Quantum variance for cubic moment of Hecke--Maass cusp forms and Eisenstein series

In this paper, we give the upper bounds on the variance for cubic moment of Hecke–Maass cusp forms and Eisenstein series respectively. For the cusp form case, the bound comes from a large sieve inequality for symmetric cubes. We also give some nontrivial bounds for higher moments of symmetric cube $L$-functions. For the Eisenstein series case, the upper bound comes from Lindelöf-on-average type bounds for various $L$-functions. In particular, we establish the sharp upper bounds for the fourth moment of $\mathrm{G}\mathrm{L}(2)\times \mathrm{G}\mathrm{L}(2)$ $L$-functions and the eighth moment of $\mathrm{G}\mathrm{L}(2)$ $L$-functions around special points $1\mathrm{/}2+i{t}_j$. Our proof is based on the work of Chandee and Li \cite{C-L20} about bounding the second moment of $\mathrm{G}\mathrm{L}(4)\times \mathrm{G}\mathrm{L}(2)$ $L$-functions.