on average bound for the first moment of $\mathrm{G}\mathrm{L}(3)\times \mathrm{G}\mathrm{L}(2)$ $L$-functions in short intervals of the subconvexity strength length, and the convexity strength upper bound for the mixed moment of $\mathrm{G}\mathrm{L}(2)$ and the triple product $L$-functions. In particular, we prove new subconvexity bounds of certain $\mathrm{G}\mathrm{L}(3)\times \mathrm{G}\mathrm{L}(2)$ $L$-functions.